math-ph0505035.tex



\documentstyle[12pt]{article}
\newtheorem{theorem}{{\sc Theorem}}
\newcommand{\bt}{\begin{theorem}}
\newcommand{\et}{\end{theorem}}
\setlength{\textheight}{600pt}
\setlength{\textwidth}{400pt}
\newcommand{\newsection}[1]{\setcounter{equation}{0} \setcounter{theorem}{0}
\section{#1}}
\renewcommand{\theequation}{\thesection .\arabic{equation}}
\renewcommand{\thetheorem}{\thesection .\arabic{theorem}}
\renewcommand{\baselinestretch}{1.5}
\newcommand{\NI}{\noindent}
\newcommand{\UL}{\underline}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\dsp}{\displaystyle}
\def \b #1 {\bf #1}
\newcommand{\IR}{I\!\!R}
\newcommand{\IN}{I\!\!N}
\newcommand{\IE}{I\!\!E}
\newcommand{\IC}{I\!\!C}
\newcommand{\IF}{I\!\!F}
\newcommand{\IL}{I\!\!L}
\newcommand{\IT}{I\!\!T}
\newcommand{\IZ}{Z\!\!\!Z}
\newcommand{\clk}{{\cal K}}
\newcommand{\cly}{{\cal Y}}
\newcommand{\cla}{{\cal A}}
\newcommand{\clz}{{\cal Z}}
\newcommand{\cli}{{\cal I}}
\newcommand{\cls}{{\cal S}}
\newcommand{\clt}{{\cal T}}
\newcommand{\clf}{{\cal F}}
\newcommand{\clg}{{\cal G}}
\newcommand{\clh}{{\cal H}}
\newcommand{\clp}{{\cal P}}
\newcommand{\clo}{{\cal O}}
\newcommand{\clb}{{\cal B}}
\newcommand{\clu}{{\cal U}}
\newcommand{\clx}{{\cal X}}
\newcommand{\cle}{{\cal E}}
\newcommand{\clj}{{\cal J}}
\newcommand{\cln}{{\cal N}}
\newcommand{\cld}{{\cal D}}
\newcommand{\cll}{{\cal L}}
\newcommand{\clc}{{\cal C}}
\newcommand{\ot}{\otimes}
\newcommand{\clv}{{\cal V}}
\newcommand{\clm}{{\cal M}}
\newcommand{\clr}{{\cal R}}
\newcommand {\pal}{\partial}
\newcommand{\dg}{\dagger}
\newcommand{\biot}{\bigotimes}
\newcommand{\op}{\oplus}
\newcommand{\lmd}{\lambda}
\newcommand{\Lmd}{\Lambda}
\newcommand {\Gma}{\Gamma}
\newcommand{\al}{\alpha}
\newcommand{\fal}{\forall}
\newcommand{\raro}{\rightarrow}
\newcommand {\Raro}{\Rightarrow}
\newcommand{\lra}{\Longrightarrow}
\newcommand{\sbs}{\subset}
\newcommand{\seq}{\subseteq}
\newcommand{\llra}{\Longleftrightarrow}
\newcommand{\vsp}{\vskip 1em}
\newcommand{\vspp}{\vskip 2em}
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
\newcommand{\ve}{\varepsilon}
\newcommand{\CC}{\centerline}
\newcommand{\ity}{\infty}
\def \qed {\hfill \vrule height6pt width 6pt depth 0pt}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ben}{\begin{eqnarray*}}
\newcommand{\een}{\end{eqnarray*}}
\pagestyle{myheadings}
\begin{document}
\thispagestyle {empty}
\sloppy

\centerline{\large \bf Phase transition and split property in quantum spin chain }

\bigskip
\bigskip
\centerline{\bf Anilesh Mohari }
\smallskip
\centerline{\bf S.N.Bose Center for Basic Sciences, }
\centerline{\bf JD Block, Sector-3, Calcutta-98 }
\centerline{\bf E-mail:anilesh@boson.bose.res.in}
\centerline{Phone: 919433263275}

\smallskip
\centerline{\bf Abstract}
\smallskip
In this exposition we investigate further the methodology developed in [Mo2] to study properties of the ground 
states of a translation invariant Hamiltonian for one lattice dimensional quantum spin chain $\cla=\otimes_{\IZ}M_d$, 
where $M_d$ is the space of $d \times d$ complex matrices. To that end we study the associated Cuntz elements [Mo2] 
representing a translation invariant pure state and find an useful criteria for the state to be in quantum detailed 
balance [Mo1]. This criteria is further explored to prove that such a pure state is split or uniformly mixing [BR,Ma2] 
if the lattice space correlation functions decay exponentially. Furthermore we also prove that a pure lattice symmetric, 
translation and $SU(2)$ gauge invariant state give rise to a canonical Popescu systems acting on a finite dimensional 
Hilbert space and thus the lattice space correlation functions of the pure state decay exponentially. The above result 
has a ready generalization where the symmetry group $SU(2)$ can be replaced by a class of simply connected compact 
Lie-group. 


\vsp
As a consequence of these results we conclude that if the ground states for an integer spin $SU(2)$ invariant 
($2s+1=d$) detailed balanced Hamiltonian is unique then the state is split. In particular if the ground state for 
integer spin anti-ferromagnetic Heisenberg chain is unique, then our main result says that the state is uniformly 
mixing and lattice space correlation functions of the ground state decay exponentially. Our main result is general 
enough to have application to other well known models such as Ising model, XY model and quasi-one dimensional 
quantum spin ladder [DR, Ma2] magnetic materials. One can also draw similar result for the Hamiltonian with a more 
general simply connected compact Lie group symmetry.  


\newpage
\newsection{ Introduction:} 

\vsp
We briefly set the standard notation and known relations in the following. The quantum spin chain
we consider here is described by a {\bf UHF} $C^*$-algebra denoted by $\cla=\otimes_{\IZ}M_d$. Here $\cla$
is the $C^*$ -completion of the infinite tensor product of the algebra ${\bf M}_d(\!C)$ of $d$ by $d$ complex matrices,
each component of the tensor product element is denoted by an integer $j$. Let $Q$ be a matrix in ${\bf M}_d(\!C)$. By
$Q^{(j)}$ we denote the element $...\otimes 1 \otimes 1 ... 1 \otimes Q \otimes 1 \otimes ... 1\otimes ,,. $, where $Q$
appears in the $j$-th component. Given a subset $\Lambda$ of $\!Z$, $\cla_{\Lambda}$ is defined as the $C^*$-subalgebra
of $\cla$ generated by all $Q^{(j)}$ with $Q \in {\bf M}_d(\!C)$, $j \in \Lambda$. We also set
$$\cla_{loc}= \bigcup_{\Lambda:|\Lambda| < \infty } \cla_{\Lambda}$$
where $|\Lambda|$ is the cardinality of $\Lambda$. Let $\omega$ be a state on $\cla$. The restriction of $\omega$
to $\cla_{\Lambda}$ is denoted by $\omega_{\Lambda}$. We also set $\omega_{R}=\omega_{[0,\infty)}$ and $\omega_{L}=
\omega_{(-\infty.0]}$. The translation $\theta_k$ is an automorphism of $\cla$ defined by $\theta_k(Q^{(j)})=Q^{(j+k)}$.
Thus $\theta_1,\theta_{-1}$ are unital $*$-endomorphism on $\cla_{R}$ and $\cla_{L}$ respectively. We say $\omega$ is
translation invariant if $\omega \circ \theta_k = \omega$ on $\cla$ ( $\omega \circ \theta_1 = \omega$ on $\cla$ ).
In such a case $(\cla_{R},\theta_1,\psi_{R})$ and $(\cla_{L},\theta_{-1},\psi_{L})$ are two unital $*$-endomorphisms 
with invariant states. It is well known that translation invariant state $\omega$ is a factor (i.e. the GNS representation 
is a factor representation ) if and only if $\mbox{limit}_{|k| \raro \infty }\omega(Q_1 \theta_k(Q_2)) \raro \omega(Q_1) 
\omega(Q_2)$ for all $Q_1,Q_2$ in $\cla$. Similar statement with appropriate direction of limit is valid for $\psi_L,\psi_R$. 
Thus for a translation invariant factor state $\omega$ of $\cla$, states $\omega_R$ and $\omega_L$ are factors too. A general 
question that is central here when can we guarantee that $\omega_R (\omega_L)$ are type-I factors? To that end we recall [BR,Ma2] 
a standard definition of a state to be split in the following.

\vsp
\NI {\bf DEFINITION 1.1: } Let $\omega$ be a translation invariant state on $\cla$. We say that $\omega$ is {\it split}
if the following condition is valid: Given any $\epsilon > 0$ there exists a $k \ge 1$ so that

\be
\mbox{sup}_{||Q|| < 1}|\omega(Q)-\omega_L \otimes \omega_R(Q)| \le \epsilon
\ee
where the above supremum is taken over all local elements $Q \in \cla_{(-\infty,-k] \cup \cla_[k, \infty)}$
with the norm less than $1$.

\vsp
Here we recall few simple facts from [BR,Ma2]. The uniform cluster condition is valid if and only if the state $\omega$ 
is quasi-equivalent to the product state $\psi_L \otimes \psi_R$ of a state $\psi_L$ of $\cla_L$ and another state 
$\psi_R$ of $\cla_R$. Thus a Gibbs state of a finite range interaction is split. On the other hand if $\omega$ is a pure 
translation invariant state, then $\omega$ is a factor state. Furthermore in such a case $\omega_R( \omega_L ) $ is 
type-I if and only if $\omega$ is also a split state. There exists both non-pure split states and non-split pure states. 
Next we present a precise definition for exponential decay.     

\vsp
\NI {\bf DEFINITION 1.2: }  Let $\omega$ be a translation invariant state on one dimensional spin chain $\cla$. We say 
the two point spacial correlation functions for $\omega$ {\it decay exponentially} if there exists a $\delta > 0$ so 
that
\be
e^{\delta k} |\omega(Q_1 \theta_k(Q_2))-\omega(Q_1)\omega(Q_2)| \raro 0
\ee
as $|k| \raro \infty$ for any local elements $Q_1,Q_2 \in \cla.$

\vsp
A translation invariant state $\omega$ is said to be in {\it detailed balance } if $\omega$ is {\it lattice symmetric } 
and {\it real} (for details see section 3 ). Our main mathematical results are the following.

\vsp
\NI {\bf THEOREM 1.3: } Let $\omega$ be a pure translation invariant detailed balance state on $\cla$. If two point 
spacial correlation function for $\omega$ decay exponentially then $\omega$ is a split state. 

\vsp
For any compact group $G$ let $g \raro v(g)$ be an irreducible representation in $\IC^d$. We say $\omega$ is 
$G-$invariant if
$\omega(Q)=\omega(...\otimes v(g) \otimes v(g) ...Q ...\otimes v(g)^* \otimes v(g)^* ..)$ for
all $Q \in \cla_{loc}$.. 

\vsp
\NI {\bf THEOREM 1.4: } Let $G$ be a simply connected compact Lie group and $g \raro v(g)$ be an irreducible 
representation in $\IC^d$. Let $\omega$ be a pure translation and $G$ invariant detailed balance state on $\cla$ 
then the following hold:

\NI (a) The matrix elements in representation $g \raro v^i_j(g)$ are real numbers;

\NI (b) Further if the invariant subspace in $\IC^d \otimes \IC^d$ of the representation $g \raro \bar{v(g)} \otimes v(g)$ is 
one dimensional then $\omega$ is a split state and special correlation function decays exponentially.

\vsp
\NI {\bf THEOREM 1.5: } Let $\omega$ be a state as in Theorem 1.3. If $\omega$ is $SU(2)$ invariant then the following 
hold:

\NI (a) $d$ is an odd integer;

\NI (b) $\omega$ is a split state and special correlation function decays exponentially.  

\vsp
Theorem 1.5 (a) says that there exists no translation and $SU(2)$ invariant real lattice symmetric pure state on 
$\cla=\otimes_{\IZ}M_d$ for $d=2s+1$ and $s={1 \over 2}, {3 \over 2}, ...$ i.e. half-odd integer spin. Theorem 1.5 
has an easy generalization valid for any compact simply connected Lie groups.  

\vsp
The paper is organized as follows. In section 2 we essentially recall from [Mo2] the representation of Cuntz algebra 
associated with a translation invariant state on quantum spin chain. In section 3 we give a brief description of 
the amalgamated Hilbert space [BJKW] described as in [Section 3 in Mo3] and also recall in details the 
properties of Popescu systems associated with a detailed balance pure translation invariant state [Mo3]. In section 
4 we prove our main mathematical result Theorem 1.3. In Section 5 we prove that any $SU(2)$ invariant integer spin 
pure state can be represented by Popescu elements $(\clk,\clm,v_i,\:1 \le i \le d,\;\Omega)$ where $\clm$ is a finite 
type-I factor acting on a finite dimensional Hilbert space $\clk$. In the last section we study ground states of a 
class of translation invariant Hamiltonians. We prove that if ground state for an integer spin $SU(2)$ invariant 
Hamiltonian is unique then the state is split and spacial correlation function decays exponentially. 


\vsp
\NI {\bf Acknowledgments:} This work was initiated when the author was visiting the department of mathematics, 
university of Iowa, for the fall semester 2005. The author gratefully acknowledge Prof. Palle E. T. Jorgensen 
inspiring participation in sharing the intricacy of the present problem. The author also thanks Dr. D. Goswami 
for his interest in this problem and comments.       

\newsection{ Cuntz algebra and translation invariant pure states:}

\vsp
First we recall that the Cuntz algebra $\clo_d ( d \in \{2,3,.., \} )$ is the universal $C^*$-algebra
generated by the elements $\{s_1,s_2,...,s_d \}$ subject to the relations:

$$s^*_is_j=\delta^i_j1$$
$$\sum_{1 \le i \le d } s_is^*_i=1.$$

\vsp
There is a canonical action of the group $U(d)$ of unitary $d \times d$ matrices on $\clo_d$ given by
$$\beta_g(s_i)=\sum_{1 \le j \le d}\overline{g^j_i}s_j$$
for $g=((g^i_j) \in U(d)$. In particular the gauge action is defined by
$$\beta_z(s_i)=zs_i,\;\;z \in \IT =S^1= \{z \in \IC: |z|=1 \}.$$
If UHF$_d$ is the fixed point subalgebra under the gauge action, then UHF$_d$ is the closure of the
linear span of all wick ordered monomials of the form
$$s_{i_1}...s_{i_k}s^*_{j_k}...s^*_{j_1}$$
which is also isomorphic to the UHF$_d$ algebra
$$M_{d^\infty}=\otimes^{\infty}_1M_d$$
so that the isomorphism carries the wick ordered monomial above into the matrix element
$$e^{i_1}_{j_1}(1)\otimes e^{i_2}_{j_2}(2) \otimes....\otimes e^{i_k}_{j_k}(k) \otimes 1 \otimes 1 ....$$
and the restriction of $\beta_g$ to $UHF_d$ is then carried into action
$$Ad(g)\otimes Ad(g) \otimes Ad(g) \otimes ....$$

\vsp
We also define the canonical endomorphism $\lambda$ on $\clo_d$ by
$$\lambda(x)=\sum_{1 \le i \le d}s_ixs^*_i$$
and the isomorphism carries $\lambda$ restricted to UHF$_d$ into the one-sided shift
$$y_1 \otimes y_2 \otimes ... \raro 1 \otimes y_1 \otimes y_2 ....$$
on $\otimes^{\infty}_1 M_d$. Note that $\lambda \beta_g = \beta_g \lambda $ on UHF$_d$.

\vsp
Let $d \in \{2,3,..,,..\}$ and $\IZ_d$ be a set of $d$ elements.  $\cli$ be the set of finite sequences
$I=(i_1,i_2,...,i_m)$ where
$i_k \in \IZ_d$ and $m \ge 1$. We also include empty set $\emptyset \in \cli$ and set $s_{\emptyset }=1=s^*_{\emptyset}$,
$s_{I}=s_{i_1}......s_{i_m} \in \clo_d $ and $s^*_{I}=s^*_{i_m}...s^*_{i_1} \in \clo_d$. In the following we recall from
[BJKW] a crucial result originated in [Po,BJP].

\vsp
\NI{\bf THEOREM 2.1: } There exists a canonical one-one correspondence between the following objects:

\vsp
\NI (a) States $\psi$ on $\clo_d$

\vsp
\NI (b) Function $C: \cli \times \cli \raro \IC$ with the following properties:

\NI (i) $C(\emptyset, \emptyset)=1$;

\NI (ii) for any function $\lambda:\cli \raro \IC$ with finite support we have
         $$\sum_{I,J \in \cli} \overline{\lambda(I)}C(I,J)\lambda(J) \ge 0$$

\NI (iii) $\sum_{i \in \IZ_d} C(Ii,Ji) = C(I,J)$ for all $I,J \in \cli$.

\vsp
\NI (c) Unitary equivalence class of objects $(\clk,\Omega,v_1,..,v_d)$ where

\NI (i) $\clk$ is a Hilbert space and $\Omega$ is an unit vector in $\clk$;

\NI (ii) $v_1,..,v_d \in \clb(\clk)$ so that $\sum_{i \in \IZ_d} v_iv^*_i=1$;

\NI (iii) the linear span of the vectors of the form $v^*_I\Omega$, where $I \in \cli$, is dense in $\clk$.

\vsp
Where the correspondence is given by a unique completely positive map $R: \clo_d \raro \clb(\clk)$ so that

\NI (i) $R(s_Is^*_J)=v_Iv^*_J;$

\NI (ii) $\psi(x)=<\Omega,R(x)\Omega>;$

\NI (iii) $\psi(s_Is^*_J)=C(I,J)=<v^*_I \Omega,v^*_J\Omega>.$

\NI (iv) For any fix $g \in U_d$ and the completely positive map $R_g:\clo_d \raro \clb(\clk)$ defined by $R_g= R
\circ \beta_g$ give rises to a Popescu system given by $(\clk,\Omega,\beta_g(v_i),..,\beta_g(v_d))$ where
$\beta_g(v_i)=\sum_{1 \le j \le d} \overline{g^i_j} v_j.$

\vsp
Now we present a commutant lifting theorem ( Theorem 5.1 in [BJKW] ).

\vsp
\NI {\bf THEOREM 2.2:} Let $v_1,v_2,...,v_d$ be a family of bounded operators on a Hilbert space
$\clk$ so that $\sum_{1 \le k \le d} v_kv_k^*=I$. Then there exists a unique up to
isomorphism Hilbert space $\clh$, a projection $P$ on $\clk$ and a family of operators
$\{S_k:,\;1 \le k \le d,\;P\}$ satisfying Cuntz relation so that
\be
PS^{*}_{k}P=S_k^*P=v^*_k
\ee
for all $1 \le k \le d$ and $\clk$ is cyclic for the representation i.e. the vectors
$\{ S_I\clk: |I| < \infty \}$ are total in $\clh$.

Moreover the following hold:

\NI (a) $\Lambda_n(P) \uparrow I$ as $n \uparrow \infty$;

\NI (b) For any $D \in \clb_{\tau}(\clk)$, $\Lambda_n(D) \raro X'$ weakly as $n \raro \infty$
for some $X'$ in the commutant $\{S_k,S^*_k: 1 \le k \le d \}'$ so that $PX'P=D$. Moreover
the self adjoint elements in the commutant $\{S_k,S^*_k: 1 \le k \le d \}'$ is isometrically
order isomorphic with the self adjoint elements in $\clb_{\tau}(\clk)$ via the
surjective map $X' \raro PX'P$, where $\clb_{\tau}(\clk)=\{x \in \clb(\clk): \sum_{1 \le k \le d }
v_kxv^*_k=x \}.$

\NI (c) $\{v_k,v^*_k,\;1 \le k \le d \}' \subseteq \clb_{\tau}(\clk)$ and equality hold if and only if
$P \in \{S_k,S_k,\;1 \le k \le d \}''$.

\vsp
\NI {\bf PROOF: } Following Popescu [Po] we define a completely positive map
$R: \clo_d \raro \clb(\clk)$ by
\be
R(s_Is^*_J)=v_Iv^*_J
\ee
for all $|I|,|J| < \infty$. The representation $S_1,..,S_d$ of $\clo_d$ on $\clh$ thus may be taken to be the Stinespring
dilation of $R$ [BR, vol-2] and uniqueness up to unitary equivalence follows from uniqueness of the Stinespring
representation. That $\clk$ is cyclic for the representation follows from the minimality property of the Stinespring
dilation. For (a) let $Q$ be the limiting projection. Then we have $\Lambda(Q)=Q$, hence $Q \in \{S_k,S^*_k \}'$
and $Q \ge P$. In particular $QS_If=S_If$ for all $f \in \clk$ and $|I| < \infty$. Hence $Q=I$ by the
cyclicity of $\clk$. For (b) essentially we defer from the argument used in Theorem 5.1 in [BJKW]. We fix any $D
\in \clb_{\tau}(\clk)$ and note that $P\Lambda_k(D)P=\tau_k(D)=D$ for any $k \ge 1$. Thus for any integers $n > m$
we have
$$\Lambda_m(P)\Lambda_n(D)\Lambda_m(P)=\Lambda_m(P\Lambda_{n-m}(D)P)=\Lambda_m(D)$$
Hence for any fix $m \ge 1$ limit $<f,\Lambda_n(D)g>$ as $n \raro \infty$ exists for all
$f,g \in \Lambda_m(P)$. Since the family of operators $\Lambda_n(D)$ is uniformly bounded and $\Lambda_m(P) \uparrow I$
as $m \raro \infty$, a standard density argument guarantees that the weak operator limit of $\Lambda_n(D)$ exists as
$n \raro \infty$. Let $X'$ be the limit. So $\Lambda(X')=X'$, by Cuntz's relation, $X' \in \{S_k,S^*_k:1 \le k \le k \}'$.
Since $P\Lambda_n(D)P=D$ for all $n \ge 1$, we also conclude that $PX'P=D$ by taking limit $n \raro \infty$. 
Conversely it is obvious that $P\{ S_k,S^*_k:\; k \ge 1 \}'P \subseteq \clb_{\tau}(\clk)$. Hence we can 
identify $P\{S_k,S^*_k:\; k \ge 1 \}'P$ with $\clb_{\tau}(\clk)$.

\vsp
Further it is obvious that $X'$ is self-adjoint if and only if $D=PX'P$ is self-adjoint. Now fix any self-adjoint
element $D \in \clb_{\tau}(\clk)$. Since identity operator on $\clk$ is an element in $\clb_{\tau}(\clk)$ for any
$\alpha \ge 0$ for which $- \alpha P  \le D \le \alpha P$, we have $\alpha \Lambda_n(P) \le \Lambda_n(D)
\le \alpha \Lambda_n(P)$ for all $n \ge 1$. By taking limit $n \raro \infty$ we
conclude that $- \alpha I \le X' \le \alpha I$, where $PX'P=D$. Since operator norm of a self-adjoint element $A$
in a Hilbert space is given by
$$||A||= \mbox{inf}_{\alpha \ge 0}\{\alpha: - \alpha I  \le A \le \alpha I \}$$
we conclude that $||X'|| \le ||D||$. $P$ being a projection, we also have $||D||=||PX'P|| \le ||X'||$. 
Thus the map is isometrically order isomorphic taking self-adjoint elements of the commutant
to self-adjoint elements of $\clb_{\tau}(\clk)$.

\vsp
We are left to prove (c). Inclusion is trivial.  For the last part note that for any invariant element $D$
in $\clb(\clk)$ there exists an element $X'$ in $\{S_k,S^*_k,\;1 \le k \le d\}'$ so that $PX'P=D$. In such a
case we verify that $Dv^*_k= PX'PS^*_kP=PX'S^*_kP=PS^*_kX'P=PS^*_kPX'P=v^*_kD$. We also have $D^* \in \clb_
{\tau}(\clk)$ and thus $D^*v^*_k=v^*_kD^*$. Hence $D \in \{v_k,v^*_k: 1 \le k \le d \}'$. Since $P\pi_{\hat{\omega}}
(\clo_d)'P = \clb(\clk)_{\tau}$, we conclude that $\clb(\clk)_{\tau} \subseteq \clm'$. Thus equality hold whenever
$P \in \{S_k,S_k^*,\;1 \le k \le d \}''$. For converse note that by commutant lifting property self-adjoint elements
of the commutant $\{S_k,S^*_k,1 \le k \le d \}'$ is order isometric with the algebra $\clm'$ via the map $X' \raro PX'P$.
Hence $P \in \{S_k,S^*_k,1 \le k \le d \}''$ by Proposition 4.2 in [BJKW]. \qed

\vsp
\NI{\bf PROPOSITION 2.3: } Let $\psi$ be a state on $\clo_d$ and $(\clh_{\pi},\pi_{\psi},\Omega_{\psi})$
be the GNS representation. Then the following hold unique up to unitary isomorphism:

\vsp
\NI (a) There exists a family $v_1,v_2,...,v_d$ of bounded operators on a Hilbert subspace
$\clk$ of $\clh_{\pi}$ with a unit vector $\Omega$ so that $\sum_{1 \le k \le d} v_kv_k^*=I$ 
and $\{ v^*_I\Omega: |I| < \infty \}$ is total in $\clk$;

\vsp
\NI (b) For any $I=(i_1,i_2,...,i_k),J=(j_1,j_2,...,j_l)$ with $|I|,|J| < \infty$ we have $\hat{\omega}(s_Is^*_J)=
<\Omega,v_Iv^*_J\Omega>$ and the vectors $\{ S_If: f \in \clk,\;|I| < \infty \}$ are total in
the GNS Hilbert space associated with $(\clo_d,\psi)$, where $S_k=\pi_{\psi}(s_k)$ and
$v^*_k=PS^*_kP$ for all $1 \le k \le d$ and $P$ is the projection on the closed subspace
generated by the vectors $\{S^*_I\Omega;\;|I| < \infty \}$.

\vsp
Conversely given a Popescu system $(\clk,v_k,\;1 \le k \le d,\Omega)$ satisfying (a)
there exists a unique state $\psi$ on $\clo_d$ so that (b) is satisfied.

Furthermore the following statements are valid:
\vsp
\NI (c)  If the normal state $\phi_0(x)=<\Omega,x \Omega>$ on the von-Neumann algebra
$\clm=\{ v_i, v^*_i \}''$ is invariant for the Markov map $\tau(x)=\sum_{1 \le k
\le d} v_ixv^*_i,\;x \in \clm$ then $\psi$ is $\lambda$ invariant and $\phi_0$ is faithful
on $\clm$.

\vsp
\NI (d) If $P \in \pi_{\psi}(\clo)''$ then $\psi$ is an ergodic state
for $(\clo_d,\lambda)$ if and only if $(\clm,\tau,\phi_0)$ is ergodic.
In such a case $\clm$ is a factor.

\vsp
\NI {\bf PROOF:} We consider the GNS space  $(\clh_{\pi},\pi_{\psi},\Omega_{\psi})$
associated with $(\clo_d,\psi)$. Set $S_i=\pi_{\psi}(s_i)$ and consider the normal state
$\psi_{\Omega}$ on $\pi_{\psi}(\clo_d)''$ defined by $\psi_{\Omega}(X) = <\Omega, X
\Omega>$, where for simplicity we use symbol $\Omega$ for $\Omega_{\psi}$. Let $P$
be the closed subspace generated by the vectors $\{S^*_I\Omega: |I| < \infty \}$. It is obvious
that $S^*_iP \subseteq P$ for all $1 \le k \le d $, thus $P$ is the minimal subspace
containing $\Omega$, invariant by all $\{ S^*_i:1 \le k \le d \}$ i.e.
\be
PS^*_kP=S^*_kP
\ee
Let $\clk$ be the range of $P$ as a Hilbert subspace of $\clh_{\psi}$,
\be
v_k=PS_kP
\ee
for $1 \le k \le d$ and $\clm$ be the von-Neumann algebra generated by $\{ v_i,v_i^* \}$.
Thus $v^*_i=S^*_iP$ and $\sum_i v_iv^*_i=\sum_i PS_iS^*_iP= P$ which is identity
operator in $\clk$. This completes the proof of (a).

\vsp
For (b) we note that
$$\psi(s_Is^*_J)=\psi_{\Omega}(S_IS^*_J)$$
$$=<\Omega,PS_IS^*_JP\Omega>=<\Omega,v_Iv^*_J\Omega>.$$
Since $\clh_{\psi}$ is spanned by the vectors $\{S_IS^*_J\Omega:|I|,|J| < \infty \}$ and
$\clk$ is spanned by the vectors $\{ S^*_J\Omega=v^*_J\Omega:|I| < \infty \}$,
$\clk$ is cyclic for $S_I$ i.e. the vectors $\{ S_I\clk: |I| < \infty \}$ spans
$\clh_{\psi}$. Uniqueness up to isomorphism follows as usual by total property of vectors $v^*_I\Omega$ in $\clk$.

\vsp
Conversely for a Popescu systems $(\clk,v_i,\Omega)$ satisfying (a), we consider
the family $(\clh,S_k,\;1 \le k \le d,P)$ of Cuntz's elements defined as in
Theorem 2.2. We claim that $\Omega$ is cyclic vector for the representation $\pi(s_i) \raro
S_i$. Note that by our construction vectors $\{S_If,f \in \clk: |I| < \infty \}$ are total
in $\clh$ and $v^*_J\Omega=S^*_J\Omega$ for all $|J| < \infty$. Thus by our hypothesis that
vectors $\{v^*_J\Omega:|I| < \infty \}$ are total in $\clk$, we verify that
vectors $\{S_IS^*_J\Omega: |I|,|J| < \infty \}$ are total in $\clh$. Hence $\Omega$ is cyclic for
the representation $s_i \raro S_i$ of $\clo_d$.

\vsp
We left to prove (c) and (d). It simple to note by (b) that $\psi \lambda=\psi$ i.e.
$$\sum_i <\Omega,S_iS_IS^*_JS^*_i\Omega>=<\Omega,S_IS^*_J\Omega>$$ for all $|I|,|J| < \infty$ if and only if
the vector state $\phi_0$ on $\clm$ is invariant. Let $p'$ be the support projection in $\clm$ for $\tau$
invariant state $\phi_0$. Thus $\phi_0(1-p')=0$ and by invariance we have $\phi_0(p'\tau(1-p')p')=\phi_0(1-p')=0$.
Since $p'\tau(1-p')p' \ge 0$, by minimality of support projection, we conclude that $p'\tau(1-p')p=0$. Hence
$p'\Omega=\Omega$ and $p'v^*_kp'=v^*_kp'$ for all $1 \le k \le d$. Thus $p'v^*_I\Omega=v^*_I\Omega$ for all
$|I| < \infty$. As $\clk$ is the closed linear span of the vectors $\{ v_I^*\Omega: |I| < \infty \}$, we conclude
that $p'=p$. In other words $\phi_0$ is faithful on $\clm$. This completes the proof for (c).

\vsp
We are left to show (d). $\Omega$ being a cyclic vector for $\pi_{\psi}(\clo_d)$, the weak$^*$ limit of the
increasing projection $\Lambda^k(P)$ is $I$. Thus by Theorem 3.6 in [Mo1] we have $(\pi_{\psi}(\clo_d)'',\Lambda,
\psi_{\Omega})$ is ergodic if and only if the reduced dynamics $(\clm,\tau,\phi_0)$ is ergodic. Last part of the
statement is an easy consequence of a Theorem of D. E. Evans [Ev], (also see [Fr], [Mo1], [BJKW]). \qed

\vsp
Before we move to next result we comment here that in general for a $\lambda$ invariant state on $\clo_d$
the normal state $\phi_0$ on $\clm=\{v_k,v_k^*:1 \le k \le d \}''$ need not be invariant for $\tau$. To that
end we consider [BR] the unique KMS state $\psi=\psi_{\beta}$ for the automorphism $\alpha_t(s_i)=e^{it}s_i$
on $\clo_d$. $\psi$ is $\lambda$ invariant and $\psi_{|}\mbox{UHF}_d$ is the unique faithful trace. $\psi$ being
a KMS state for an automorphism, the normal state induced by the cyclic vector on $\pi_{\psi}(\clo_d)''$ is also
separating for $\pi(\clo_d)''$. As $\psi \beta_z = \psi$ for all $z \in S^1$ we note that $<\Omega,\pi(s_I)\Omega>=0$
for all $|I| \ge 1$. In particular $<\Omega,v_I^*\Omega>=0$ where $(v_i)$ are defined as in Proposition 2.3 and thus
$v_i\Omega=0$ for all $1 \le i \le d$. Hence $\Omega$ is not separating for $\clm$. Thus by Proposition
2.3 (c) we conclude that $\phi_0$ is not $\tau$ invariant. This example also indicates that the support projection
of a $\lambda$ invariant state $\psi$ in $\pi(\clo_d)''$ need not be equal to the minimal sub-harmonic projection
$P_0$ i.e. the closed span of vectors $\{ S_I^*\Omega: |I| < \infty \}$.

\vsp
\NI{\bf PROPOSITION 2.4:} Let $(\clh_{\pi},\pi_{\psi},\Omega)$ be the GNS representation of a $\lambda$ invariant state
$\psi$ on $\clo_d$ and $P$ be the support projection of the normal state $\psi_{\Omega}(X)=<\Omega,X\Omega>$ in the
von-Neumann algebra $\pi_{\psi}(\clo_d)''$. Then the following hold:

\vsp
\NI (a) $P$ is a sub-harmonic projection for the endomorphism $\Lambda(X)=\sum_k S_kXS^*_k$ on $\pi_{\psi}(\clo_d)''$
i.e. $\Lambda(P) \ge P$ satisfying the following:

\NI (i) $\Lambda_n(P) \uparrow I$ as $n \uparrow \infty$;

\NI (ii) $PS^*_kP=S^*_kP,\;\;1 \le k \le d$, where $S_k=\pi_{\hat{\omega}}(s_k)$;

\NI (iii) $\sum_{1 \le k \le d} v_kv_k^*=I$ where $v_k=PS_kP$ for $1 \le k \le d$;

\vsp
\NI (b) For any $I=(i_1,i_2,...,i_k),J=(j_1,j_2,...,j_l)$ with $|I|,|J| < \infty$ we have $\hat{\omega}(s_Is^*_J) =
<\Omega,v_Iv^*_J\Omega>$ and the vectors $\{ S_If: f \in \clk,\;|I| < \infty \}$ are total in $\clh_{\pi_{\hat{\omega}}}$;

\vsp
\NI (c) The von-Neumann algebra $\clm=P\pi_{\psi}(\clo_d)''P$, acting on the Hilbert space
$\clk=P \clh_{\pi}$, is generated by $\{v_k,v^*_k:1 \le k \le d \}$ and the normal state
$\phi_0(x)=<\Omega,x \Omega>$ is faithful on the von-Neumann algebra $\clm$.

\NI (d) The self-adjoint part of the commutant of $\pi_{\hat{\omega}}(\clo_d)'$ is norm and order isomorphic to the 
space of self-adjoint fixed points of the completely positive map $\tau$. The isomorphism takes 
$X' \in \pi_{\hat{\omega}}(\clo_d)'$ onto $PX'P \in \clb_{\tau}(\clk)$, where $\clb_{\tau}(\clk) = 
\{ x \in \clb(\clk): \sum_kv_kxv^*_k=x \}$. Furthermore $\clm'=\clb_{\tau}(\clk)$.

\vsp
Conversely let $\clm$ be a von-Neumann algebra generated by a family $\{v_k: 1 \le k \le d \}$ of bounded operators
on a Hilbert space $\clk$ so that $\sum_kv_kv_k^*=1$ and the commutant $\clm'=\{x \in \clb(\clk): \sum_k v_kxv_k^*=x
\}$. Then the Popescu dilation $(\clh,P,S_k,\;1 \le k \le d)$ described as in Theorem 2.2 satisfies the following:

\NI (i) $P \in \{S_k,S^*_k,\;1 \le k \le d \}''$;

\NI (ii) For any faithful normal invariant state $\phi_0$ on $\clm$ there exists a state $\psi$ on $\clo_d$ defined by
$$\psi(s_Is^*_J)=\phi_0(v_Iv^*_J),\;|I|,|J| < \infty $$
so that the GNS space associated with $(\clm,\phi_0)$ is the support projection
for $\psi$ in $\pi_{\psi}(\clo_d)''$ satisfying (a)-(d).

\vsp
\NI (e) $\phi_0$ is a normal invariant state for $\tau$ on $\clm$. Furthermore the following statements are equivalent:

\NI (i) $\psi$ is an ergodic state;

\NI (ii) $(\clm,\tau,\phi_0)$ is ergodic;

\NI (iii) $\clm$ is a factor.

\vsp
\NI {\bf PROOF: } $\Lambda(P)$ is also a projection in $\pi_{\psi}(\clo_d)''$ so that
$\psi_{\Omega}(\Lambda(P))=1$ by invariance property. Thus we have $\Lambda(P) \ge P$ i.e.
$P\Lambda(I-P)P=0$. Hence we have
\be
PS^*_kP=S^*_kP
\ee
Moreover by $\lambda$ invariance property we also note that the faithful normal state
$\phi_0(x)=<\Omega,x\Omega>$ on the von-Neumann algebra $\clm=P\pi_{\psi}(\clo_d)''P$
is invariant for the reduce Markov map [Mo1] on $\clm$ given by
\be
\tau(x)=P\Lambda(PxP)P
\ee
\vsp
We claim that $\mbox{lim}_{n \uparrow \infty}\Lambda^n(P)= I$. That $\{ \Lambda^n(P): n \ge 1 \}$
is a sequence of increasing projections follows from sub-harmonic property of $P$ and endomorphism
property of $\Lambda$. Let the limiting projection be $Y$. Then $\Lambda(Y)=Y$ and so $Y \in
\{S_k,S^*_k \}'$. Since by our construction GNS Hilbert space $\clh_{\pi_{\hat{\omega}}}$ is
generated by $S_IS^*_J\Omega$, $Y$ is a scaler, being a non-zero projection, it is
the identity operator in $\clh_{\pi_{\psi}}$.

\vsp
Now it is routine to verify (a) (b) and (c). For the first part of (d) we appeal to Theorem 2.2. For the last part
note that for any invariant element $D$ in $\clb(\clk)$ there exists an element $X'$ in $\pi(\clo_d)'$ so that $PX'P=D$.
Since $P \in \pi(\clo_d)''$ we note that $(1-P)X'P=0$.
Now since $X' \in \{ S_k,S^*_k \}'$, we verify that $Dv^*_k= PXPS^*_kP=PXS^*_kP=PS^*_kXP=PS^*_kPXP=v^*_kD$. Since 
$D^* \in \clb_{\tau}(\clk)$ we also have $D^*v^*_k=v^*_kD^*$. Thus $D \in \{v_k,v^*_k: 1 \le k \le d \}'=\clm'$. Since
$P\pi_{\hat{\omega}}(\clo_d)'P=
\clb(\clk)_{\tau}$, we conclude that $\clb(\clk)_{\tau} \subseteq \clm'$. The reverse inclusion is trivial.
This completes the proof for (d).

\vsp
For the converse part of (i), since by our assumption and commutant lifting property self-adjoint elements of the 
commutant
$\{S_k,S^*_k,1 \le k \le d \}'$ is order isometric with the algebra $\clm'$ via the map $X' \raro PX'P$,
$P \in \{S_k,S^*_k,1 \le k \le d \}''$ by Proposition 4.2 in [BJKW]. For (ii) without loss of generality assume
that $\phi_0(x)=<\Omega,x\Omega>$ for all $x \in \clm$ and $\Omega$ is a cyclic and separating vector for $\clm$.
We are left to show that $\Omega$ is a cyclic vector for the representation $\pi(s_i) \raro S_i$. To that end
let $Y \in \pi_{\psi})\clo_d)'$ be the projection on the subspace generated by the vectors 
$\{S_IS^*_J\Omega:  |I|,|J| < \infty \}$. Note that $P$ being an element in $\pi(\clo_d)''$, $Y$ also commutes 
with all the element $P\pi(\clo_d)''P=P \clm P$. Hence $Yx\Omega=x\Omega$ for all $x \in \clm$. Thus $Y \ge P$.
Since $\Lambda_n(P) \uparrow I$ as $n \uparrow \infty$ by our construction, we conclude that $Y=\Lambda_n(Y)
\ge \Lambda_n(P) \uparrow I$ as $n \uparrow \infty$. Hence $Y=I$. In other words
$\Omega$ is cyclic for the representation $s_i \raro S_i$. This completes the proof for (ii).

\vsp
The first part of (e) is routine. By Theorem 3.6 in [Mo1] Markov semigroup $(\clm,\tau,\phi_0)$ is ergodic if and
only if $(\pi_{\psi}(\clo_d)'',\Lambda,\psi_{\Omega})$ is ergodic ( here recall by (a) that
$\Lambda_n(P) \uparrow I$ as $n \uparrow \infty$. By a standard result [Fr, also BJKW] $(\clm,\tau,\phi_0)$
is ergodic if and only if $\clm$ is a factor. This completes the proof. \qed

\vsp
The following two propositions are essentially easy adaptation of results appeared in [BJKW, Section 6 and Section 7],
stated here in our present framework. A slight variation of the following proposition is crucial for the main results
(Theorem 3.4) of the next section.

\vsp
\NI {\bf PROPOSITION 2.5: } Let $\psi$ be a $\lambda$ invariant factor state on $\clo_d$ and $(\clh_{\pi},\pi,\Omega)$
be it's GNS representation. Then the following hold:

\NI (a) The closed subgroup $H=\{z \in S^1: \psi \beta_z =\psi \}$ is equal to $\{z \in S^1: \beta_z \mbox{extends to an
automorphism of } \pi(\clo_d)'' \} $.

\NI (b) Let $\clo_d^{H}$ be the fixed point sub-algebra in $\clo_d$ under the gauge group $\{ \beta_z: z \in H \}$. Then
$\pi(\clo_d^{H})'' = \pi(\mbox{UHF}_d)''$.

\NI (c) If $H$ is a finite cyclic group of $k$ many elements and $\pi(\mbox{UHF}_d)''$ is a factor,
then $\pi(\clo_d)'' \bigcap \pi(\mbox{UHF}_d)' \equiv \IC^m$ where $1 \le m \le k$.

\vsp
\NI {\bf PROOF: } It is simple that $H$ is a closed subgroup. For any fix $z \in H $ we define unitary operator $U_z$
extending the map $\pi(x)\Omega \raro \pi(\beta_z(x))\Omega$ and check that the map $X \raro U_zXU^*_z$ extends $\beta_z$
to an automorphism of $\pi(\clo_d)''$. For the converse we will use the hypothesis that $\psi$ is a $\lambda$-invariant
factor state and $\beta_z \lambda= \lambda \beta_z$ to guarantee that $\psi \beta_z (X) = {1 \over n}
\sum_{1 \le k \le n} \psi \lambda^k \beta_z(X) = {1 \over n} \sum_{1 \le k \le n} \psi \beta_z \lambda^k(X)
\raro \psi(X)$ as $n \raro \infty$ for any $X \in \pi(\clo_d)''$, where we have used the same symbol $\beta_z$
for the extension. Hence $z \in H$.

\vsp
For any $z_1,z_2 \in S^1$ we extend both $\psi \beta_{z_1}$ and $\psi \beta_{z_2}$ to its inductive limit state on $\clo_d^*$
using the canonical endomorphism $\clo_d \raro^{\lambda} \clo_d$. Inductive limit state being an affine map, their inductive limit states
are also factors. The inductive limit of the canonical endomorphsim became an automorphism. Now we appeal a standard result in
non-commutative ergodic theory to conclude that their inductive limit states are either same or orthogonal. In the following
instead of working with $\clo_d$ we should be working with the inductive limit $C^*$ algebra and their inductive limit states.
For simplicity of notation we still use $UHF_d,\clo_d$ for its inductive limit of $\clo_d \raro^{\lambda} \clo_d$
and $UHF_d \raro^{\lambda} UHF_d$ respectively and so its inductive limit states.

\vsp
Now $H$ being a closed subgroup of $S^1$, it is either entire $S^1$ or a finite subgroup
$\{exp({ 2i \pi l \over k})|l=0,1,...,k-1 \}$ where the integer $k \ge 1$. If $H=S^1$ we have nothing to prove
for (b). When $H$ is a finite closed subgroup, we identify $[0,1)$ with $S^1$ by the usual map and note that
if $\beta_t$ is restricted to $t \in [0,{1 \over k})$, then by scaling we check that $\beta_t$ defines a
representation of $S^1$ in automorphisms of $\clo^H_d$. Now we consider the direct integral representation
$\pi'$ defined by
$$\pi'= \int^{\oplus}_{[0,{1 \over k})} dt \pi_{|_{\clo^H_d}} \beta_t $$
of $\clo_d^{H}$ on $\clh_{|_{\clo_d^H}}\otimes L^2([0,{1 \over k})\;)$, where $\clh_{|_{\clo_d^H}}$ is the cyclic
space of $\pi(\clo_d^H)$ generated by $\Omega$. Interesting point here to note that the new representation $\pi'$
is $(\beta_t)$ covariant i.e. $\pi' \beta_t=\beta_t \pi' $, hence by simplicity of the $C^*$ algebra $\clo_d$ we
conclude that $$\pi'(\mbox{UHF}_d)'' = \pi'(\clo_d^{H})''^{\beta_t}$$

\vsp
By exploring the hypothesis that $\psi$ is a factor state, we also have as in Lemma 6.11 in [BJKW]
$I \otimes L^{\infty}([0,{1 \over k})\;) \subset \pi'(\clo_d^H)''$. Hence we also have
$$\pi'(\clo_d^H)''=\pi(\clo_d^H)'' \otimes L^{\infty}([0,{1 \over k})\;).$$
Since $\beta_t$ is acting as translation on $I \otimes L^{\infty}([0,{1 \over k})\;)$ which being
an ergodic action, we have
$$\pi'(\mbox{UHF}_d)'' = \pi(\clo_d^{H})'' \otimes 1$$
Since $\pi'(\mbox{UHF}_d)'' = \pi(\mbox{UHF}_d)'' \otimes 1$, we conclude
that $\pi(\mbox{UHF}_d)'' = \pi(\clo_d^{H})''$.

\vsp
The statement (c) follows from Lemma 7.12 in [BKKW]. \qed

\vsp
Let $\omega'$ be an $\lambda$-invariant state on the $\mbox{UHF}_d$ sub-algebra of $\clo_d$. Following [BJKW, section 7],
we consider the set
$$K_{\omega'}= \{ \psi: \psi \mbox{ is a state on } \clo_d \mbox{ such that } \psi \lambda =
\psi \mbox{ and } \psi_{|\mbox{UHF}_d} = \omega' \}$$

By taking invariant mean on an extension of $\omega'$ to $\clo_d$, we verify that $K_{\omega'}$ is non empty and
$K_{\omega'}$ is clearly convex and compact in the weak topology. In case $\omega'$ is an ergodic state ( extremal state )
$K_{\omega'}$ is a face in the $\lambda$ invariant states. Before we proceed to the next section here we recall Lemma 7.4
of [BJKW] in the following proposition.

\vsp
\NI {\bf PROPOSITION 2.6:} Let $\omega'$ be ergodic. Then $\psi \in K_{\omega'}$ is an extremal point in
$K_{\omega'}$ if and only if $\psi$ is a factor state and moreover any other extremal point in $K_{\omega'}$
have the form $\psi \beta_z$ for some $z \in S^1$.

\vsp
\NI {\bf PROOF:} Though Proposition 7.4 in [BJKW] appeared in a different set up, same proof goes through for
the present case. We omit the details and refer to the original work for a proof.  \qed

\newsection{ Dual Popescu system and pure translation invariant states: }

\vsp
In this section we review the amalgamated Hilbert space developed in [BJKW] and prove a powerful criteria
for a translation invariant factor state to be pure.

\vsp
To that end let $\clm$ be a von-Neumann algebra acting on a Hilbert space $\clk$ and $\{v_k,\;1 \le k \le d\}$
be a family of bounded operators on $\clk$ so that $\clm=\{v_k,v_k^*,\;1 \le k \le d \}''$ and
$\sum_kv_kv^*_k=1$. Furthermore let $\Omega$ be a cyclic and separating vector for $\clm$ so that the normal state
$\phi_0(x)=<\Omega,x\Omega>$ on $\clm$ is invariant for the Markov map $\tau$ on $\clm$ defined by $\tau(x)=\sum_kv_kxv^*_k$
for $x \in \clm$. Let $\omega$ be the translation invariant state on UHF$_d =\otimes_{\IZ}M_d$ defined by
$$\omega( e^{i_1}_{j_1}(l) \otimes e^{i_2}_{j_2}(l+1) \otimes ....\otimes e^{i_n}_{j_n}(l+n-1) )=\phi_0(v_Iv^*_J)$$
where $e^i_j(l)$ is the elementary matrix at lattice sight $l \in \IZ$.

\vsp
We set $\tilde{v}_k = \overline{ \clj \sigma_{i \over 2}(v^*_k) \clj } \in \clm'$ ( see [BJKW] for details )
where $\clj$ and $\sigma=(\sigma_t,\;t \in \IR)$ are Tomita's conjugation operator and modular automorphisms
associated with $\phi_0$.

By  KMS relation [BR vol-1] we verify that
$$\sum_k \tilde{v}_k \tilde{v}_k^*=1$$
and
\be
\phi_0(v_Iv^*_J)= \phi_0(\tilde{v}_{\tilde{I}}\tilde{v}^*_{\tilde{J}})
\ee
where $\tilde{I}=(i_n,..,i_2,i_1)$ if $I=(i_1,i_2,...,i_n)$. Moreover $\tilde{v}^*_I\Omega =
\clj \sigma_{i \over 2}(v_{\tilde{I}})^*\clj\Omega= \clj \Delta^{1 \over 2}v_{\tilde{I}}\Omega
=v^*_{\tilde{I}}\Omega$.

\vsp
Let $(\clh,P,S_k,\;1 \le k \le d )$ and $(\tilde{\clh},P,\tilde{S}_k,\;1 \le k \le d)$ be the Popescu dilation described
as in Theorem 2.2 associated with $(\clk,v_k,\;1 \le k \le d)$ and $\clk,\tilde{v}_k,\;1 \le k \le d)$ respectively.
Following [BJKW] we consider the amalgamated tensor product $\clh \otimes_{\clk} \tilde{\clh}$ of $\clh$ with
$\tilde{\clh}$ over the joint subspace $\clk$. It is the completion of the quotient of the set $$\IC I \otimes
\IC \bar{I} \otimes \clk,$$ where $I,\bar{I}$ both consist of all finite sequences with elements in $\{1,2, ..,d \}$,
by the equivalence relation defined by a semi-inner product defined on the set by requiring
$$<I \otimes \bar{I} \otimes f,IJ \otimes \bar{I}\bar{J} \otimes g>=<f,v_J\tilde{v}_{\bar{J}}g>,$$
$$<I \otimes \bar{I}\bar{J} \otimes f,IJ \otimes \bar{I} \otimes g>=<\tilde{v}_{\bar{J}}f,v_Jg>$$
and all inner product that are not of these form are zero. We also define
two commuting representations $(S_i)$ and $(\tilde{S}_i)$ of $\clo_d$ on
$\clh \otimes_{\clk} \tilde{\clh}$ by the following prescription:
$$S_I\lambda(J \otimes \bar{J} \otimes f)=\lambda(IJ \otimes \bar{J} \otimes f),$$
$$\tilde{S}_{\bar{I}}\lambda(J \otimes \bar{J} \otimes f)=\lambda(J \otimes \bar{J}\bar{I} \otimes f),$$
where $\lambda$ is the quotient map from the index set to the Hilbert space. Note that the subspace generated by
$\lambda(I \otimes \emptyset \otimes \clk)$ can be identified with $\clh$ and earlier $S_I$ can be identified
with the restriction of $S_I$ defined here. Same is valid for $\tilde{S}_{\bar{I}}$. The subspace $\clk$ is
identified here with $\lambda(\emptyset \otimes \emptyset \otimes \clk)$.
Thus $\clk$ is a cyclic subspace for the representation $$s_i \otimes \tilde{s}_j \raro S_i \tilde{S}_j$$
of $\clo_d \otimes \tilde{\clo}_d$ in the amalgamated Hilbert space. Let $P$ be the projection on $\clk$. Then we
have
$$S_i^*P=PS_i^*P=v_i^*$$
$$\tilde{S}_i^*P=P\tilde{S}_i^*P=\tilde{v}^*_i$$
for all $1 \le i \le d$.

\vsp
We start with a simple proposition.

\vsp
\NI {\bf PROPOSITION 3.1: } The following hold:

\NI (a) The vectors $\{ \tilde{v}^*_I\Omega : |I| < \infty \}$ are total in $\clk$ if and only the
vectors $\{ v^*_I\Omega : |I| < \infty \}$ are total in $\clk$;

\NI (b) For any $1 \le i,j \le d$ and $|I|,|J|< \infty$ and $|\bar{I}|,|\bar{J}| < \infty$
$$<\Omega,\tilde{S}_{\bar{I}}\tilde{S}^*_{\bar{J}} S_iS_IS^*_JS^*_j \Omega>=<\Omega,
\tilde{S}_i \tilde{S}_{\bar{I}}\tilde{S}^*_{\bar{J}}\tilde{S}^*_jS_IS^*_J \Omega>;$$

\NI (c) The vector state $\psi_{\Omega}$ on $$\tilde{\mbox{UHF}}_d \otimes \mbox{UHF}_d \equiv
\otimes_{-\infty}^0 M_d \otimes_1^{\infty}M_d \equiv \otimes_{\IZ} M_d$$ is equal to $\omega$;

\NI (d) $\pi(\clo_d \otimes \tilde{\clo}_d)''= \clb(\clh \otimes_{\clk} \tilde{\clh})$ if and only if
$\{x \in \clb(\clk): \tau(x)=x,\; \tilde{\tau}(x)=x \}= \{zI: z \in  \IC \}$.

\vsp
\NI {\bf PROOF: } By our construction $P\tilde{S}^*_iP\Omega =\tilde{v}^*_i\Omega=v_i^*\Omega=PS_i^*P\Omega$ for each
$1 \le i \le d$. Thus (a) follows trivially as $v^*_I\Omega=\tilde{v}^*_{\tilde{I}}\Omega$
for all $|I| < \infty$. Now (b) and (c) follows by repeated application of $\tilde{S}^*_i\Omega=S^*_i\Omega$ and
commuting property of the two representation $\pi(\clo_d \otimes I)$ and $\pi(I \otimes \tilde{\clo}_d)$. The last
statement (d) follows from a more general fact proved below
that the commutant of $\pi(\clo_d \otimes \tilde{\clo}_d)''$ is
order isomorphic with the set $\{x \in \clb(\clk): \tau(x)=x,\; \tilde{\tau}(x)=x \}= \{zI: z \in  \IC \}$ via the
map $X \raro PXP$ where $X$ is the weak$^*$ limit of $\{\Lambda^m\tilde{\Lambda}^n(x)$ as $(m,n) \raro
(\infty,\infty)$. For details let $Y$ be the strong limit of increasing sequence of projections
$(\Lambda \tilde{\Lambda})^n(P)$ as $n \raro \infty$. Then $Y$ commutes with $S_i\tilde{S}_j, S^*_i\tilde{S}^*_j$
for all $1 \le i,j \le d$. As $\Lambda(P)) \ge P$, we also have $\Lambda(Y) \ge Y$. Hence $(1-Y)S_i^*Y=0$. As $Y$ commutes
with $S_i\tilde{S}_j$ we get $(1-Y)S_i^*S_i\tilde{S}_jY=0$ i.e. $(1-Y)\tilde{S}_jY=0$ for all $1 \le j \le d$. By symmetry
of the argument we get $(1-Y)S_iY=0$ for all $1 \le i \le d$. Hence $Y$ commutes with $\pi(\clo_d)''$ and by symmetry of
the argument $Y$ commutes as well with $\pi(\tilde{\clo}_d)''$. As $Yf=f$ for all $f \in \clk$, by cyclicity of the
representation we conclude that $Y=I$ in $\clh \otimes_{\clk} \tilde{\clh}$.

\vsp
Let $x \in \clb(\clk)$ so that $\tau(x)=x$ and $\tilde{\tau}(x) = x $ then as in the proof of Proposition 2.2
we also check that $(\Lambda \tilde{\Lambda})^k(P) \Lambda^m \tilde{\Lambda}^n(x) (\Lambda \tilde{\Lambda})^k(P)$ is
independent of $m,n$ as long as $m,n \ge k$. Hence weak$^*$ limit $\Lambda^m 
\tilde{\Lambda}^n(x) \raro X$ exists as $m,n \raro \infty$ and $\Lambda(X)=X=\tilde{\Lambda}(X)$
and furthermore $PXP=x$. Thus $X \in \pi(\clo_d \otimes \tilde{\clo}_d)'$ and that the map $X \raro PXP$ is an 
order-isomorphic, from the set of self adjoint elements in the commutant of $\pi(\clo_d \otimes \tilde{\clo}_d)''$ 
to the self adjoint elements in the set $\{x \in \clb(\clk): \tau(x)=x=\tilde{\tau}(x) \}$, follows as in 
Proposition 2.2. So (d) follows.  \qed

\vsp
Now we will be more specific in our starting Popescu systems in order to explore the representation $\pi$ of
$\clo_d \otimes \tilde{\clo}_d)$ in the amalgamated Hilbert space $\clh \otimes_{\clk} \tilde{\clh}$.
To that end let $\omega$ be a translation invariant extremal state on $\cla$ and we fix any extremal
point $\psi \in K_{\omega'}$. In the following we will eventually consider the Popescu systems $(\clk,\clm,v_k,1 \le k
\le d,\Omega)$ described as in Proposition 2.4 and associated dual Popescu space.

\vsp
In case we consider the Popescu systems $(\clk,\clm,v_k,1 \le k \le d,\Omega)$ described as in Proposition 2.3, we
need an extra assumption that $\phi_0$ is $\tau$ invariant in order to define the dual Popescu elements $(\clk,\clm',
\tilde{v}_k,1 \le k \le d,\Omega)$. In such a case by our construction the common subspace $\clk = \mbox{closed span of}
\{S_I^*\Omega: |I| < \infty \}$ is cyclic for the representation $\pi$ of $\clo_d \otimes \tilde{\clo}_d$ in $\clh
\otimes_{\clk} \tilde{\clh}$. For the Popescu systems described in Proposition 2.3, the set of vectors
$\{S^*_I\Omega: |I| < \infty \}$ being total in $\clk$, we also conclude that $\Omega$ is a cyclic vector for the
representation $\pi$ of $\clo_d \otimes \tilde{\clo}_d$. Note here that $\pi(\clo_d)''$ restricted to the Hilbert
subspace $\clh$ of $\clh \otimes_{\clk} \tilde{\clh}$ is isomorphic with the GNS space associated with $(\clo_d,\psi)$.
Same hold for $\pi(\tilde{\clo}_d)''$ as the set of vectors $\{\tilde{v}^*_I\Omega: |I| < \infty \}$ are also total
in $\clk$.

\vsp
However for the Popescu systems $(\clk,\clm,v_k,1 \le k \le d,\Omega)$ described as in Proposition 2.4, as
$\Omega$ is separating for $\clm$ and thus following above we have a representation $\pi: \clo_d \otimes \tilde{\clo}_d$
in the amalgamated space over the common subspace $\clk$ associated with the support projection $P$ of
the state $\psi$ in $\pi(\clo_d)''$. Let $E \in \pi(\clo_d \otimes I)'', \tilde{E} \in \pi(I \otimes \tilde{\clo}_d)''$
be the support projections of the states $\psi,\tilde{\psi}$ respectively where
$$\psi(s_Is^*_J)=<\Omega,S_IS^*_J\Omega>$$
and
$$\tilde{\psi}(\tilde{s}_I \tilde{s}^*_J)=<\Omega,\tilde{S}_I\tilde{S}^*_J\Omega>$$
for $|I|,|J| < \infty$. $\clh$ being an invariant subspace of $(S_i)$ and $\clm'= \{ x \in \clb(\clk):
\sum_k v_kxv_k^*= x \}$ by Lemma 7.5 in [BJKW] we recall that $P_{{|}_{\clh}}=E_{{|}_{\clh}}$.
Hence $ES_IS^*_J\Omega=PS_IS^*_J\Omega=v_Iv^*_J\Omega$ for all $|I|,|J| < \infty \}$. Since $\Omega$
is cyclic for $\clm$ and $E \in \pi(\clo_d \otimes \tilde{\clo}_d)''$ we conclude that $\Omega$ is cyclic
for the representation $\pi$ of $\clo_d \otimes \tilde{\clo}_d$.

\vsp
It is also obvious that $\pi(\clo_d)''$ restricted to the Hilbert subspace $\clh$ is isomorphic with the GNS space
associated with $(\clo_d,\psi)$. However same property for $\pi(\tilde{\clo}_d)''$ is not
immediate as $\Omega$ may not be a cyclic vector for $\pi(\tilde{\clo}_d)''$ unless
$\tilde{E}_{{|}_{\tilde{\clh}}}=P_{{||_{\tilde{\clh}}}}$. Thus this property can be ensured
by Lemma 7.5 in [BJKW] applying to the dual representation $(\tilde{S}_i)$ provided we assume
that $\{x \in \clb(\clk): \sum_k \tilde{v}_kx\tilde{v}^*_k=x \} = \clm$. This additional condition
is rather deep and will lead us to a far reaching consequence on the state $\omega$. In the following
we prove a crucial step for that goal.

\vsp
\NI {\bf PROPOSITION 3.2: } Let $\omega$ be an extremal translation invariant state on $\cla$ and $\psi$
be an extremal point in $K_{\omega'}$. In the following we consider the amalgamated representation $\pi$ of 
$\clo_d \otimes \tilde{\clo}_d$ in $\clh \otimes \tilde{\clh}$ as described in Proposition 3.1 with 
Popescu systems $(\clk,\clm,v_k,\;1 \le k \le d)$ associated with the support projection as defined 
in Proposition 2.4. Then the following are true:

\vsp
\NI (a) $\pi(\clo_d \otimes \tilde{\clo}_d)''= \clb(\clh \otimes_{\clk} \tilde{\clh})$. Furthermore $\pi(\clo_d)''$ and
$\pi(\tilde{\clo}_d)''$ are factors and the following sets are equal:

\NI (i) $H=\{ z \in S^1: \psi \beta_z = \psi \}$;

\NI (ii) $H_{\pi}  = \{ z : \beta_z \mbox{ extends to an automorphisms of } \pi(\clo_d)'' \}$;

\NI (iii) $\tilde{H}_{\pi}= \{ z : \beta_z \mbox{ extends to an automorphisms of } \pi(\tilde{\clo}_d)'' \}$.
Moreover $\pi(I \otimes \mbox{UHF}_d)''$ and $\pi(\mbox{UHF}_d \otimes I)''$ are factors.

\NI (b) $z \raro U_z$ is the unitary representation of $H$ in the Hilbert space $\clh \otimes_{\clk} \tilde{\clh}$
defined by
$U_z(\pi(s_i \otimes \tilde{s}_j)\Omega=\pi(zs_i \otimes z\tilde{s}_j)\Omega$

\NI (c) The centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ is invariant by the canonical endomorphisms
$\Lambda(X)=\sum_i S_iXS_i^*$ and $\tilde{\Lambda}(X) = \sum_i \tilde{S}_iX \tilde{S}^*_i$. Moreover for each $i$ the
surjective map $X \raro S^*_iXS_i$ keeps the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ is invariant.
Same holds for the map $X \raro \tilde{S}^*_iX\tilde{S}_i$.

\vsp
\NI {\bf PROOF: } $P$ being the support projection by Proposition 2.4 we have $\{x \in \clb(\clk): \sum_k v_k x v_k^*= 
x\} = \clm'$. That $(\clm',\tilde{\tau},\phi_0)$ is ergodic follows from a general result [Mo1] ( see also [BJKW] for 
a different proof ) as $(\clm,\tau,\phi_0)$ is ergodic. Hence
$\{x \in \clb(\clk): \tau(x)=\tilde{\tau}(x)=x \}= \IC$. Hence by Proposition 3.1 we conclude that
$\pi(\clo_d \otimes \tilde{\clo}_d)''= \clb(\clh \otimes_{\clk} \tilde{\clh})$

\vsp
That both $\pi(\clo_d)''$ and $\pi(\tilde{\clo}_d)''$ are factors follows trivially as
$\pi(\clo_d \otimes \tilde{\clo}_d)''=\clb(\clh \otimes_{\clk} \tilde{\clh})$. By our discussion above we first recall
that $\Omega$ is a cyclic vector for the representation of $\pi(\clo_d \otimes \tilde{\clo}_d)$. Let $G = \{ z=(z_1,z_2)
\in S^1 \times S^1: \beta_z \mbox{
extends to an automorphism of } \pi(\clo_d \otimes \tilde{\clo}_d)'' \}$ be the closed subgroup where
$$\beta_z(s_i \otimes \tilde{s}_j)=z_1 s_i \otimes z_2 \tilde{s}_j.$$

\vsp
By repeated application of the fact that $\pi(\clo_d)$ commutes with $\pi(\tilde{\clo}_d)$ and 
$S_i^*\Omega=\tilde{S}^*_i\Omega$ as in Proposition 3.1 (b) we verify that $\psi \beta_{(z,z)}=\psi$ on 
$\clo_d \otimes \tilde{\clo}_d$ for all $z \in H$. For $z \in H $ we set unitary operator $U_z \pi(x \otimes y) 
\Omega = \pi(\beta_z(z) \otimes \beta_z(y))\Omega$ for all $x \in \clo_d$ and $y \in \tilde{\clo}_d$. Tues we have 
$U_z\pi(s_i)U_z^*=z\pi(s_i)$ and also $U_z\pi(\tilde{s}_i)U_z^*=z \tilde{s}_i.$ By taking it's restriction to 
$\pi(\clo_d)''$ and $\pi(\tilde{\clo}_d)''$ respectively we check that $H \subseteq \tilde{H}_{\pi} $ and 
$H \subseteq H_{\pi}$.

\vsp
For the converse let $z \in H_{\pi}$ and we use the same symbol $\beta_z$ for the extension to an automorphism of
$\pi(\clo_d)''$. Since $\lambda$ commutes with $\beta_z$ on $\clo_d$, the canonical endomorphism $\Lambda$ defined
by $\Lambda(X) = \sum_k S_kXS_k^*$ also commutes with extension of $\beta_z$ on $\pi(\clo_d)''$. Note that the map
$\pi(x)_{{|}_{\clh}} \raro \pi(\beta_z(x))_{{|}_{\clh}} $ for $x \in \clo_d$ is a well defined linear $*$-homomorphism.
Since same is true for $\bar{z}$ and $\beta_z\beta_{\bar{z}}=I$, the map is an isomorphism. Hence $\beta_z$ extends
uniquely to an automorphism of $\pi(\clo_d)''_{{|}_{\clh}}$ commuting with the restriction of the canonical endomorphism
on $\clh$. Since $\pi(\clo_d)''_{{|}_{\clh}}$ is a factor, we conclude as in Proposition 2.5 (a) that $z \in H$. Thus
$H_{\pi} \subseteq H$. As $\pi(\tilde{\clo}_d)''$ is also a factor, we also have $\tilde{H}_{\pi} \subseteq H$. Hence we
have $H=H_{\pi}=\tilde{H}_{\pi}$ and $G \subseteq H \times H$.

\vsp
For the second part of (a) we will adopt the argument used for Proposition 2.5. To that end we first note that $\Omega$
being a cyclic vector for the representation $\tilde{\clo}_d \otimes \clo_d$ in the Hilbert space $\tilde{\clh} \otimes_{\clk}
\clh $, by Lemma 7.11 in [BJKW] (note that the proof only needs the cyclic property ) the representation of UHF$_d$ on
$\tilde{\clh} \otimes_{\clk} \clh $ is quasi-equivalent to it's sub-representation on the cyclic space generated by $\Omega$.
On the other hand by our hypothesis that $\omega$ is a factor state, Power's theorem [Po1] ensures that the state $\omega'$
(i.e. the restriction of $\omega$ to $\cla_R$ which is identified here with $\mbox{UHF}_d$ ) is also a factor state
on $\mbox{UHF}_d$. Hence quasi-equivalence ensures that $\pi(\mbox{UHF}_d \otimes I)''$ is a factor. We also note that
the argument used in Lemma 7.11 in [BJKW] is symmetric i.e. same argument is also hold true for $\tilde{\mbox{UHF}}_d$.
Thus $\pi(I \otimes \tilde{\mbox{UHF}}_d)''$ is also a factor.

\vsp
As $\Lambda(E)$ commutes with $\pi(\lambda(\mbox{UHF}_d) \otimes \tilde{\mbox{UHF}}_d)''$ and
$\{ S_iS^*_j: 1 \le i,j \le d \}$ we verify that $\Lambda(E)$ is also an element in the centre of
$\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ as $\beta_z\Lambda=\Lambda\beta_z$.
For the last statement consider the map $X \raro S_i^*XS_i$. As $\beta_z(S^*_iXS_i) = S^*_i\beta_z(X)S_i$ for
all $z \in G$, it preserves $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ and clearly
onto. Hence all that we need to show that $S^*_iES_i$ is an element in the commutant whenever
$E$ is an element in the centre. To that end note that $S_i^*ES_iS_i^*XS_i=S_i^*S_iS_i^*EXS_i=S_i^*XES_i=
S^*_iXS_iS_i^*ES_i$. Thus onto property of the map ensures that $S_i^*ES_i$ is an element in the centre of
$\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$. This completes the proof of (c). \qed

\vsp
One interesting problem here how to describe the von-Neumann algebra $\cli$ consists of invariant elements of
the gauge action $\{ \beta_z: z \in H \}$ in $\clb(\clh \otimes_{\clk} \tilde{\clh})$. A general result
due to E. Stormer [So] says that the algebra of invariant elements are von-Neumann algebra of type-I with
completely atomic centre. On the other hand invariant elements  $\cli$ contains $\pi(\mbox{UHF}_d \otimes
\tilde{\mbox{UHF}}_d)''$. In the following we prove that the equality hold if and only if $\pi(\clo_d \otimes
\tilde{\mbox{UHF}}_d)''=\pi(\mbox{UHF}_d \otimes \tilde{\clo}_d)''= \clb(\clh \otimes_{\clk} \tilde{\clh})$. Moreover
such equality hold if and only if $\omega$ is pure on $\cla$. However as a first step we describe the
center of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ by exploring Cuntz relation that it is also non-atomic
for a factor state $\omega$. In the following we give an explicit description.

\vsp
\NI {\bf PROPOSITION 3.3 : } Let $(\clo_d,\psi)$ be as in Proposition 3.2 with Popescu system $(\clk,\clm,v_k,\Omega)$ 
is taken as in Proposition 2.4. If $\omega$ is a factor state $\cla$ then the centre of $\pi(\mbox{UHF}_d \otimes 
\tilde{\mbox{UHF}}_d)''$ is completely atomic and the element $E_0=[\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)' 
\vee \pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''\Omega]$ is a minimal projection in the centre of $\pi(\mbox{UHF}_d 
\otimes \tilde{\mbox{UHF}}_d)''$ and centre is invariant for both $\Lambda$ and $\tilde{\Lambda}$. Furthermore the 
following hold:

\NI (a) The centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ has the following two disjoint
possibilities:

\NI (i) There exists a positive integer $n \ge 1$ such that the centre is generated by the family of minimal
orthogonal projections $\{ \Lambda_k(E_0): 0 \le k \le n-1 \}$ where $n \ge 1$ is the least positive integer so that
$\Lambda^n(E_0)=E_0$.

\NI (ii) The family of minimal nonzero orthogonal projections $\{ E_k: k \in \IZ \}$ where $E_k= \Lambda^k(E_0)$
for $k \ge 0$ and $E_k= S^*_IE_0S_I$ for $k < 0$ where $|I|=-k$ and independent of multi-index $I$ generates
the centre. In such a case $H=S^1$ and $E_0=E'$, where $E'=[\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''\Omega]$.

\vsp
\NI (b) $\Lambda(E)=\tilde{\Lambda}(E) $ for any $E$ in the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$

\NI (c) If $\Lambda(E_0)=E_0$ then $E_0=1$.

\vsp
\NI {\bf PROOF: } Let $E' \in \pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)'$ be the projection on the subspace generated
by the vectors $\{ S_IS^*_J\tilde{S}_{I'}S^*_{J'}\Omega,\; |I|=|J|,|I'|=|J'| < \infty \}$ and $\pi_{\Omega}$ be the restriction 
of the representation $\pi$ of $\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d$ to the cyclic subspace $\clh_{\Omega}$ generated by
$\Omega$. Identifying $\cla$ with $\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$ we check that $\pi_{\omega}$ is unitary
equivalent with $\pi_{\Omega}$. Thus $\pi_{\Omega}$ is a factor representation.

\vsp
For any projection $E$ in the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ we note that $EE'=E'EE'$
is an element in the centre of $\pi_{\Omega}(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$. $\omega$ being a factor 
state we conclude that
\be
EE' = \omega(E)E'
\ee
for any projection $E$ in the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$. Thus we have
\be
EE_0 = \omega(E)E_0
\ee
for all element $E$ in the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$. Since $EE'$ is a projection
we have $\omega(E)=\omega(E)^2$ as $E' \neq 0$. Thus $\omega(E)=1$ or $0$. So for such an element $E$
the following are true:

\NI (i) If $E \le E_0$ then either $E=0$ or $E=E_0$

\NI (ii) $\omega(E)=1$ if and only if $E \ge E_0$

\NI (iii) $\omega(E)=0$ if and only if $EE_0=0$.

\vsp
As $\Lambda(E_0)$ is a projection in the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$, either
$\omega(\Lambda(E_0))=1$ or $0$. So we have either $\Lambda(E_0) \ge E_0$ or $\Lambda(E_0)E_0=0$. In case
$\Lambda(E_0) \ge E_0$ we have $S^*_iE_0S_i \le E_0$ for all $1 \le i \le d$. However by (i) as $S^*_iE_0S_i$ being a
non-zero projection in the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$, we conclude
that $E_0=\Lambda(E_0)$.

\vsp
Furthermore if $\Lambda(E_0)E_0=0$, we have $\Lambda(E_0) \le I-E_0$ and by Cuntz's relation we check that
$E_0 \le I - S^*_iE_0S_i$ and $S^*_jS^*_iE_0S_iS_j \le I- S^*_jE_0S_j$ for all $1 \le i,j \le d$.
So we also have $E_0S^*_jS^*_iE_0S_iS_jE_0 \le E_0- E_0S^*_jE_0S_jE_0=E_0$. Thus we have either
$E_0S^*_jS^*_iE_0S_iS_jE_0= 0$ or $E_0S^*_jS^*_iE_0S_iS_jE_0=E_0$ as $S^*_jS^*_iE_0S_iS_j$ is an element
in the centre by (b). So either we have $\Lambda^2(E_0)E_0=0$ or $\Lambda^2(E_0) \le E_0$. $\Lambda$ being an injective
map we either have $\Lambda^2(E_0)E_0=0$ or $\Lambda^2(E_0)= E_0.$

\vsp
More generally we check that if $\Lambda(E_0)E_0=0, \Lambda^2(E_0)E_0=0,..\Lambda^n(E_0)E_0=0$ for some $n \ge 1$
then either $\Lambda^{n+1}(E_0)E_0=0$ or $\Lambda^{n+1}(E_0)=E_0$. To verify that first we check that in such a
case $E_0 \le I- S^*_IE_0S_I$ for all $|I|=n$ and then following the same steps as before to check that $S^*_iS^*_IE_0S_IS_i
\le I-S^*E_0S_i$ for all $i$. Thus we have $E_0S^*_iS^*_IE_0S_IS_iE_0 \le E_0$ and arguing as before we complete the proof
of the claim that either $\Lambda^{n+1}(E_0)E_0=0$ or $\Lambda^{n+1}(E_0)=E_0.$ We summarize now by saying that if
$\Lambda(E_0)E_0=0$ then either we have $E_0\Lambda^n(E_0)=0$ for all $n \ge 1$ or there exists an integer $n \ge 2$
so that $E_0\Lambda_k(E_0)=0$ for all $1 \le k \le n-1$ but $\Lambda^n(E_0)=E_0$. Since $S_I^*\Omega=\tilde{S}^*_{\tilde{I}}
\Omega$ we verify that $\omega(\tilde{\Lambda}^n(E_0))=\omega(\Lambda^n(E_0))$. Thus the sequence of orthogonal projections
$E_0, \tilde{\Lambda}(E_0), \tilde{\Lambda}^2(E_0),...$ are also periodic with same period or aperiodic according as the
sequence of orthogonal projections $E_0,\Lambda(E_0),...$. By the above we summarize now by saying either $\Lambda(E_0)
\ge E_0$ ( hence $\tilde{\Lambda}(E_0) \ge E_0$) or the sequence of orthogonal projections $E_0, \tilde{\Lambda}(E_0),
\tilde{\Lambda}^2(E_0),...$ are also periodic with same period $n \ge 2$ or aperiodic according as the sequence of
orthogonal projections $E_0,\Lambda(E_0),...$.

\vsp
Let $\pi_k,\;k \ge 0$ be the representation $\pi$ of $\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d$ restricted to the
subspace $\Lambda^k(E_0)$. The representation $\pi_0$ of $ \mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d$ is quasi-equivalent
to the representation $\pi$ of $\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d$ restricted to $E'$. $\omega$ being a factor state,
$\pi_0$ is a factor representation. Now we fix any $k \ge 1$ and let $X$ be an element in the centre of
$\pi_k(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$. Then for any $|I|=k$, $S^*_IXS_I$ is an element in the centre
of $\pi_0(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$. Further $S_I^*XS_I= S^*_IXS_IS^*JS_J = S^*_JXS_J$ for all
$|J|=|I|=k$. $\pi_0$ being a factor representation, we have $S^*_IXS_I= c E_0$ for some scaler $c$ independent of
the multi-index we choose $|I|=k$. Hence $c \Lambda_k(E_0) = \sum _{|J|=k } S_JS^*_IXS_IS_J^*=
\sum _{|J|=k } S_JS^*_IS_IS_J^*X= X$ as $X$ is an element in the centre of
$\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$. Thus for each $k \ge 1$, $\pi_k$ is a factor
representation as $\pi_0$ is.

\vsp
Let $\pi_k,\; k \le -1$ be the representation $\pi$ of $\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d$ restricted to the
subspace $S^*_IE_0S_I$ where $I$ is any multi-index $|I|=-k$. $E_0$ being an element in the centre of $\mbox{UHF}_d 
\otimes \tilde{\mbox{UHF}}_d$, the projection $E_k=S^*_IE_0S_I$ is independent of the multi-index. Going along the same 
line as above, we verify that for each $k \le 0$, $\pi_k$ is a factor representation of $\mbox{UHF}_d \otimes 
\tilde{\mbox{UHF}}_d$.

\vsp
We note that $\Lambda(E_0)\tilde{\Lambda}(E_0) \neq 0$. Otherwise we have $<S_i\Omega,\tilde{S}_j\Omega>=0$
for all $i,j$ and so $<S^*_i\Omega,\tilde{S}^*_j\Omega>=0$ for all $i,j$ as $\pi(\clo_d)''$ commutes with
$\pi(\tilde{\clo}_d)''$. However $\tilde{S}^*_j\Omega=S^*_j\Omega$ which leads a contradiction. Hence
$\Lambda(E_0)\tilde{\Lambda}(E_0) \neq 0$. As $\pi$ restricted to $\Lambda(E_0)$ is a factor state we
conclude that $\Lambda(E_0)=\tilde{\Lambda}(E_0).$ Using commuting property of the endomorphisms $\Lambda$
and $\tilde{\Lambda}$, we verify by a simple induction method that $\Lambda^m(E_0) = \tilde{\Lambda}^m(E_0)$
for all $m \ge 1$. By Cuntz's relation and commuting property of two Cuntz's representation we also get from
the equality $\Lambda^n(E_0)=\tilde{\Lambda}^n(E_0)$ that $S^*_IE_0S_I = \tilde{S}^*_JE_0 \tilde{S}_J$ for
any $|I|=|J|=n$.

\vsp
Now we consider the case where $E_0,\Lambda(E_0),..\Lambda^n(E_0),..$ is a sequence of aperiodic orthogonal
projection. We set
$$E_k=\Lambda^k(E_0)\;\; \mbox{for all}\;\; k \ge 1$$
and
$$E_k = S^*_IE_0S_I\;\; \mbox{for all}\;\; k \le -1,\;\;\mbox{where}\;\; |I|=-k$$
As $S_IS_J^*$ with $|I|=|J|$ commutes with $E_0$ being an element in the centre of $\pi(\mbox{UHF}_d
\otimes \tilde{\mbox{UHF}}_d)''$ we verify that the definition of $E_k$ is independent of multi-index $I$ that 
we choose. Further $\Lambda^n(E_0)=\tilde{\Lambda}^n(E_0)$ ensures that $S_I\tilde{S}_J^*$ commutes with $E_0$ 
for all $|I|=|J|$. Hence we get $E_k=S^*_IE_0S_I\tilde{S}^*_J\tilde{S}_J=\tilde{S}^*_JE_0\tilde{S}_J$ for all 
$|J|=|I|$. Hence we have $\Lambda(E_k) = \tilde{\Lambda}(E_k) = E_{k+1}$ for all $k \in \!Z$. We also claim that
$\{ E_k: k \in \!Z \}$ is an orthogonal family of non-zero projections. To that end we choose any two elements
say $E_k,E_m,\;k \ne m$ and use endomorphism $\Lambda^n$ for $n$ large enough so that both $n+k \ge 0 , n+m  \ge 0$
to conclude that $\Lambda^n(E_kE_m) = E_{k+n}E_{k+m}=0$ as $k+n \ne k+m$. $\Lambda$ being an injective map we get
the required orthogonal property. Thus $\sum_{ k \in \!Z} E_k$ being an invariant projection for both $\Lambda$ and
$\tilde{\Lambda}$ we get by cyclicity of $\Omega$ that $\sum_{ k \in \!Z} E_k=I$.

\vsp
We also set
$$U_z= \sum_{n \in \IZ } z^n E_n,\;\;z \in S^1 $$
and verify that $U_zS_iU_z^*=z S_i$ and $U_z \tilde{S}_i U_z^*= z \tilde{S}_i$ for all $1 \le i \le d$. Hence
$H=S^1$ as $U_z\Omega=\Omega$ for all $z \in S^1$.
However cyclicity of the vector $\Omega$ for $\pi(\clo_d \otimes \tilde{\clo}_d)''$ ensures that
$E_0=E'$ and $E_k= \Lambda_k(E')$ for all $k \ge 1$ and $E_k= S^*_IE'S_I$ for $k < 0$, where $|I|=-k$.
For an explicit proof note that the new family of projections $F_k=\Lambda^k(E')$ for $k \ge 0$ and
$F_k= S^*_IE'S_I$ for all $k < 0$ are orthogonal as they are subspace of another orthogonal family $E_k$
but their sum is an invariant projection for $\Lambda$ and $\tilde{\Lambda}$. Hence by cyclicity of
$\Omega$ it is equal to $I$. Hence we get the required equality.  \qed

\vsp
Now we prove an important step for the central point of the main result in this section.

\vsp
\NI {\bf PROPOSITION 3.4: }  Let $(\clo_d,\psi)$ be as in Proposition 3.3 with Popescu system 
$(\clk,\clm,v_k,\Omega)$ is taken as in Proposition 2.4. Then the following statements are equivalent:

\NI (a) $\omega$ is a pure state on $\cla$;

\NI (b) $\pi(\clo_d \otimes \tilde{\mbox{UHF}}_d)''= \pi(\mbox{UHF}_d \otimes \tilde{\clo}_d)''= \clb(\clh \otimes
\tilde{\clh})$;

\NI (c) The von-Neumann algebra generated by $\{ \beta_z: z \in H \}$ invariant elements is equal to
$\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$;

\vsp
In such a case the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ is equal to $\{ U_z: z \in H \}''$.

\vsp
\NI {\bf PROOF: } We will prove that (a) implies (b) , (b) implies (c) and then (c) implies (a).

\vsp
Now we explore the pure property of $\omega$ to prove that each representation $\pi_k$ of
$\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d$ (defined in Proposition 3.3 ) is pure. For $k=0$ we 
have nothing to prove. Now fix any $k \in \IZ$ and let $X$ be an element in the commutant of 
$\pi_k(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$. If $k \ge 1$ we check that $S^*_IXS_I$ is an 
element in commutant of $\pi_0(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$, where 
$|I|=k$. Hence $S_I^*XS_I$ is a scaler independent of $I$ as $\pi_0$ is pure. We use commuting
property of $X$ with $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ to conclude that $X= c \Lambda^k(E_0)$
for some scaler $c$. If $k \le -1$ we employ the same method with endomorphism $\Lambda^{-k}$ to
pull back $X$ to an element in the commutant of $\pi_0(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$.
We omit the details as it is similar to what we have done in Proposition 3.3.

\vsp
Hence the family of projection $\{ E_k: k \in \IZ \}$ is a maximal abelian algebra in the commutant of
$\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$ and $\pi$ decomposes into irreducible representations 
$\pi_k$. Since the family of $\{ E_k: k \in \IZ \}$ is also a factor decomposition of $\pi$, commutant of
$\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$ is generated by the family of projections $\{ E_k: k \in \IZ \}$.
Thus for any $X \in \pi(\clo_d \otimes \tilde{\mbox{UHF}}_d)'$ we have $X= \sum_k c_k E_k$ and as $\Lambda(X)=X$
we get $c_k=c_{k+1}$. Hence $\pi(\clo_d \otimes \tilde{\mbox{UHF}}_d)''= \clb(\clh \otimes_{\clk} \tilde{\clh})$.
Similarly we also have $\pi(\mbox{UHF}_d \otimes \tilde{\clo}_d )'' = \clb(\clh \otimes_{\clk} \tilde{\clh})$.
This completes the proof that (a) implies (b).

\vsp
That (b) implies (c) follows as in the proof for Proposition 2.5. Now we assume (c). In such a case by Stormers
theorem [So], $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ is type-I von-Neumann algebra with centre completely
atomic. The projection $E'=[\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''\Omega]$ is an $(\beta_z:z \in H)$ invariant 
element and thus by (c) $E'$ is an element in the centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''.$ As $E' 
\le E_0=$ and
$E_0$ is minimal we conclude that $E'=E_0$, hence the representation $\pi_0$ of
$\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d$ i.e. the restriction of $\pi$ to $E'$
is a type-I factor representation. Hence $\omega$ is a type-I translation invariant factor state on $\cla$.
Now we refer to [BJKW, Lemma 7.13 ] to conclude that $\omega$ is pure. This completes the proof that (c)
implies (a).

\vsp
We claim that the centre is equal to $\{ U_z: z \in H \}''$. As $\beta_w(U_z)=U_z$ for all $z,w \in H$, we have
$U_z \in \pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ by (c). Hence $\{U_z: z \in H \}''$
is a subset of the centre. Conversely if $x$ is an element in the centre of
$\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ then $E_kxE_k \in \pi_k(\mbox{UHF}_d \otimes
\tilde{\mbox{UHF}}_d)'$. Since each $\pi_k$ is pure, we conclude that $E_kxE_k= c_kE_k$ for some $c_k$.
Hence $x= \sum_k c_kE_k $ is an element in $\{ U_z: z \in H \}''$.   \qed

\vsp
The following proposition is a crucial step that will be used repeatedly.

\vsp
\NI {\bf PROPOSITION 3.5: } Let $\omega$ be an extremal state on $\cla$ and $\psi$ be an extremal element
$\psi$ in $K_{\omega}$. We consider the Popescu elements $(\clk,v_k:1 \le k \le m, \clm,\Omega)$
as in Proposition 2.4 for the dual Popescu elements and associated amalgamated representation $\pi$ of
$\clo_d \otimes \tilde{\clo}_d$ as described in Proposition 3.1. Then the following hold:

\NI (a) $\pi(\clo_d \otimes \tilde{\clo}_d)''= \clb(\clh \otimes_{\clk} \tilde{\clh})$;

\NI (b) $Q=E\tilde{E}$ is the support projection of the state $\psi$ in $\pi(\clo_d)''\tilde{E}$
and also in $\pi(\tilde{\clo}_d)''E$ where $E$ and $\tilde{E}$ are the support projections of the state $\psi$
in $\pi(\clo_d)''$ and $\pi(\tilde{\clo}_d)''$ respectively.

\NI (c) If $\omega$ is pure on $\cla$ then the following hold:

\NI (i) $Q=P$,

\NI (ii) $[\pi(\clo_d)''\Omega]=\tilde{E}$ and $[\pi(\tilde{\clo}_d)''\Omega]=E$.

\NI (d) If $\omega$ is as in (c) and then the following statements are equivalent:

\NI (i) $\clm'= \tilde{\clm}$ where $\clm= \{PS_iP: 1 \le i \le d \}''$ and $\tilde{\clm}=\{P\tilde{S}_iP: 1 \le i \le d
\}''$;

\NI (ii) $\pi(\clo_d)'=\pi(\tilde{\clo}_d)''$.

\vsp
\NI {\bf PROOF: } For (a) we will apply Proposition 3.1 (d). To that end let $x \in \clb(\clk)$ be any element for which
$\tau(x)=x=\tilde{\tau}(x)$. Since $P$ is the support projection of the state $\psi$ in $\pi(\clo_d)''$
by Proposition 2.4 the equality $x=\tau(x)$ ensures that $x \in \clm'$ and also $(\clm,\tau,\phi_0)$ is ergodic. 
However by a general result of Frigerio [Fr]
( see also appendix of [BJKW], [Mo1] for a ready proof ) $(\clm',\tilde{\tau},\phi_0)$ is also ergodic. 
Hence the other equality i.e. $x=\tilde{\tau}(x)$ ensures that $x$ is a scaler as $x \in \clm'$.

\vsp
By our construction in general $\pi(\tilde{\clo}_d)'' \subseteq \pi(\clo_d)' $. Before we aim to prove
the reverse inclusion we state few general points now. To that end let $E$ and $\tilde{E}$ be the support projections of
the state $\psi$ in $\pi(\clo_d)''$ and $\pi(\tilde{\clo}_d)''$ respectively. We set von-Neumann algebras
$\cln_1=\pi(\clo_d)'E$ and $\cln_2=\pi(\tilde{\clo}_d)''E$. Note that it is enough for $\pi(\clo_d)'= \pi(\tilde{\clo}_d)''$ 
if we prove that $\cln_2=\cln_1$. That it is enough follows as $E$ being the support projection of the state on the 
factor $\pi(\clo_d)''$, we have $E \ge [\pi(\tilde{\clo}_d)\Omega]$ and hence $\Lambda^n(E) \uparrow I$ as $n \raro \infty$ 
because $\Omega$ is cyclic for $\pi(\clo_d \otimes \tilde{\clo}_d)$ in $\clh \otimes \tilde{\clh}$. Hence two operators in
$\pi(\clo_d)'$ are same if their actions are same on $E$. Further we note that $Q=E\tilde{E} \in \cln_2 \subseteq \cln_1$
and claim that $Q$ is the support projection of the state $\psi$ in $\cln_2$. To that end let $xE \ge 0$ for some $x \in 
\pi(\tilde{\clo}_d)''$ so that $\psi(QxQ)=0$. As $\Lambda^k(xE) \ge 0$ for all $k \ge 1$ and $\Lambda^k(E) \raro I$
we conclude that $x \ge 0$. As $E \Omega=\Omega$ and thus $\psi(\tilde{E}x\tilde{E})=\psi(QxQ)=0$, we conclude
$\tilde{E}x\tilde{E}=0$, $\tilde{E}$ being the support projection for $\pi(\tilde{\clo}_d)''$. Hence $QxQ=0$.
As $\psi(Q)=1$, we complete the proof of the claim that $Q$ is the support of $\psi$ in $\cln_2$. Similarly $Q$ 
is also the support projection of the state $\psi$ in $\pi(\clo_d)''\tilde{E}$. As $E \in \pi(\clo_d)''$ and 
$\tilde{E} \in \pi(\tilde{\clo}_d)''$ we check that von-Neumann algebras $\clm_1 = Q \pi(\clo_d)'' Q$ and 
$\tilde{\clm}_1 = Q \pi(\tilde{\clo}_d) Q$ acting on $Q$ satisfies $\tilde{\clm}_1 \subseteq \clm_1'$.


\vsp
Now we explore that $\pi(\clo_d \otimes \tilde{\clo}_d)''=\clb(\clh \otimes_{\clk} \tilde{\clh})$ and note
that in such a case $Q\pi( \clo_d \otimes \tilde{\clo}_d )''Q$ is the set of all bounded operators on the
Hilbert subspace $Q$. As $E \in \pi(\clo_d)''$ and $\tilde{E} \in \pi(\tilde{\clo}_d)''$ we check that together
$\clm_1 = Q \pi(\clo_d)'' Q$ and $\tilde{\clm}_1 = Q \pi(\tilde{\clo}_d) Q$ generate all bounded operators on $Q$.
Thus both $\clm_1$ and $\tilde{\clm}_1$ are factors. The canonical states $\psi$ on $\clm_1$ and
$\tilde{\clm}_1$ are faithful and normal.


\vsp
We set $l_k=QS_kQ$ and $\tilde{l}_k=Q\tilde{S}_kQ,\; 1 \le k \le d$ and recall that $v_k=PS_kP$ and
$\tilde{v}_k=P\tilde{S}_kP,\; 1 \le k \le d$. We note that $Pl_kP=v_k$ and $P\tilde{l}_kP = \tilde{v}_k$
where we recall by our construction $P$ is the support projection of the state $\psi$ in $\pi(\clo_d)''_
{|}[\pi(\clo_d)\Omega]$. $Q$ being the support projection of $\pi(\clo_d)\tilde{E}$, by Theorem 2.4 applied to
Cuntz elements $\{S_i\tilde{E}: 1 \le i \le d \}$, $\pi(\clo_d)'\tilde{E}$ is order isomorphic with $\clm'_1$
via the map $X \raro QXQ$. As the projection $F=[\pi(\clo_d)''\Omega] \in \pi(\clo_d)'$, we check that
the element $QF\tilde{E}Q \in \clm_1'$. However $QF\tilde{E}Q=E\tilde{E}F\tilde{E}E=QPQ=P$ and thus $P \in \clm_1'$.


\vsp
Now we aim to prove (c). To that end, going along the line of Proposition 2.5, we note that
$\beta=\{ \beta_z: z \in H \}$ invariant elements in $\pi(\clo_d \otimes \tilde{\mbox{UHF}}_d)''$
are $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ i.e. $\{ U_z: z \in H \}'=
\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ and by a Theorem of E. Stormer [So]
$\pi(\clo_d \otimes \tilde{\mbox{UHF}}_d)''$ is type-I von-Neumann algebra with
completely atomic centre. In particular the family $\{U_z:  z \in H \}''$ is the
centre of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$. Let $U_z= \sum_k z^k E_k$ be
the spectral decomposition where either $0 \le  k \le n$ or $- \infty \le k \le \infty $
according as $H$ is a cyclic subgroup of $S^1$ or $H=S^1$. That $E_0 = [\pi(\mbox{UHF}_d \otimes
\tilde{\mbox{UHF}}_d)''\Omega]$ follows as $E_0$ is a minimal projection in the centre. Hence
$\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ restricted to $E_0$ is pure.

\vsp
As $\beta_z(F)=F$ for all $z \in H$, $F \in \pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$. Now the hard part
is to show that $F \in \pi(\tilde{\mbox{UHF}}_d)''$.

\vsp
We claim that $E_0\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''E_0 \bigcap (E_0\pi(\mbox{UHF})E_0)' =
E_0\pi(\tilde{\mbox{UHF}}_d)''E_0$. To that end note that for each fix $m,n \ge 0$, von-Neumann algebra
$\Lambda^m\tilde{\Lambda}^n(\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$ is also a type-I von-Neumann
sub-algebra of $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$ with centre
$\{U_z: z \in H \}''$. Thus the relative commutant of $\Lambda^m\tilde{\Lambda}^n(\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)$
i.e. $\{S_IS^*_J\tilde{S}_{I'}\tilde{S}_{J'}^*: |I|=|J|=m, |I'|=|J'|=n \}'' \vee \{U_z:z \in H\}''$ is a type-I von-Neumann algebra
with centre $\{U_z: z \in H \}''$.

\vsp
Hence $\{E_0S_IS_J^*E_0: |I|=|J|=m \}''$ is type-I sub-factor of a type-I factor
$\{E_0S_IS^*_J\tilde{S}\tilde{S}^*_{J'}E_0: |I|=|J|=m, |I'|=|J'|=n \}''$ acting on the subspace $E_0$. As subfactors
$\{E_0S_IS_J^*E_0: |I|=|J|=m \}''$ and $\{E_0S_{I'}S^*_{J'}E_0: |I'|=|J'|=n \}''$ are type-I, we conclude by a standard
argument that
$\{E_0S_IS^*_J\tilde{S}\tilde{S}_{J'}E_0: |I|=|J|=m, |I'|=|J'|=n \}'' \bigcap
\{E_0S_IS_J^*E_0: |I|=|J|=m \}'= \{ E_0S_{I'}S^*_{J'}E_0: |I'|=|J'|=n \}''$. Now by taking limit $n \raro \infty$ and then
$m \raro \infty$ we complete the proof of our claim. Same argument is valid if we replace $E_0$ by any $E_k$. Hence relative
commutant of $\pi(\mbox{UHF}_d)''$ in $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ is equal to
$\pi(\tilde{\mbox{UHF}}_d)'' \vee \{ U_z : z \in H \}''$. Thus for each $k$ we have $PE_k \in E_kQ\pi(\tilde{\mbox{UHF}}_d)''QE_k$
and so $P \in Q\pi(\tilde{\mbox{UHF}}_d)''Q$. As $F$ is the strong limit of $\Lambda^n(P)$ as $n \raro \infty$,
we conclude that $F \in \tilde{E}\pi(\tilde{\mbox{UHF}}_d)''\tilde{E}$. $\tilde{E} ( \ge F )$ being the support
projection of $\psi$ in $\pi(\tilde{\mbox{UHF}}_d)''$, we conclude that $F=\tilde{E}$. Hence $Q=E\tilde{E}=EF=P$.

\vsp
Thus $\Omega$ is a cyclic and separating vector for both $\clm_1$ and $\tilde{\clm}_1$. As $\tilde{\Lambda}^n(P) =
\tilde{\Lambda}^n(\tilde{E}) E \uparrow E$ as $n \raro \infty$, we get $E=[\pi(\clo_d)'\Omega]=[\pi(\tilde{\clo}_d)''\Omega]$
and similarly $\tilde{E}=[\pi(\tilde{\clo}_d)'\Omega]=[\pi(\clo_d)''\Omega]$. As $E\tilde{E}=P$ and  $\pi(\clo_d)''$ is
isomorphic with $\pi(\clo_d)''_{| \clh}$ we conclude that $P$ is the support projection of the state $\psi$ in
$\pi(\clo_d)''|_{\clh}$. Following the same steps for the dual Popescu system we conclude also that $P$ is the
support projection of the state $\psi$ in $\pi(\tilde{\clo}_d)''_{|\tilde{\clh}}$. This completes the proof of (c).

\vsp
Now we are left to prove (d). $\omega$ being pure we have $P=Q$ by (c). That (ii) implies (i) is obvious. Now we aim to
prove the converse. We appeal to the commutant lifting Theorem 2.2 for Cuntz elements $(\tilde{S}_iE)$ to conclude that the
self adjoint elements in $E\pi(\tilde{\clo}_d)'E$ are order isomorphic with self adjoint elements of $Q\pi(\tilde{\clo}_d)'Q$
( here we have used that $Q$ is the support projection of the state $\psi$ in $\cln_1$ ) which is equal to $\clm_1$ by 
our assumption that $\clm= \tilde{\clm}$ and $P=Q$. However $Q\pi(\clo)_d)''Q=\clm_1$ and $\pi(\clo_d)
\subseteq \pi(\tilde{\clo}_d)'$. As $\clm_1$ is a von-Neumann algebra, we conclude by order isomorphic property
that self adjoint elements in $E\pi(\tilde{\clo}_d)'E$ are equal to the self adjoint elements in $E\pi(\clo_d)''$.
Since both are von-Neumann algebras in their own right, we conclude $\cln_1=\cln_2$. This completes the proof for
(ii) in (d) by our starting observation. \qed

\vsp
The merit of the results in Proposition 3.5 is well understood once we note that for a factor state
$\omega$ if $\clm'_1=\tilde{\clm}_1$, then $\Omega$ is also cyclic for $\clm_1$ and thus $P=Q$. However
toward the end of this section we show that the canonical trace on $\cla$ indeed gives an example where
$\clm'_1$ is not equal to $\tilde{\clm}_1$. Thus a more reasonable question that one puts is it true for
pure state $\omega$. To that end in the following we introduce few more symmetry on $\omega$.

\vsp
Let $\psi$ be a $\lambda$-invariant state on $\clo_d$ and $\tilde{\psi}$ be the state on $\clo_d$ defined by
$$\tilde{\psi}(s_Is_J^*)=\psi(s_{\tilde{J}}s_{\tilde{I}}^*)$$
for all $|I|,|J| < \infty$ and $(\clh_{\tilde{\psi}},\pi_{\tilde{\psi}},\Omega_{\tilde{\psi}})$ be the GNS
space associated with $(\clo_d,\tilde{\psi})$. That $\tilde{\psi}$ is well defined follows once we check
by (3.1) that $$\psi(s_{\tilde{J}}s_{\tilde{I}}^*)=\phi_0(v_{\tilde{J}}v_{\tilde{I}}^*) =
\phi_0(\tilde{v}_I\tilde{v}_J^*)$$ and appeal to Proposition 2.1 by observing that cyclicity condition i.e.
the closed linear span $P_0$ of the set of vectors $\{\tilde{v}_I^*\Omega: |I| < \infty \}$ is $\clk$, can be
ensured if not by taking a new set of Popescu elements $\{P_0\tilde{v}_kP_0:1 \le k \le d \}$.

\vsp
Similarly for any translation invariant state $\omega$ on $\cla$
we set translation invariant state $\tilde{\omega}$ on $\cla$ by
$$\tilde{\omega}(e^{i_1}_{j_1}(l) \otimes e^{i_2}_{j_2}(l+1) \otimes  ... \otimes e^{i_n}_{j_n}(l+n-1)) =
\omega(e^{i_n}_{j_n}(l)... \otimes e^{i_2}_{j_2}(l+1) \otimes e^{i_1}_{j_1}(l+n-1))$$
Note first that the map $\psi \raro \tilde{\psi}$ is a one to one and onto map in the convex set of $\lambda$ invariant
state on $\clo_d$. In particular the map $\psi \raro \tilde{\psi}$ takes an element from $K_{\omega}$ to $K_{\tilde{\omega}}$
and the map is once more one to one and onto. Hence for any extremal point $\psi \in K_{\omega}$, $\tilde{\psi}$ is also an
extremal point in $K_{\tilde{\omega}}$. Using Power's criteria we also verify here that $\omega$ is an extremal state if
and only if $\tilde{\omega}$ is an extremal state. However such a conclusion for a pure state $\omega$ seems not
so obvious. For the time being we have the following useful proposition.

\vsp
\NI {\bf PROPOSITION 3.6: } Let $\omega$ be an extremal translation invariant state on $\cla$ and $\psi \raro \tilde{\psi}$
be the map defined for $\lambda$ invariant states on $\clo_d$. Then the following hold:

\NI (a) $\psi \in K_{\omega}$ is a factor state if and only if $\tilde{\psi} \in K_{\tilde{\omega}}$ is a factor state.

\NI (b) A Popescu systems $(\clk,\clm,v_k,\Omega)$ of $\psi$ satisfies Proposition 2.4 with
$(\pi_{\psi}(s_k),\;1 \le k \le d,\;P,\;\Omega)$ i.e. the projection $P$ on the subspace $\clk$
is the support projection of the state $\psi$ in $\pi(\clo_d)''$ and $v_i=P\pi_{\psi}(s_i)P$
for all $1 \le i \le d$, then the dual Popescu systems $(\clk,\clm',\tilde{v}_k,\Omega)$ satisfies
Proposition 2.4 with $(\pi_{\tilde{\psi}}(s_k),\;1 \le k \le d,\;P,\;\Omega)$ i.e. the projection
$P$ on the subspace $\clk$ is the support projection of the state $\tilde{\psi}$ in
$\pi_{\tilde{\psi}}(\clo_d)''$ and $\tilde{v}_i=P\pi_{\tilde{\psi}}(s_i)P$ for all
$1 \le i \le d$, if and only if $\{x \in \clb(\clk): \sum_k \tilde{v}_kx \tilde{v}_k^* = x  \} = \clm$.

\vsp
\NI {\bf PROOF: } Since $\omega$ is an extremal translation invariant state, by Power's criteria $\tilde{\omega}$ is 
also an extremal state. As an extremal point of $K_{\omega}$ is map to an extremal point in $K_{\tilde{\omega}}$ by one to one property of the map $\psi \raro \tilde{\psi}$, we conclude by Proposition 2.6 that $\psi$ is a factor state if and only if $\tilde{\psi}$
is a factor state. (b) follows by the converse part of the Proposition 2.4 applied to the dual
Popescu systems $(\clk,\clm',\tilde{v}_k,\Omega)$. This completes the proof \qed

\vsp
We will show toward the end of this section that for a translation invariant factor state $\omega$ in general
$\tilde{\clm}$ need not be equal to $\clm'$ and also in Proposition 3.6 (b) $P$ need not be the support projection
of the state $\tilde{\psi}$ in $\pi(\tilde{\clo}_d)''$. To that end we introduce few more additional symmetry on $\omega$.

\vsp
If $Q = Q^{(l)}_0 \otimes Q^{(l+1)}_1 \otimes ....\otimes Q^{(l+m)}_m $ we set
$\tilde{Q} = Q^{(-l-m+1)}_m \otimes Q^{(-l-m+2)}_{m-1} \otimes ..\otimes Q^{(-l+1)}_0$
where $Q_0,Q_1,...,Q_m$ are arbitrary elements in $M_d$. Note that it is the refection
around the point ${1 \over 2}$. We define $\tilde{Q}$ by extending linearly to any
$Q \in \cla_{loc}$. For a state $\omega$ on UHF$_d$ $C^*$ algebra $\otimes_{\IZ} M_d$
we define a state $\tilde{\omega}$ on $\otimes_{\IZ} M_d$ by the following prescription
\be
\tilde{\omega}(Q)= \omega(\tilde{Q})
\ee
Thus the state $\tilde{\omega}$ is translation invariant, ergodic, factor state if and only if $\omega$ is
translation invariant, ergodic, factor state respectively. We say $\omega$ is {\it lattice symmetric } if
$\tilde{\omega}=\omega$.

\vsp
For a $\lambda$ invariant state $\psi$ on $\clo_d$ we define as before a $\lambda$ invariant state
$\tilde{\psi}$ by
\be
\tilde{\psi}(s_Is^*_J)=\psi(s_{\tilde{I}}s^*_{\tilde{J}})
\ee
for all $|I|,|J| < \infty.$ It is obvious that $\psi \in K_{\omega'}$ if and only if $\tilde{\psi}
\in K_{\tilde{\omega}'}$ and the map $\psi \raro \tilde{\psi}$ is an affine map. In particular an
extremal point in $K_{\omega'}$ is also mapped to an extremal point of $K_{\tilde{\omega}'}$. It is also
clear that $\tilde{\psi} \in K_{\omega'}$ if and only if $\omega$ is lattice symmetric. Hence a lattice symmetric
state $\omega$ determines an affine map $\psi \raro \tilde{\psi}$ on the compact convex set $K_{\omega'}$.
Furthermore, if $\omega$ is also extremal on $\cla$, then the affine map, being continuous on the set of extremal
elements in $K_{\omega'}$, which can be identified with $S^1 / H \equiv S^1$ or $\{1 \}$ ( by Proposition 2.6 )
$z \raro \psi \beta_z$ being fixing an extremal element $\psi \in K_{\omega'}$ for the time being ).


\vsp
There exists $z_0 \in S^1$ so that $\tilde{\psi}=\psi \beta_{z_0}$ and as $\tilde{\beta \beta_z}=\tilde{\beta} \beta_z$
for all $z \in S^1$, we get the affine map taking $\psi \beta_z \raro \psi \beta_{z_0} \beta_z $ and thus
determines a continuous one to one and onto map on $S^1 / H$ and as $\tilde{\tilde{\psi}}=\psi$ its
inverse is itself. Thus either the affine map has a fixed point or $z_0^2=1$ i.e. it is a rotation map
by an angle $2\pi$ ( Here we have identified $S^1 / H $ with $S^1$ in case $H \neq S^1$ ). Thus there exists
an extremal element $\psi \in K_{\omega'}$ so that either $\tilde{\psi}=\psi \beta_{\zeta}$ where
$\zeta $ is either $1$ or $-1$ where we recall that we have identified $S^1 / H =S^1$ when $H \neq S^1$. Note that
if we wish remove the identification, then for $H=\{ z : z^n =1 \}$ for some $n \ge 1$, $\zeta$ is either $1$ or
or $exp^{\pi i \over n }$. Note that in case $H=S_1$ then $\tilde{\psi}=\psi$ for $\psi \in K_{\omega'}$ as
$K_{\omega}$ is a singleton set by Proposition 2.6.

\vsp
\NI {\bf PROPOSITION 3.7: } Let $\omega$ be a translation invariant lattice symmetric state on $\cla$. Then the following
hold:

\NI (a) If $\omega$ is also an extremal translation invariant state on $\cla$ then $H=\{z \in S^1: \psi \beta_z =\psi \}$
is independ of $\psi \in K_{\omega'}$.

\NI (b) If $H=\{z: z^n =1 \}$ for some $n \ge 0$ then $\tilde{\psi}=\psi \beta_{\zeta}$ for all $\psi \in K_{\omega'}$ where
$\zeta$ is fixed either $1$ or $exp^{\pi i \over n }$. Let $(\clh,S_k,\;1 \le k \le d,\Omega)$ be the GNS space associated
with $(\clo_d,\psi)$, $P$ be the support projection of the state $\psi$ in $\pi(\clo_d)''$ and $\clk= P \clh$ with Popescu
systems $(\clk,\clm,v_k, \;1 \le k \le d,\;\Omega)$ as in Proposition 2.4 where $v_k=PS_kP$ and associated normal state $\phi_0$
on $\clm=\{ v_k,v_k^*: 1 \le k \le d \}''$ is invariant for $\tau(x)=\sum_k v_k x v_k^*$. Let $(\tilde{\clh},S_k
\;1 \le k \le d,\Omega)$ be the Popescu minimal dilation in Theorem 2.2 of the dual Popescu systems
$(\clk,\tilde{\clm},\tilde{v}_k,\;1 \le k \le d,\;\Omega)$ defined in Proposition 3.2. Then there exists a
unitary operator $U_{\zeta}: \clh \otimes_{\clk} \tilde{\clh}$ so that
\be
U_{\zeta}^*=U_{\bar{\zeta}}, U_{\zeta}\Omega=\Omega,\;\;\;U_{\zeta} S_k U_{\zeta}^*= \beta_{\bar{\zeta}}(\tilde{S}_k)
\ee
for all $1 \le k \le d$. Furthermore if $\omega$ is also pure then there exists a unitary operator $u: \clk \raro \clk$
so that
\be
u_{\zeta}\Omega=\Omega, \;\;\; u_{\zeta}v_ku_{\zeta}^*= \beta_{\bar{\zeta}}(\tilde{v}_k)
\ee
for all $1 \le k \le d$ and $u_{\zeta} \clj u_{\zeta}^*=\clj,\;\;u_{\zeta} \Delta^{1 \over 2} u_{\zeta}^*=\Delta^{-{1 \over 2}}$
and $u_{\zeta}^*=u_{\bar{\zeta}}$. Moreover in any case $\clm' = \tilde{\clm}$ and $u_{\zeta} \clm u_{\zeta}^* = \clm'$.
If $\zeta=1$ then $u_{\zeta}$ is self-adjoint and the other hand if $\zeta \ne 1$ then $u_{\zeta}^{2n}$ is self adjoint.

\NI (c) Further if $H=S^1$ then $K_{\omega'}$ is having only one element $\psi$, so $\psi=\tilde{\psi}$ and
(3.6) and (3.7) is valid with $\zeta=1$.

\vsp
\NI {\bf PROOF: } (a) follows by Proposition 2.6. Now we aim to prove (b). For existence of an extremal state $\psi \in K_{\omega'}$
so that $\tilde{\psi}=\psi \beta_{\zeta}$ we refer to the paragraph preceding the statement of this proposition. As $\tilde{(\psi \beta_z)}
=\tilde{\psi} \beta_z$ for all $z \in S^1$, a simple application of Proposition 2.6 says that $\tilde{\psi} = \psi \beta_{\zeta} $ for
all extremal points in $K_{\omega'}$ if it holds for one extremal element. Hence existence part in (b) is true by Krein-Millmann theorem.


\vsp
Thus we define
$U_{\zeta}:\clh \otimes_{\clk} \tilde{\clh} \raro \clh \otimes_{\clk} \tilde{\clh}$ by
$$U_{\zeta}: S_IS^*_J\tilde{S}_{I'}\tilde{S}_{J'}^*\Omega \raro
\beta_{\bar{\zeta}}(S_{I'}S_{J'}^*\tilde{S}_I\tilde{S}^*_J) \tilde{\Omega}$$
That $U_{\zeta}$ is an unitary operator follows from (3.1) and (3.3). By our construction we also have
$U_{\zeta}S_k = \beta_{\bar{\zeta}}(\tilde{S}_k)U_{\zeta} $ for all $1 \le k \le d$.
In particular $U_{\zeta}\pi(\clo_d)''U_{\zeta}^*=\pi(\tilde{\clo}_d)''$.

\vsp
Now for a pure state $\omega$ by Proposition 3.5 we have $P=Q$ and hence $UPU^*=UQU^*=Q=P$ which ensures an unitary
operator $u=PUP$ on $\clk$ and a routine calculation shows that
\be
u_{\zeta}v^*_k u_{\zeta}^*= \beta_{\bar{\zeta}}(\tilde{v}^*_k)
\ee
for all $1 \le k \le d$.As $U_{\zeta}^*=U_{\bar{\zeta}}$ we have $u_{\zeta}^*=u_{\bar{\zeta}}$. If $\zeta \ne 1$, then
$\zeta^{2n}=1$ and thus $U^{2n}_{\zeta}$ is inverse of its own. Thus $u^{2n}_{\zeta}$ is self-adjoint.

\vsp
In the following we consider the case $\zeta=1$ for simplicity of notation and otherwise for the case $\zeta \ne 1$ very
little modification is needed in the symbols or simply reset notation including the phase factor on the right hand side
of the Popescu elements.

\vsp
We denote $u_1=u$ in the following for simplicity. It is simple to verify now the following steps
$uSv_Iv_J^*\Omega=uv_Jv_I^*\Omega=\tilde{v}_J\tilde{v}^*_I\Omega=F\tilde{v}_I\tilde{v}^*_J\Omega$
where $Sx\Omega=x^*\Omega,\;x \clm$ and $Fx'\Omega=x'^*\Omega,\;x' \in \clm'$ are the Tomita's
conjugate operator. Hence $u \clj \Delta^{1 \over 2}=\clj \Delta^{-{1 \over 2}}u$, i.e $u \clj u^*
u\Delta^{1 \over 2}u^* = \clj \Delta^{-{1 \over 2}}$ and by uniqueness of polar decomposition we
conclude that $u \clj u^* = \clj$ and $u\Delta^{1 \over 2}u^*= \Delta^{-{1 \over 2}}$. We are left to
prove $\tilde{\clm}=\clm'$. It is obvious that $\tilde{\clm} \subseteq \clm'$. However $\clm'=\clj \clm \clj=
\clj u \tilde{\clm} u^* \clj= u \clj \tilde{\clm} \clj u \subseteq u \clj \clm' \clj u^*= u \clm u^*= \tilde{\clm}$.
Note that in the third identity we have used that $u$ commutes with $\clj$. Hence $\clm'=\tilde{\clm}$. \qed

\vsp
Before we move to the main result of this section we introduce another useful concept.
If $Q= Q^{(l)}_0 \otimes Q^{(l+1)}_1 \otimes ....\otimes Q^{(l+m)}_m$ we set
$Q^t={Q^t}^{(l)}_0 \otimes {Q^t}^{(l+1)}_1 \otimes ..\otimes {Q^t}^{(l+m)}_m$
where $Q_0,Q_1,...,Q_m$ are arbitrary elements in $M_d$ and $Q_0^t,Q^t_1,..$ stands for transpose
(not complex conjugate) of $Q_0,Q_1,..$ respectively. We define $Q^t$ by extending linearly for any
$Q \in \cla_{loc}$. For a state $\omega$ on UHF$_d$ $C^*$ algebra $\otimes_{\IZ} M_d$ we define
a state $\bar{\omega}$ on $\otimes_{\IZ} M_d$ by the following prescription
\be
\bar{\omega}(Q) =
\omega(Q^t)
\ee
Thus the state $\bar{\omega}$ is translation invariant, ergodic, factor state if and only if $\omega$ is
translation invariant, ergodic, factor state respectively. We say $\omega$ is {\it real } if
$\bar{\omega}=\omega$. In this section we study a translation invariant real state.

\vsp
For a $\lambda$ invariant state $\psi$ on $\clo_d$ we define a $\lambda$ invariant state $\bar{\psi}$ by
\be
\bar{\psi}(s_Is^*_J)=\psi(s_Js^*_I)
\ee
for all $|I|,|J| < \infty$ and extend linearly. That it defines a state follows as
for an element $x= \sum c(I,J)s_Is^*_J$ we have $\bar{\psi}(x^*x)=\psi(y^*y) \ge 0$
where $y= \sum \overline{c(I,J)}s_Js_I^*$. It is obvious that $\psi \in K_{\omega'}$ if and only if $\bar{\psi}
\in K_{\bar{\omega}'}$ and the map $\psi \raro \bar{\psi}$ is an affine map. In particular an
extremal point in $K_{\omega'}$ is also mapped to an extremal point in $K_{\bar{\omega}'}$. It is also
clear that $\bar{\psi} \in K_{\omega'}$ if and only if $\omega$ is real. Hence a real
state $\omega$ determines an affine map $\psi \raro \bar{\psi}$ on the compact convex set $K_{\omega'}$.
Furthermore, if $\omega$ is also extremal on $\cla$, then the affine map, being continuous on the set of extremal
elements in $K_{\omega'}$, which can be identified with $S^1 / H \equiv S^1$ or $\{ 1 \}$ ( by Proposition 2.6 )
by fixing an extremal element $\psi \in K_{\omega'}$. Now $\bar{\psi}=\psi \beta_{z_0}$ and $\bar{\psi \beta_z}=
\bar{\psi} \beta_{\bar{z}}$ for all $z \in S^1$, the affine map takes $\psi \beta_z \raro \psi \beta_{z_0 \bar{z}}$
and further its inverse is itself. If $z_0=1$ we get the map fixes a point. Even otherwise we can choose
$z \in S^1$ so that $z^2=z_0$ and for such a choice we get an extremal element fixed by the map. What is also
importent here to note that we can as well choose $z$ so that $z^2=-z_0$, if so then $\psi \beta_z $ gets maped into
$\psi \beta_{\bar{z}}$. Thus in any case we also have an extremal element $\psi \in K_{\omega'}$ so that
$\bar{\psi}=\psi \beta_{-1}$.

\vsp
Thus going back to the original set up, we sum up the above by saying that if  $H=\{z: z^n= 1 \} \subseteq S^1$ and
$\zeta \in \{1,exp^{i\pi \over n } \}$ then there exists an extremal element $\psi \in K_{\omega'}$ so that $\tilde{\psi}
=\psi \beta_{\zeta}$.

\vsp
However unlike the lattice symmetric property we note here that $\overline{ \psi \circ \beta_z } =
\bar{\psi} \circ \beta_{\bar{z}}$ for any $z \in S^1$ and thus for a real $\psi \in K_{\omega'}$
( $\psi$ is called real if $\bar{\psi}=\psi$ ) $\psi \beta_z$
is also real if and only if $\psi \beta_{z^2}=\psi$. The closed subgroup $H=\{ z \in S^1: \psi\beta_z =\psi \}$
of $S^1$ is either a cyclic group of $n \ge 1$ elements or the entire $S^1$. For example if $H$ is the trivial
subgroup, there exists only two extremal real states in $K_{\omega'}$ when $\omega$ is an extremal state.
Same can be said about the set of extremal elements $\{\psi \in K_{\omega'}: \bar{\psi}=\psi \beta_{\zeta} \}$.

\vsp
\NI {\bf PROPOSITION 3.8: } Let $\omega$ be a translation invariant real factor state on $\otimes_{\IZ}M_d$. Then the
following hold:

\NI (a) if $H=\{z: z^n= 1 \} \subseteq S^1$ and $\zeta \in \{1,exp^{i\pi \over n } \}$ then there exists an extremal
element $\psi \in K_{\omega'}$ so that $\tilde{\psi}=\psi \beta_{\zeta}$. Let $(\clh,\pi_{\psi}(s_k)=S_k,\;1 \le k \le d,
\Omega)$ be the GNS representation of $(\clo_d,\psi)$, $P$ be the support projection of the state $\psi$ in $\pi(\clo_d)''$
and $(\clk,\clm,v_k,\;1 \le k \le d, \Omega)$ be the associated Popescu systems as in Proposition 2.4.
Let $\bar{v}_k= \clj v_k \clj$ for all $1 \le k \le d$ and
$(\bar{\clh},\bar{S}_k,P_0,\Omega)$ be the Popescu minimal dilation as described in Theorem 2.2 associated with
the systems $(\clk,\clm',\bar{v}_k,\;1 \le k \le d,\Omega)$. Then there exists a unitary
operator
$W_{\zeta}: \clh \raro \bar{\clh}$ so that
\be
W_{\zeta}\Omega=\Omega,\;\;\;W_{\zeta}S_kW_{\zeta}^* = \beta_{\bar{\zeta}}(\bar{S}_k)
\ee
for all $1 \le k \le d$. Furthermore $P$ is the support projection of the state $\bar{\psi}$ in $\bar{\pi}(\clo_d)''$
and there exists a unitary operator $w_{\zeta}$ on $\clk$ so that
\be
w_{\zeta}\Omega=\Omega, \;\;\; w_{\zeta}v_kw_{\zeta}^* = \beta_{\bar{\zeta}}(\bar{v}_k)= \clj \beta_{\zeta}(v_k) \clj
\ee
for all $1 \le k \le d$ and  $w_{\zeta} \clj w_{\zeta}^*=\clj$ and $w_{\zeta} \Delta^{1 \over 2} w_{\zeta}^* =
\Delta^{-{1 \over 2}}$. Moreover if $\clm$ is a factor then $w_{\zeta}^*=w_{\bar{\zeta}}$.

\NI (b) If $H=S^1$, $K_{\omega'}$ is a set with unique elememt $\psi$ so that $\tilde{\psi}=\psi$ and relations
(3.11) and (3.12) are valid with $\zeta=1$.


\vsp
\NI {\bf PROOF: } For existence part in (a) we refer the paragraph above preceeded the statement of the proposition.
We fix a state $\psi \in K_{\omega'}$ so that $\bar{\psi}=\psi  \beta_{\zeta}$ and
\vsp
define $W:\clh \raro \bar{\clh}$ by
$$W_{\zeta}: S_IS^*_J\Omega \raro \beta_{\bar{\zeta}}(\bar{S}^*_{I}\bar{S}^*_{J})\Omega$$
That $W_{\zeta}$ is a unitary operator follows from (3.10) and thus
$W_{\zeta}S_k= \beta_{\bar{\zeta}}(\bar{S}_k)W_{\zeta}$ for all $1 \le k \le d$. For simplicity of notation we take the case $\zeta=1$
as very little modification is needed to include the case when $\zeta \ne 1$ or reset Cuntz elements by
absorbing the phase factor in the following computation and use notation $W$ for $W_{\zeta}$.

\vsp
$P$ being the support projection we have by Proposition 2.4 that $\clm'= \{ x \in \clb(\clh): \sum_k v_k x v_k^*= x \}$
and thus $\clm = \{ x \in \clb(\clk): \sum_k \clj v_k \clj x \clj v_k^* \clj = x \}$. Hence by the converse part of
Proposition 2.4 we conclude that $P$ is also the support projection of the state $\bar{\psi}$ in $\bar{\pi}(\clo_d)''$.
Hence $W_{\zeta}PW_{\zeta}^*=P$. Thus we define an unitary operator $w_{\zeta}: \clk \raro \clk$ by
$w_{\zeta}= PW_{\zeta}P $ and verify that
$$\bar{v}^*_k = P\bar{S}^*_kP$$
$$ = PW_{\zeta} \beta_{\zeta}(S^*_k)W_{\zeta}^*P = PW_{\zeta}P \beta_{\zeta}(S^*_k)PW_{\zeta}^*P$$
$$= PW_{\zeta}P\beta_{\zeta}(v^*_k)PW_{\zeta}^*P=w_{\zeta}\beta_{\zeta}(v^*_k)w_{\zeta}^*.$$

\vsp
We recall that Tomita's conjugate linear operators $S,F$ defined as in [BR] are
the closure of the linear operators defined by $S:x\Omega \raro x^*\Omega$ for $x \in \clm$ and $F:y\Omega \raro y^*\Omega$
for $y \in \clm'$. We check the following relations for $\zeta=1$ for simplicity
$wSv_Iv^*_J\Omega=wv_Jv^*_I\Omega= \bar{v}_J \bar{v}^*_I\Omega= F \bar{v}_I\bar{v}^*_J\Omega=
Fw v_Iv^*_J\Omega$ for $|I|,|J| < \infty$. Since such vectors are total, we have $wS=Fw$ on the domain of $S$. Thus
$wSw^*=F$ on the domain of $F$. We write $S=\clj \Delta^{1 \over 2}$ as the unique polar decomposition. Then $F=S^* =
\Delta^{1 \over 2} \clj= \clj \Delta^{-{1 \over 2}}$. Hence $w \clj w^*w\Delta^{1 \over 2}w^*=\clj \Delta^{-{1 \over 2}}$.
By the uniqueness of polar decomposition we get $w \clj w^*=\clj$ and $w \Delta^{1 \over 2} w^*= \Delta^{-{1 \over 2}}$.
Same algebra is valid in case $\zeta \ne 1$ if we reset the Popescu elements on the right hand side absorbing the phase
factor.


\vsp
Now we are going to show that $w_1$ is self-adjoint if $\zeta =1$ and also
if $\zeta \ne 1$, $w^{2n}_{\zeta}$ is self-adjoint. We give the proof in the following for the case
$\zeta=1$ as the proof follows same steps for $\zeta \ne 1$ where we need to replace $w_{\zeta}^{2n}$ in
the place of $w_1$. In the following we take $\zeta=1$ and use notation $w$ for $w_1$. We claim that
\be
w \bar{v} ^*_kw^*=v^*_k
\ee
for all $1 \le k \le d$.

\vsp
Note by (3.12) and Tomita's theorem that $w \clm w^* = \clm'$. However by Tomita's theorem we also have $\clj w \clm w^* \clj
= \clm$ and as $\clj$ commutes with $w$, we conclude that $w \clm' w^* = \clm$. Further the separating property of the vector
$\Omega$ for $\clm$ ensures that (3.13) hold if we verify the following identities:
$$w\bar{v}^*_kw^*\Omega = w \bar{v}^*_k\Omega$$
$$=w \clj v^*_k\Omega =\clj w v^*_kw\Omega = \clj \bar{v}^*_k\Omega= v^*_k\Omega $$
Thus $w\bar{v}^*_kw^*=v^*_k$. Hence $w^2 \in \clm'$ and as $w$ commutes with $\clj$, $w^2 \in \clm$.
$\omega$ being an extremal element in $K_{\omega'}$ we have $\clm \vee \tilde{\clm}=\clb(\clk)$ by Proposition 3.5
and as $\tilde{\clm} \subseteq \clm'$, we get that $\clm$ is a factor. Thus for a factor $\clm$, $w^2$ is a scaler.
Since $w\Omega=\Omega$ we get $w^*=w$. This completes the proof. \qed

\vsp
A state $\omega$ on $\otimes_{\IZ}M_d$ is said be in detailed balance if $\omega$ is both lattice symmetric and real.
In the following proposition as before we identified once more $S^1 / H \equiv S^1$ in case $H \neq S^1$ and set $\zeta$
be the least value in $S^1 \ H \equiv S^1 $ so that $\zeta^2 \in H$ ( see Proposition 3.7 for details ) .


\vsp
\NI {\bf PROPOSITION 3.9: } Let $\omega$ be a translation invariant extremal lattice symmetric state on the UHF$_d$
algebra $\otimes_{\IZ}M_d$. Then the following are equivalent:

\NI (a) $\omega$ is in detailed balance;

\NI (b) There exists an extremal element $\psi \in K_{\omega'}$ so that $\tilde{\psi}=\psi \beta_{\zeta}$ and
$\bar{\psi}=\psi \beta_{\zeta}$, where $\zeta$ is fixed number either $1$ or $exp^{i\pi \over n }$ ( where
$H=\{z : z^n= 1 \}$ ).

\vsp
Furthermore if $\omega$ is a pure state then the following hold:

\NI (c) There exists a Popescu elements $(\clk,v_k,\;1 \le k \le d,\Omega)$ so that $\omega=\omega_v$ with
$v_k= \clj \tilde{v}_k \clj $
for all $1 \le k \le d$.

\NI (d) The map $\clj: \clh \otimes_{\clk} \tilde{\clh} \raro \clh \otimes_{\clk} \tilde{\clh}$ defined by
$\pi(s_Is^*_J\tilde{s}_{I'}\tilde{s}^*_{J'})\Omega
\raro \pi(s_{J'}s_{I'}^*\tilde{s}_{J}\tilde{s}^*_{I})\Omega,
\;|I|,|J|,|I'|,|J'| < \infty $ extends the map $\clj: \clk  \raro  \clk$ to an anti-unitary map so that $\clj
\pi(s_i) \clj = \pi(\tilde{s}_i)$ for all $ 1 \le i \le d$.


\vsp
\NI {\bf PROOF: } Since $\omega$ is symmetric, by Proposition 3.7 $\psi=\tilde{\psi} \beta_{\zeta}$
for all $\psi \in K_{\omega'}$ where $\zeta$ is fixed number either $1$ or $exp^{i \pi \over n}$ for some $n \ge 1$.
Thus by Proposition 3.8 for there exists an extremal element $\psi \in K_{\omega'}$ so that $\psi=\bar{\psi}\beta_{\zeta}
=\tilde{\psi}\beta_{\zeta}$. This proves that (a) implies (b). That (b) implies (a) is obvious.

\vsp
Now we aim to prove the last statements. For simplicity of notation we consider the case $\zeta=1$ and leave it
to reader to check that a little modification needed to include the case $\zeta \neq 1$ and all the algebra stays
valid if $\tilde{v}_k$ is replaced by $\beta_{\zeta}(\tilde{v}_k)$. To that end we fix an extremal point
$\psi \in K_{\omega'}$ so that $\psi=\tilde{\psi}=\bar{\psi}$. We consider the Popescu system $(\clk,\clm,v_k,
\;1 \le k \le d,\Omega)$ as in Proposition 2.4 associated with $\psi$. Thus by Proposition 3.7 and Proposition 3.8
there exists unitary operators $u_{\zeta},w_{\zeta}$ on $\clk$ so that
$$u_{\zeta}v_ku_{\zeta}^*=\beta_{\bar{\zeta}}(\tilde{v}_k)$$
$$w_{\zeta}v_kw_{\zeta}^* = \beta_{\bar{\zeta}}(v_k) = \clj \beta_{\zeta}(v_k) \clj $$
where $u_{\zeta}^*=u_{\bar{\zeta}},\;\;u_{\zeta} \clj u_{\zeta}^*=\clj \;,w_{\zeta}^*=w_{\bar{\zeta}}
\;w_{\zeta} \clj w_{\zeta}^*=\clj$ and
$u_{\zeta} \Delta^{1 \over 2}u_{\zeta}^*=w_{\zeta} \Delta^{1 \over 2}w^*_{\zeta}=\Delta^{-{1 \over 2}}$.
Thus
\be
u_{\bar{\zeta}} w_{\zeta}v_kw_{\zeta}^* u_{\bar{\zeta}}^*=u_{\bar{\zeta}} \clj \beta_{\zeta}(v_k) \clj u_{\bar{\zeta}}^*
= \clj u_{\bar{\zeta}} \beta_{\zeta}(v_k) u_{\bar{\zeta}}^* \clj = \clj \beta_{\bar{\zeta}}(\beta_{\zeta}(\tilde{v}_k)\clj=
\clj \tilde{v}_k \clj
\ee
We also compute that
\be
w_{\zeta}u_{\bar{\zeta}}v_ku_{\bar{\zeta}}^*w_{\zeta}^* = w_{\zeta} \beta_{\zeta}(\tilde{v}_k) w_{\zeta}^* =
\clj \tilde{v}_k \clj
\ee
and
\be
w_{\bar{\zeta}}u_{\zeta}v_ku_{\zeta}^*w_{\bar{\zeta}}^* = w_{\bar{\zeta}} \beta_{\bar{\zeta}}(\tilde{v}_k) w_{\bar{\zeta}}^* =
\clj \tilde{v}_k \clj
\ee

\vsp
By Theorem 3.2 for a factor state $\omega$ we also have $\clm \vee \tilde{\clm} = \clb(\clk)$. As $\tilde{\clm} \subseteq \clm'$,
in particular we note that $\clm$ is a factor. So $u_{\zeta}w_{\zeta}^*u^*_{\zeta}w_{\zeta} \in \clm'$ commuting also with $\clj$
and thus a scaler as $\clm$ being a factor. As $u_{\zeta}\Omega=w_{\zeta}\Omega=\Omega$, we conclude that $u^*_{\zeta}$ and
$u_{\zeta}$ commutes with $w_{\zeta}$.

\vsp
Now we set $v_{\zeta}=u_{\bar{\zeta}}w_{\zeta}$ is an unitary operator commuting with both $\clj$ and
$\Delta^{1 \over 2}$. That $v_{\zeta}$ commuting with $\Delta^{1 \over 2}$ follows as $u^*_{\zeta}w_{\zeta}\Delta^{1 \over 2}=
u^*_{\zeta}\Delta^{- {1 \over 2}}w_{\zeta}= \Delta^{1 \over 2} u^*_{\zeta}w_{\zeta}$.

\vsp
Next claim we make now that $v_{\zeta}$ is a self-adjoint element. To that end note that (3.15) and (3.16) together says
that $v^2_{\zeta} \in \clm'$ as $v_{\zeta}$ commutes with $\clj$, $v_{\zeta}^2$ is an element in the centre of $\clm$.
The centre being trivial as $\omega$ is a factor state and $v_{\zeta}\Omega=\Omega$, we conclude that $v^2_{\zeta}$ is
the unit operator. Hence $v_{\zeta}$ is a self-adjoint element.

\vsp
Our next claim that $v_{\zeta}=1$. As a firts state we aim to show that $v_{\zeta}$ is an element in the centre of $\clm$
and for simplicity we use the notation $v$ for $v_{\zeta}$ in the following computation. To that end let $\theta$ be an unitary
element in $\clm'$ and by (3.14) we also have
\be
\theta v \theta^* v_k \theta v \theta^* = \clj \tilde{v}_k \clj
\ee
By symmetry we also have
\be
\theta v \theta^* \clj  \tilde{v}_k  \clj \theta v \theta^*  = v_k
\ee
The automorphism $\alpha_{\theta}: x \raro \theta v \theta^*  x \theta v \theta^*$ on $\clm$ is independent of $\theta$
and equal to $\alpha_1(x)=vxv$. Since the automorphism $\alpha_1$ preserves $\phi_0$, it commutes with Tomita's
modular automorphism group and conjugation action. Thus in particular (3.17) can be rewritten as
\be
\theta v \theta^*  \tilde{v}_k  \theta v \theta^*  = \clj v_k \clj
\ee
Thus the unitary operator $v^* \theta v \theta^*$ commutes with both $\{v_k: 1 \le k \le d \}$ and $\{\tilde{v}_k: 1 \le k
\le d \}$. $\omega$ being a factor state by Proposition 3.2 and our starting remark we have
$\clm \vee \tilde{\clm} =\clb(\clk)$ and thus $\theta v \theta^* = \mu v$ where $\mu$ is a scaler of modulus $1$.
However $v^*=v$ and so we get $\mu=\bar{\mu}=1$.
$\theta$ being an arbitrary unitary element in $\clm'$, we conclude that $v \in \clm$. As $v=\clj v \clj \in \clm'$ and $\clm$
is a factor, $v$ is a scaler multiple of $1$. As $v\Omega=\Omega$, we get $v=1$. This
completes the proof for (c). The last statement (d) follows by a routine calculation.  \qed

\vsp
\NI {\bf THEOREM 3.10:} Let $\omega$ be a translation invariant factor state on $\cla$ and $\psi$ be an
extremal point $K_{\omega'}$. We consider the representation $\pi$ of $\clo_d \otimes \tilde{\clo}_d$ described as in
Proposition 3.2. Then the following hold:

\NI (a) If $\omega$ is pure then we have the following duality relation
$$\pi_{\omega}(\cla_R)''= \pi_{\omega}(\cla_L)'$$
where $\cla_L,\cla_R$ are $C^*$-subalgebras of $\cla$ defined as in section 1.

\NI (b) If $\omega$ is also real and lattice symmetric then $H \subseteq \{1, -1 \}$ and $\omega=\omega_v$
where Popescu elements $(\clk,v_k;1 \le k \le d,\Omega)$ satisfies the dual relation
$v_k = \Delta^{1 \over 2}v_k^*\Delta^{-{1 \over 2}}$ for all $1 \le k \le d$.

\vsp
\NI {\bf PROOF:} For (a) we refer to the proof of Proposition 3.5 where we have shown that
$E_0\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''E_0 \bigcap (E_0\pi(\mbox{UHF}_d)E_0)'=
E_0\pi(\tilde{\mbox{UHF}}_d)''E_0$ while giving the proof for Proposition 3.5 (c).

\vsp
For (b) we recall that by Proposition 3.9 we have an extremal
element $\psi$ so that associated Popescu elements satisfies the following:
$$\beta_{\zeta}(v_k)= \clj \tilde{v}_k \clj= \Delta^{1 \over 2}v_k^* \Delta^{-{1 \over 2}}$$
for all $1 \le k \le d$ and $\zeta \in S^1$ so that $\zeta^2 \in H$. As $\beta_z(v_k)=zv_k$ for
all $z \in H$ and $\beta_z(\tilde{v}_k)= z\tilde{v}_k$ for all $z \in H$, we
conclude that $z^2=1$ for all $z \in H$. Thus $H \subseteq \{1, -1\}$. \qed
\vsp
The rest of the results in this section is not directly related with the main application discussed in section 4 and
section 5. Partly it is motivated to highlight what we have gained compare to results obtained in [BJKW].
Before we proceed to the next section, here we briefly review and refine the main result obtained in Section
7 of [BJKW]. As before let $\omega$ be an extremal translation invariant state on $\cla$ and $\psi$ be an extremal
point in $K_{\omega'}$. Let $P$ be the support projection of the state $\psi$ in $\pi(\clo_d)''$ and consider Popescu
systems $(\clk,\clm,v_k,\;1 \le k \le d,\Omega)$ associated with $(\clo_d,\psi)$ as defined
in Proposition 2.4. Thus we have $\clm'= \{ x \in \clb(\clk): \sum_k v_kxv_k^*= x \}$ by Proposition 2.4 (d).
However it is not automatic that the dual Popescu systems $(\clk,\tilde{\clm},\tilde{v}_k,\; 1 \le k \le d,\Omega)$
satisfy $\tilde{\clm}' = \{ x \in \clb(\clk): \sum_k \tilde{v}_k x \tilde{v}^*_k=x \}$. As an illustration of this point
we take $\omega$ to be the canonical trace on $\cla$ and Theorem 7.1 needs some refinement.
In fact by reviewing the main steps which leads to Theorem 7.1 in [BJKW] we prove here that
the property hold if and only if $\omega$ is a pure state modulo an additional assumption that
$\omega$ is also lattice symmetric for sufficiency. In the following we give the detailed of the result.

\vsp
\NI {\bf THEOREM 3.11: }  Let $\omega$ be a translation invariant lattice symmetric extremal state on $\cla$. Then
the following are equivalent:

\NI (a) $\omega$ is pure.

\NI (b) We fix any extremal state in $\psi \in K_{\omega'}$ and associated elements $(\clh,S_k,\;1 \le k \le d,P,\Omega)$
and $(\clk,\;\clm,\;v_k,\;1 \le k \le d,\phi_0)$ described as in Proposition 2.4. We consider the dual Popescu elements
$(\clk,\tilde{\clm},\tilde{v}_k,\;1 \le k \le d, \phi_0)$ and its dilated Cuntz's elements $(\tilde{S}_k,\;1
\le k \le d, P,\Omega)$ as in Theorem 2.2. Then the following equivalent statements are true:

\NI (i) $P \in \{ \tilde{S}_i,\tilde{S}_i^*: 1 \le i \le d \}''$;

\NI (ii) $\clm = \{x \in \clb(\clk): \sum_k \tilde{v}_k x \tilde{v}_k^*= x \}$.

\vsp
\NI {\bf PROOF: } That (a) implies (b) follows from Proposition 3.4 irrespective weather $\omega$ is lattice
symmetric or not lattice symmetric. However for the converse the condition (i) ensures that $P$ is the support projection of the state
$\psi$ in $\pi(\tilde{\clo}_d)''$. Hence $P= \tilde{E}\tilde{F}$ where $\tilde{F}=[\pi(\tilde{\clo}_d)''\Omega]$.
We recall from Proposition 3.7 that a lattice symmetric factor state ensures that $UEU^*=\tilde{E}$ and $UFU^*=\tilde{F}$
and thus $UPU^*=P$. Hence $u\clm u^*= \tilde{\clm}$ where $u=PUP$ as in Proposition 3.7 (c). Now following the same
step as in Proposition 3.7 (c) we conclude that $\tilde{\clm}=\clm'$. Hence by Proposition 3.5 (d) we have
$\pi(\clo_d)''= \pi(\tilde{\clo}_d)'$. $\beta_z$ invariant elements in $\pi(\clo_d)''$ is $\pi(\mbox{UHF}_d)''$ and
as $\pi(\mbox{UHF}_d)''$ is a factor, $\pi(\clo_d)''\bigcap \pi(\mbox{UHF}_d)'$ is
a commutative algebra by Arveson's spectral theorem ( see also Lemma 7.12 in [BJKW]). Hence its commutant
i.e. $\pi(\clo_d)' \vee \pi(\mbox{UHF}_d)''= \pi(\tilde{\clo}_d)'' \vee \pi(\mbox{UHF}_d)''$ is a type-I von-Neumann algebra.
Thus $\beta_z$ invariant elements i.e. $\pi(\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d)''$ is a type-I von-Neumann algebra by
Proposition 7.14 in [BJKW]. Thus $\omega$ is a type-I translation invariant state on $\cla$. Now we appeal to Lemma 7.13 in
[BJKW] to complete the proof that (a) and (b) are equivalent. \qed

\vsp
The results (b) implies (a) in Theorem 3.11 are in a sense a corrected and generalized version of Theorem 7.11 in [BJKW].
Purity of the state is also related to Kolmogorov's property [Mo1] of the canonical Markov semigroup $(\clm,\tau,\phi_0)$
appeared in Proposition 2.4. For more details and related results we refer to [Mo2].

\vsp
The merit of the results in Proposition 3.5 is well understood once we notice that for a factor state
$\omega$ if $\clm'_1=\tilde{\clm}_1$, then $\Omega$ is also cyclic for $\clm_1$ and thus $P=Q$. Thus
for a lattice symmetric factor state, we have $UPU^*=P$. Thus $P$
is the support projection of the state $\psi$ in $\pi(\tilde{\clo}_d)''$ when restricted to the
cyclic subspace generated by $\Omega$. Hence by Theorem 3.11 such a situation ensures that
$\omega$ is a pure state. As the canonical trace on $\cla$ is a lattice symmetric factor
state, we conclude that in general $\clm'_1$ need not be equal to $\tilde{\clm}_1$. A similar argument
also says that in general $\clm'$ need not be equal to $\tilde{\clm}$. The lattice symmetric property of
a pure state $\omega$ is an elegant sufficient condition in order to guarantee that $\clm'_1=\tilde{\clm}_1$.
However it remains an open question weather pure property itself is enough to ensure the equality.

\newsection{ Detailed balance translation invariant pure state and split property: }

\vsp
Let $\omega$ be a translation invariant real lattice symmetric pure state on $\cla$ as in last section. We fix an 
extremal element $\psi \in K_{\omega'}$ so that $\psi=\bar{\psi}=\tilde{\psi}$ and consider the Popescu elements 
$(\clk,\clm,v_i,\Omega)$ as in Proposition 3.9. $P$ being the support projection of a factor state $\psi$ we have 
$\clm = P\pi(\clo_d)''P = \{ v_k,v^*_k: 1 \le k \le d \}''$ ( Proposition 2.4 ).  So the dual Popescu elements 
$(\clk,\clm',\tilde{v}_k,1 \le k \le d, \Omega)$ satisfy the relation $\tilde{v}_k=\clj v_k \clj$ 
for all $1 \le k \le d$ and so we have
\be
\phi_0(\clj x \clj \tau(y))=\phi_0(\clj \tau(x) \clj y)
\ee
for all $x,y \in \clm$. A direct proof for (4.1) also follows from Proposition 3.1 (b) as $P=E\tilde{E}$ and  
$$<\Omega,\tilde{v}_{I'}\tilde{v}^*_{J'}v_Iv^*_J\Omega>=
<\Omega,\tilde{E}E\tilde{S}_{I'}\tilde{S}^*_{J'}E \tilde{E}S_iS_IS^*_JS^*_j \tilde{E}E \Omega>$$
$$=<\Omega,\tilde{S}_{I'}\tilde{S}^*_{J'} S_IS^*_J\Omega>$$
for all $|I|,|J|,|I'|,|J'| < \infty$.

\vsp
Now we also set von-Neumann algebra $\cla_0=P\pi_{\phi}(\mbox{UHF}_d)''P$. That $\cla_0$ is indeed 
a von-Neumann algebra follows once we recall that $\beta_z(P)=P$ for all $z \in H$ which ensures that 
$P \in \pi_{\phi}(\mbox{UHF}_d)''$ (Proposition 2.5). 
Pure property of the state $\omega$ says that $\pi(\mbox{UHF}_d \otimes \tilde{\clo}_d)''=\clb(\clh \otimes_{\clk} 
\tilde{\clh})$ (Proposition 3.4) and so in particular $\cla_0 \vee \tilde{\clm}=\clb(\clk)$. Thus $\cla_0$ is a factor.  

\vsp
Further as $\cla_0$ is the $\{\beta_z:\;z \in H \}$ invariant elements of $\clm$, the norm one projection 
$x \raro \int_{z \in H}\beta_z(x)dz$ from $\clm$ onto $\cla_0$ preserves the faithful normal state $\phi_0$. 
So by Takesaki's theorem modular group associated with $\phi_0$ preserves $\cla_0$. Further since 
$\beta_z(\tau(x))=\tau(\beta_z(x))$ for all $x \in \clm$, the restriction of the completely positive map 
$\tau(x)=\sum_k v_kxv_k^*$ to $\cla_0$ is a well defined map on $\cla_0$. Hence the completely 
positive map $\tau(x)=\sum_k v_kxv_k^*$ on $\cla_0$ is also KMS symmetric i.e. 
$$<<x,\tau(y)>>=<<\tau(x),y>>$$ 
where $x,y \in \cla_0$ and $<<x,y>>=\phi_0(x^*\sigma_{i \over 2}(y))$ and $(\sigma_t)$ is the modular automorphism
group on $\cla_0$ associated with $\phi_0$. 

\vsp
We now fix a translation invariant real lattice symmetric pure state $\omega$ and explore KMS-symmetric property of 
$(\cla_0,\tau,\phi_0)$ and the extended Tomita's conjugation operator $\clj$ on $\clh \otimes_{\clk} \tilde{\clh}$ 
to study the relation between split property and exponential decaying property of spacial correlation functions of 
$\omega$. 

\vsp
For any fix $n \ge 1$ let $Q \in \cla_{[-k+1,k]}$. We write  
$$Q = \sum_{|I|=|J|=|I'|=|J'|=n} q(I',J'|I,J)\tilde{S}_{I'}\tilde{S}^*_{J'}S_IS^*_J $$
and $q$ be the matrix $q=((q(I',J'|I,J) ))$ of order $d^{2n} \times d^{2n}$. \qed 

\vsp
\NI {\bf PROPOSITION 4.1:} The matrix norm of $q$ is equal to operator norm of $Q$ in $\cla_{[-n+1,n]}$.

\vsp
\NI {\bf PROOF: } Note that the operator norm of $Q$ is equal to the matrix norm of $\hat{q}$ 
where $\hat{q}=(( \hat{q}(I',I|J',J) ))$ is a $d^{2n} \times d^{2n}$ matrix with $\hat{q}(I',I|J',J)=
q(I',J'|I,J)$. Note that the map $L(q)=\hat{q}$ is linear and identity preserving. Moreover $L^2(q)=q$.
Thus $||L||=1$. Hence $||q||=||\hat{q}||$ \qed   

\vsp
\NI {\bf PROPOSITION 4.2:} Let $\omega$ be a translation invariant lattice symmetric pure state on UHF$_d$ 
$\otimes_{\IZ}M_d$. Then there exists an extremal point $\psi \in K_{\omega'}$ so that 
$\psi=\tilde{\psi}=\bar{\psi}$ and the associated Popescu systems $(\clh,S_k,\;1 \le k \le d,\Omega)$ 
and $(\clh,\tilde{S}_k,\;1 \le k \le d,\Omega)$ described in Proposition 3.2 and Proposition 3.9 satisfies 
the following:

\NI (a) For any $n \ge 1$ and $Q \in \cla_{[-n+1,n]}$ we write 
$$Q=\sum_{|I'|=|J'|=|I|=|J|=n} q(I',J'|I,J)\tilde{S}^*_{I'}\tilde{S}^*_{J'}S^*_IS_J$$ and 
set a notation for simplicity as $$\theta_k(Q)= \sum_{|I|=|J|=|I'|=|J'|=n} q(I',J'|I,J)
\tilde{\Lambda}_{k+1}(\tilde{S}_{I'}\tilde{S}^*_{J'})\Lambda_k(S_IS^*_J).$$
Then $\theta_k(Q) \in \cla_{(-\infty,-k] \bigcup [k,\infty)}$. 

\NI (b) $Q=\clj Q \clj$ if and only if $q(I',J'|I,J) = \overline{q(I,J|I',J')};$

\NI (c) If the matrix $q=(( q(I',J'|I,J) ))$ is non-negative then there exists a matrix $b=(( b(I',J'|I,J) ))$ 
so that $q=b^*b$ and then 
$$q=PQP=\sum_{|K|=|K'|=n}\clj x_{K,K'} \clj x_{K,K'}$$ 
where $x_{K,K'} = \sum_{I,J:\;|I|=|J|=n} b(K,K'|I,J)v_Iv^*_J$

\vsp
\NI (d) In such a case i.e. if $Q=\clj Q \clj$ the following hold:

\NI (i) $\omega(Q)=\sum_{|K|=|K'|=n} \phi_0(\clj x_{K,K'} \clj x_{K,K'})$

\NI (ii) $\omega(\theta_k(Q))=\sum_{|K|=|K'|=n}\phi_0(\clj x_{K,K'} \clj \tau_{2k}(x_{K,K'})).$

\vsp
\NI {\bf PROOF: } Since the elements $\tilde{S}_{I'}\tilde{S}^*_{J'}S_IS^*_J: |I|=|J|=|I'|=|J'|=n$ forms  
an linear independent basis for $\cla_{[-n+1,n]}$, (a) follows. (b) is also a simple consequence of linear independence 
of the basis elements and the relation 
$\clj \tilde{S}_{I'}\tilde{S}^*_{J'}S_IS^*_J \clj = S_{I'}S^*_{J'}\tilde{S}_I\tilde{S}^*_J$ as described in 
Proposition 3.9. 

For (c) we write $Q=\sum_{|K|=|K'|=n} \clj Q_{K,K'} \clj Q_{K,K'}$ where $Q_{K.K'}= 
\sum_{I,J:\;|I|=|J|=n} b(K,K'|I,J)S_IS^*_J$. $\omega$ being pure we have ( Proposition 3.5) 
$P=E \tilde{E}$ where $E$ and $\tilde{E}$ are support projection of $\psi$ in $\pi(\clo_d)''$ 
and $\pi(\tilde{\clo}_d)''$ respectively. So for any $X \in \pi(\clo_d)''$ and $Y \in \pi(\tilde{\clo}_d)''$ 
we have $PXYP=\tilde{E}EXY\tilde{E}E= \tilde{E}EYE\tilde{E}X\tilde{E}E=PXPYP$. Thus (c) follows as $\omega'(Q) 
= \psi_0(q)$ by Proposition 3.1 (b). For (d) we use (a) and (c). This completes the proof. \qed   

\vsp
For a pure translation invariant state $\cla_0 \vee \tilde{\clm}= \clb(\clk)$ and so $\clm \bigcap \cla'_0$ is trivial. 
However the inclusion $\cla_0 \subseteq \clm$ need not be an equality in general unless $H$ is trivial. The unique ground 
state of $XY$ model in absence of magnetic field give rise to a non-split translation invariant real lattice symmetric pure 
state $\omega$ and further $H=\{1,-1\}$ [Mo2].   

\vsp
\NI {\bf PROPOSITION 4.3: } Let $\omega$, a translation invariant pure state on $\cla$, be in detailed balance. Then the
following are equivalent:

\NI (a) $\omega$ is decaying exponentially.

\NI (b) The spectrum of $T-|\Omega><\Omega|$ is a subset of $[-\alpha,\alpha]$ for some $0 \le \alpha < 1$
where $T$ is the self-adjoint contractive operator defined by 
$$Tx\Omega= \tau(x)\Omega,\;\;x \in \cla_0$$ 
on the KMS-Hilbert space $<<x,y>>=\phi_0(x^*\sigma_{i \over 2}(y)>>$. 

\vsp
\NI {\bf PROOF: } Since $T^k x \Omega = \tau_k(x)\Omega$ and 
for any $L \in \cla_L$ and $R \in \cla_R$ we have 
$\omega'(L \theta_k(R)) = \phi_0(\clj y \clj \tau_k(x)) = << y,T^kx >>$ 
where $x=PRP$ and $y=\clj PLP \clj$ are elements in $\clm$. Since $P\cla_RP=\clm$ and 
$P\cla_LP=\clm'$, we conclude that (a) hold if and only if $e^{k \delta}|<f,T^kg>-<f,\Omega><\Omega,g>| \raro 0$ 
as $k \raro \infty$ for any vectors $f,g$ in a dense subset $\cld$ of the KMS Hilbert space.  

\vsp
That (b) implies (a) is obvious since $e^{k\delta}\alpha^k=(e^{\delta}\alpha)^k \raro 0$ whenever we choose 
a $\delta > 0$ so that $e^{\delta}\alpha < 1$ where $\alpha < 1 $.  

\vsp 
For the converse suppose that (a) hold and $T^2-|\Omega><\Omega|$ is not bounded away from $1$. Since
$T^2-|\Omega><\Omega|$ is a positive self-adjoint contractive operator, for each $n \ge 1$, we find a unit 
vector $f_n$ in the Hilbert space so that $E_{[1-1/n,1]}f_n=f_n$ and $f_n \in \cld$, where $E$ is the spectral 
family of the positive self-adjoint operator $T^2-|\Omega><\Omega|$ and in order to ensure $f_n \in \cld$ we 
also note that $E\cld$ is dense in $E$ for any projection $E$. 

\vsp
Thus by exponential decay there exists a $\delta > 0$ so that 
$$e^{2k \delta} (1-{1 \over n})^k \le e^{2k \delta} \int_{[0,1]}s^k<f_n,dE_sf_n>= 
e^{2k \delta }<f_n,[T^{2k}-|\Omega><\Omega|]f_n> \raro 0$$ 
as $k \raro \infty$ for each $n \ge 1$. Hence $e^{2\delta}(1-{1 \over n}) < 1$. Since $n$ is any 
integer, we have $e^{2 \delta} \le 1$. This contradicts that $\delta > 0$. 
This completes the proof. \qed

\vsp
Now we are set to state our main result. For any $Q \in \cla$ we set $\clj(Q)=\clj Q \clj $. Note that $\alpha^2=I$. 
Any element $Q={1 \over 2}(Q+\clj(Q))+{1 \over 2}(Q-\clj(Q))$ is a sum of an even element in 
$\{ Q: \clj(Q)=Q \}$ and an odd element in $\{Q: \clj(Q)=-Q \}$. Moreover $iQ$ is an 
even element if $Q$ is an odd element. Also note that $||Q_{even}|| \le ||Q||$ and $||Q_{odd}|| 
\le ||Q||$. Hence it is enough if we verify (1.1) for all even elements for split property. 
We fix any $n \ge 1$ and an even element $Q \in \cla_{[-k+1,k}$. We write as in Proposition 4.2 
$Q=\sum_{|I'|=|J'|=|I|=|J|=n}q(I',J'|I,J)\tilde{S}^*_{I'}\tilde{S}_{J'}S^*_IS_J$. The matrix 
$q=(q(I',J'|I,J)$ is symmetric and thus $q=q_+-q_-$ where $q_+$ and $q_-$ are the unique 
non-negative matrix contributing it's positive and negative parts of $q$. Hence $||q_+|| \le ||q||$ 
and $||q_-|| \le ||q||$. We set a notation for simplicity that
$$\theta_k(Q)= \sum_{|I|=|J|=|I'|=|J'|=n} q(J',I'|I,J)\tilde{\Lambda}_k(\tilde{S}_{I'}\tilde{S}^*_{J'})
\Lambda_{k+1}(S_IS^*_J)$$ which is an element in $\cla_{(-\infty,-k] \bigcup [k,\infty)}$ and by Proposition
4.2 (d) $$\omega(\theta_k(Q))=\sum_{K=K'=n}\phi_0(\clj x_{K,K'} \clj \tau_{2k+1}(x_{K,K'}))$$ provided
$q=(q(I',J'|I,J)$ is positive, where $PQP=\sum_{|K|=|K'|=n}\clj x_{K,K'}\clj x_{K,K'}$ and $x_{K,K'}=
\sum_{I,J}b(K,K'|I,J)v_Iv^*_J$ and $q=b^*b$. 
Thus in such a case we have by Proposition 4.2 (d) that 
$$|\omega'(\theta_k(Q))-\omega_L\otimes \omega_R(\theta_k(Q))| =
\sum_{K=K'=n}\psi_0(\clj x_{K,K'} \clj (\tau_{2k+1}-\psi_0)(x_{K,K'}))$$
$$\le \alpha^{2k}\omega'(Q) \le \alpha^{2k}||\hat{q}|| \le \alpha^{2k}||q|| $$   
provided $||T-|\Omega><\Omega||| \le \alpha$. In the last identity we have used 
Proposition 4.1. 

\vsp
Hence for an arbitrary $Q$ for which $\clj(Q)=Q$ we have 
$$ \omega'(\theta_k(Q))-\omega_L\otimes \omega_R(\theta_k(Q))| \le \alpha^{2k}(||q_+||+||q_-||) \le 2 \alpha^{2k}||q|| 
= 2\alpha^{2k}||Q||$$
where in the last identity we have used once more Proposition 4.1. Thus we have arrived at our main result. 

\vsp
\NI {\bf THEOREM 4.4: } Let $\omega$ be a translation invariant real lattice symmetric pure state on $\cla$ . If 
the special correlation function of $\omega$ decays exponentially then $\omega$ is split.

\vsp
For a pure translation invariant split state $\omega$, it is well known [Ma2,BJP] that $\clm=\cla_0$ and $\clm$ is 
a type-I factor. In case $\clm$ is a finite type-I factor, it is evident that the spacial correlation function 
decays exponentially as the criteria given in Proposition 4.3 can be verified easily as the contractive operator 
$T$ acting on a finite dimensional Hilbert space $\clk$ and $\{Tf=f: f \in \clk \} = \IC \Omega$. Such a pure state 
is described is a quantum Markov state [Ac], which is also known widely in the literature as a valence bound state 
[AL]. Exponential decaying special correlation function have a direct relevance to experimental evidence [DR]. The 
central aim in the next section is to include an elegant criteria for a lattice symmetric pure state to be 
a {\it quantum Markov state} equivalently {\it valence bound state }.    

\newsection{ Gauge and translation invariant pure states: }

\vsp
Let $G$ be a compact group and $g \raro v(g)$ be a $d-$dimensional unitary representation of $G$. By $\gamma_g$ we denote
the product action of $G$ on the infinite tensor product $\cla$ induced by $v(g)$,
$$\gamma_g(Q)=(..\otimes v(g) \otimes v(g)\otimes v(g)...)Q(...\otimes v(g)^*\otimes v(g)^*\otimes v(g)^*...)$$
for any $Q \in \cla$. We recall now that the canonical action of the group $S(d)$ of $d \times d$ matrices on
$\clo_d$ is given by
$$\beta_{v(g)}(s_j)=\sum_{1 \le i \le d} s_i v(g)^i_j $$
and thus
$$\beta_{v(g)}(s^*_j) = \sum_{1 \le i \le d} \bar{v(g)^i_j} s^*_i $$

\vsp
Note that $v(g)|e_i><e_j|v(g)^*=|v(g)e_i><v(g)e_j| = \sum_{k,l} v(g)^l_i \bar{v(g)}^k_j|e_l><e_k|$, where $e_1,..,e_d$
are the standard basis for $\IC^d$. Identifying $|e_i><e_j|$ with $s_is^*_j$ we verify that on $\cla_R$ the gauge action
$\beta_{v(g)}$ of the Cuntz algebra $\clo_d$ and $\gamma_g$ coincide i.e. $\gamma_g(Q)=\beta_{v(g)}(Q)$
for all $Q \in \cla_R$.

\vsp
\NI{ \bf PROPOSITION 5.1: } Let $\omega$ be a translation invariant factor state on $\cla$. Suppose that
$\omega$ is $G-$invariant,
$$\omega(\gamma_g(Q))=\omega(Q) \; \mbox{for all } g \in G \mbox{ and any } Q \in \cla. $$
Let $\psi$ be an extremal point in $K_{\omega'}$ and $(\clk,\clm,v_k,\;1 \le k \le d,\phi_0)$ be the Popescu system
associated with $(\clh, S_i=\pi(s_i),\Omega)$ described as in Proposition 2.4. Then we have the following:

\NI (a) There exists a unitary representation $g \raro U(g)$ in $\clb(\clh)$ and a representation $g \raro \zeta(g)$ 
so that
\be
U(g)S_iU(g)^*= \zeta(g) \beta_{v(g)}(S_i),\;1 \le i \le d
\ee
for all $g \in G$ and

\NI (b) There exists a unitary representation $g \raro u(g)$ in $\clb(\clk)$ so that $u(g)\clm u(g)^*=\clm$ for
all $g \in G$ and $\phi_0(u(g)xu(g)^*)=\phi_0(x)$ for all $x \in \clm$. Furthermore the operator
$V^*=(v^*_1,..,v^*_d)^{tr}: \clk \raro \IC^d \otimes \clk $ is an isometry which intertwines the representation
of $G$,
\be
( \zeta(g) v(g) \otimes u(g) )V^*=V^*u(g)
\ee
for all $g \in G$, where $g \raro \zeta(g)$ is representation of $G$ in $U(1)$.

\NI (c) $\clj u(g) \clj=u(g)$ and $\Delta^{it}u(g)\Delta^{-it}=u(g)$ for all $g \in G$.

\NI (d) Furthermore if $\omega$ is also pure real lattice symmetric and the family $\{v_k: 1 \le k \le d \}$ operators
are linearly independent (i.e. $\sum_k c_kv_k=0$ if and only if $c_k=0$ for all $1 \le k \le d$ ). Then $g \raro
\zeta(g)v(g)$ has a matrix representation with respect to an orthonormal basis in $\IC^d$ so that each entries
are real.  

\vsp
\NI {\bf PROOF: } $\omega$ being a factor state by Proposition 3.3, $H$ is a closed subgroup of $S^1$. Thus $H$ is either
$S^1$ or a finite cyclic subgroup. We also recall that $\lambda \beta_g = \beta_g \lambda$ for all $g \in G$ and $\omega$
being $G$-invariant we have $\psi \beta_g \in K_{\omega'}$ for all $\psi \in K_{\omega'}$ and $g \in G$.

\vsp
In case $H=S^1$, by Proposition 2.6 $K_{\omega'}$ is having a unique element and thus by our starting remark we have $\psi 
\beta_g = \psi$ for the unique extremal element $\psi \in K_{\omega'}$. In such a case we define unitary operator 
$U(g)\pi(x)\Omega=\pi(\beta_g(x))\Omega$ and verify (a) with $\zeta(g)=1$ for all $g \in G$.

\vsp
Now we are left to deal with the more delicate case. Let $H = \{z: z^n=1 \}''$ for some $n \ge 1$. In such a case by Proposition
2.5 and Proposition 3.2 (a) we have $\pi(\clo^H_d)''=\pi(\mbox{UHF}_d)''$ and $\pi(\tilde{\clo}^H_d)''=\pi(\tilde{\mbox{UHF}}_d)''$.
Thus for any $0 \le k \le n-1$ orthogonal projection $E_k$ is spanned by the vectors
$\{\tilde{S}_{I'}\tilde{S}^*_{J'}S_IS^*_JS^*_K\Omega: |I'|=|J'|,|I|=|J|,\; \mbox{and}\; |K|=k \}.$
We set unitary operator $U(g)'$ on $E_k: k \ge 0$ by 
$$U(g)'\pi(\tilde{s}_{I'}\tilde{s}^*_{J'}s_Is^*_Js^*_K)\Omega =
\pi(\beta_{v(g)}(\tilde{s}_{I'}\tilde{s}^*_{J'}s_Is^*_Js^*_K))\Omega$$
where $|I'|=|J'|,|I|=|J|$ and $|K|=k$. It is a routine work to check that $U(g)'$ is indeed an inner product preserving map
on the total vectors in $E_k$. The family $\{ E_k: 0 \le k \le n-1 \}$ being an orthogonal family projection with $\sum_k E_k= I$,
$U(g)'$ extends uniquely to an unitary operator on $\tilde{\clh} \otimes_{\clk} \clh$. It is obvious by our
construction that $g \raro U(g)'$ is a representation of $G$ in $\clh \otimes_{\clk} \tilde{\clh}$.

\vsp
For each $g \in G$ the Popescu element $(\clh,\beta_{v(g)}(S_k),1 \le k \le d,\;\Omega)$ determines an extremal point
$\psi_g \in K_{\omega'}$ and thus by Proposition 2.6 there exists a complex number $\zeta(g)$ with modulus $1$
so that $\psi_{g}=\psi \beta_{\zeta(g)}$. Note that for another such a choice $\zeta'(g)$, we have $\bar{\zeta(g)}\zeta(g)' \in H$.
As $H$ is a finite cyclic subgroup of $S^1$, we have a unique choice once we take $\zeta(g)$ to be an element in the group
$S^1 / H$ which we identify with $S^1$. That $g \raro \zeta(g)$ is a representation of $G$ in $S^1=\{z \in \IC: |z|=1 \}$
follows as the choice in $S^1 / H$ of $\zeta(g)$ is unique. Hence there exists an unitary operator $U(g)$ and a representation
$g \raro \zeta(g)$ in $S^1$ so that
$$U(g)\Omega=\Omega,\;U(g)S_iU(g)^* = \zeta(g)\beta_{v(g)}(S_i)$$
for all $1 \le i \le d$. Thus 
$U(g) = \sum_k \zeta(g)^k U(g)'E_k$ for all $g \in G$ as their actions on any typical vector
$S_IS^*_JS_K\Omega,\;|I|=|J|,|K|=k < \infty $ 
are same. Both $g \raro U'(g)$ and $g \raro \zeta(g)$ being representations of $G$, we conclude that
$g \raro U(g)$ is an unitary representation of $G$. 

\vsp
The above covariance relation ensures that $U(g)\pi(\clo_d)''U(g)^* = \pi(\clo_d)''$ for all $g \in G$ and thus also
$U(g)\pi(\clo_d)'U(g)^*=\pi(\clo_d)'$ for all $g \in G$. Now it is also routine work to check that $U(g)FU(g)^*=F$, where
$F=\{\pi(\clo_d)''\Omega]$ and $U(g)EU(g)^*=E$ where $E=[\pi(\clo_d)'\Omega]$ is the support projection of the state
$\psi$ in $\pi(\clo_d)''$. Hence the support projection $P=EF$ of the state $\psi$ in the cyclic subspace $F$ is
also $G$ invariant i.e. $U(g)PU(g)^*=P$ for all $g \in G$. Thus we define $g \raro u(g)=PU(g)P$ a unitary representation
of $g$ in $\clk$. Hence we have $u(g)v_ju(g)^*=\zeta(g) \beta_{v(g)}(v_j) = \zeta(g)v_i v(g)^i_j$ for all
$1 \le k \le d$. By taking adjoint we get
$u(g)v^*_ju(g)^*=\bar{\zeta(g)}\bar{v(g)^i_j} v^*_i$ for all $1 \le j \le d$.

\vsp
We are now left to prove (c). To that end we first verify that $S_0u(g)=u(g)S_0$ as their actions on
any typical vector $v_Iv^*_J\Omega$ are same, where $S_0x\Omega=x^*\Omega$ for $x \in \clm$. Hence by
uniqueness of the polar decomposition we conclude that (c) hold.

\vsp
For (d) we fix an extremal element $\psi \in K_{\omega'}$ as described in Proposition 3.9 so that $\psi= \tilde{\psi}
=\bar{\psi}$ and consider the Popescu elements $\{v_k,1 \le k \le d \}$. Then we have
$v_k= \clj \tilde{v}_k \clj$ for all $1 \le k \le k$. By the covariance relation we have
$\sum_{1 \le j \le d}[\zeta(g)v^i_j(g)-\bar{\zeta(g)v^i_j(g)}]v_j=0$ for all $1 \le i \le d$.
By linear independence we conclude that there exists an orthonormal basis for $\IC^d$ so that the
matrix representation $(\zeta(g)v^i_j(g))$ of $\zeta(g)v(g)$ with respect to the basis having real
entries. \qed

\vsp
One-dimensional representation being trivial [Ki] for a simply connected Lie group, we have $\zeta(g)=1$ for all 
$g \in G$ in Proposition 5.1 once $G$ is so. In such a case  $U(g)\pi(x)U(g)^*=\pi(\beta_g(x)),\;x \in \clo_d$ for all
$g \in G$ i.e. the representation $\pi$ of $\clo_d$ is covariant with respect to $G$.
The general class of covariant representation of a compact group is well studied . We recall that for a compact group
$G$ acting ergodically on a $C^*$-algebra, there is unique faithful $G$-invariant state which is in fact a trace.
Since $\clo_d$ does not admit a trace, $\beta_g$ can not be $G$-ergodic for any compact group $G$.
Thus a more reasonable question that one ask how about $\pi(\mbox{UHF}_d)''$ i.e. Does the group $G$ acts on
$\pi(\mbox{UHF}_d)''$ ergodically? By our earlier remark $\omega'$ is necessarily the unique trace. A necessary condition
for $G$-ergodicity is irreducibility of the representation $g \raro v(g)$. For a quick proof let $p$ be a projection on $\IC^d$
so that $v(g)pv(g)^*=p$ for all $g \in G$ and $\omega_p$ be a positive functional defined by $\omega_p(x)=
\omega(p^0xp^0)$ for all $x \in \cla$ where $p^0=p \otimes 1 \otimes 1 ...$ be the projection at the site $0$.
It is clear the $\omega_p$ is also $G$ invariant and thus for a $G$-ergodic action we have $\omega_p(x)=
\omega(p^0)\omega(x)$ for all $x \in \mbox{UHF}_d$. Hence $\omega(p^0xp^0)=\omega(p^0)\omega(x)$. $\omega$ be
the faithful trace we conclude that $p^0=\omega(p^0)I$. Hence $\omega(p^0)$ is $0$ or $1$. Hence $p^0$ is either
$1$ or $0$. So $p$ is either the identity operator or zero projection on $\IC^d$. However the converse statement
in general is not true. As a simple example we consider the unique KMS state $\phi_{\beta}$ on $\clo_d$ at the
inverse temperature $\beta=ln(d)$ and $G=SU(2)$. The element $h_0=S^{0}_xS^{1}_x + S^{0}_yS^{1}_y+S^{0}_zS^{1}_z$
is a $G$-invariant non scaler element in $\mbox{UHF}_d$ where $S^{i}_x,S^{i}_y,S^{i}_z$ are representation in
$\IC^d$ of Pauli's spin matrices $\sigma_x,\sigma_y$ and $\sigma_z$ respectively at lattice site $i \in \IZ$. Thus
the converse statement is not true in general for the unique trace $\omega$ on $\cla$ even if $g \raro v(g)$ is
irreducible. Thus a more refined and interesting problem when $G$ acts ergodically in the dynamical system
$(\clm,\alpha_g,\phi_0)$ of $*$-automorphism where $\alpha_g(x)=u(g)xu(g)^*,x \in \clm \}$. If so i.e. $\phi_0$ is
$G$-ergodic i.e. $\{x \in \clm: \alpha_g(x)=x,\;g \in G \} = \IC$ then $\phi_0$ is a trace on $\clm$ [HLS] and thus
$\clm$ is either a type-II$_1$ factor or a finite type-I factor. In case $\clm$ is a finite type-I factor, the state
$\omega$ on $\cla$ is pure and $\omega'$ on $\cla_R$ is a type-I factor state [BJP,BJKW,Ma1]. On the other hand it is
not hard to argue that $\clm$ can not be a type-II$_1$ factor if $\omega$ is a pure state [see Mo4] for details. Thus
we formulate the following refined question as a general problem:


\vsp
\NI {\bf PROBLEM 5.2: } Let $G$ be a simply connected compact Lie group and $g \raro v(g)$ in $\IC^d$ be an irreducible
representation of $G$. The questions that are central now:

\NI (a) How to classify the space of translation and $G$-invariant pure states on $\cla = \otimes_{\IZ}M_d$ by 
Popescu elements $(\clk,\clm,v_k,\alpha_g,\phi_0)$ solving $G$-covariant systems described in Proposition 5.1 (e). 

\NI (b) When does a translation and $G$-invariant pure state ensure that $(\clm,\alpha_g,\phi_0)$ is 
$G$-ergodic i.e. $\{x \in \clm: \alpha_g(x)=x,\; g \in G \} =\IC$?

\vsp
In the following proposition we take a close look at the above problem for an important class of simply connected
compact Lie-group $G$ and translation invariant pure state $\omega$.

\vsp
\NI {\bf THEOREM 5.3:} Let $G$ be a simply connected compact Lie group and $g \raro v^i_j(g)$ be an irreducible representation
on $\IC^d$ given in a basis $(e_i)$ for $\IC^d$ so that the invariance vectors of $\IC^d \otimes \IC^d$ with respect to the 
representation $g \raro \bar{v(g)} \otimes v(g)$ is one dimensional, where $d$ is an integer. Let $\omega$ be a translation 
and $\{\beta_g: g \in G \}$ invariant real ( with respect to the basis $(e_i)$ ) lattice symmetric pure state on $\cla = 
\otimes_{\IZ}M_d$. Then there exists an extremal element $\psi \in K_{\omega'}$ so that $\psi=\bar{\psi}=\tilde{\psi}$ and 
the associated Popescu elements $(\clk,\clm,v_k,1 \le k \le d,\Omega)$ described in Proposition 5.1 satisfies the following:

\NI (a) The matrix elements in the representation $g \raro v^i_j(g)$
are real numbers;

\NI (b) $\Delta=I,\;\; \{v_k^*=v_k: 1 \le k \le d \}$; 

\NI (c) $\clm$ is a finite type-I$_n$ for some $n \ge 1$ and the unique normalized trace $\phi_0$ on $\clm$ is 
strongly mixing for $\tau: \clm \raro \clm$, where $(\clm,\tau,\phi_0)$ is defined as in Proposition 2.4;

\NI (d) The pure state $\omega$ is split and two point spacial correlation functions are decaying exponentially;

\NI (e) There exists an unique representation of $G$, $g \raro u'(g) \in \clm$ so that
$$u'(g)v_ku'(g)^*= \sum_{1 \le j \le d} v^k_j(g)v_j$$
for all $1 \le k \le d$. Furthermore the unique representation $g \raro u'(g)$ is irreducible i.e. 
$(\clm,\alpha_g,\phi_0)$ is $G$-ergodic. 

\vsp
\NI {\bf PROOF: } Let $\psi$ be an extremal element in $K_{\omega'}$ so $\psi=\bar{\psi}=\tilde{\psi}$ and 
$(\clk,\clm,v_i,\Omega)$ be the associated Popescu element on the support projection of the state $\psi$ as 
in Proposition 5.1. Thus we have $\tilde{v}_k= \clj v_k \clj $ where we recall that $\tilde{v}_k =
\overline{\clj \Delta^{-{1 \over 2}}v^*_k\Delta^{1 \over 2} \clj}$.

\vsp
As a first step we explore the hypothesis that the invariant vectors in $\IC^d \otimes \IC^d$ of the 
representation $g \raro \bar{v(g)} \otimes v(g)$ of $G$ is one dimensional. 

\vsp
By appealing to the assumption with invariant vectors $((\phi_0(v_i^*v_j)))$ and 
$((<\Omega,S_i^*S_j\Omega>))$ we have
$$\phi_0(v^*_iv_j)= \delta^i_j {\lambda \over d}$$
for all $1 \le i,j \le d$ and scaler $\lambda > 0$. That the scaler $\lambda$ is indeed non-zero follows 
by separating property of $\Omega$ ( otherwise we will have $v_k=0$ for all $k$ ).  In particular we get the 
vectors $\{v_i\Omega: 1 \le i \le d \}$ are linearly independent and so by separating property of $\Omega$ for 
$\clm$, $\{v_k:\;1 \le k \le d \}$ are linearly independent. Thus by Proposition 5.1 (d) the matrix elements 
in the representation $g \raro v^i_j(g)$ are real numbers.  

\vsp
By (a) vectors $((<\Omega,S^*_iS_j\Omega>))$ and $((\phi_0(v_iv^*_j) ))$ are also invariant for the product 
representation $g \raro \bar{v(g)} \otimes v(g)$ and thus we also have
$$\phi_0(v_iv^*_j) = {1 \over d}\delta^i_j$$
for all $1 \le i,j \le d$, where we have used Popescu's relation $\sum_kv_kv_k^*=1$.   

\vsp
We recall that $\overline{\clj \Delta^{1 \over 2}v_k\Delta^{-{1 \over 2}}\clj} = \tilde{v}_k^*$ and
by Proposition 3.9 we also have $\clj \tilde{v}^*_k \clj = v^*_k$ for all $1 \le k \le d$. 

\vsp
Thus we have 
$$\Delta^{-{1 \over 2}}v^*_k = v_k\Delta^{-{1 \over 2}}$$
for all $k$, where the equality is to be interpreted as a bilinear form on the dense set $\clm\Omega$ 
and it is well defined as the domain of $\Delta^{1 \over 2}$ contains $\clm\Omega$ as $S$ do. Since 
$\sum_k v_kv_k^*=1$, we have 
$$\sum_k v_k\Delta^{-{1 \over 2}}v_k = \Delta^{-{1 \over 2}}$$

\vsp
By taking formal adjoint on the dense set $\clm\Omega$ we also get
$$\sum_k v_k^*\Delta^{-{1 \over 2}}v^*_k= \Delta^{-{1 \over 2}}$$

\vsp
We check now that for each $1 \le k \le d$
$$<\Omega,v_k^*\Delta^{-{1 \over 2}}v_k^*\Omega>$$
$$=<\Omega,v_k^* \clj \clj \Delta^{-{1 \over 2}}\tilde{v}_k^*\Omega>\;\;\mbox{as}\;\;v_k^*\Omega=\tilde{v}_k^*\Omega$$
$$=<\Omega, v_k^* \clj \tilde{v}_k\Omega>$$
$$=<\Omega, v_k^*v_k \Omega > = {\lambda \over d}$$
Hence by summing over all $k$ we get $\lambda=<\Omega,\Delta^{-{1 \over 2}}\Omega>=1$. Hence 
$$\phi_0(v_iv_j^*) =\phi_0(v^*_iv_j) = {\delta^i_j \over d}$$ 
for all $1 \le i,j \le d$.  

\vsp
However $\phi_0(v^*_iv_j) = <\clj \Delta^{1 \over 2}v^*_i \Omega, \clj \Delta^{1 \over 2}v^*_j\Omega>$
and so we have 
$$<v_i^* \Omega, \Delta v^*_j\Omega>=\delta^i_j {1 \over d}$$ 
for all $1 \le i,j \le d$. 

\vsp
On the other hand we have 
$$<\Omega,v_i \Delta v^*_j\Omega> $$
$$= <\tilde{v}^*_i\Omega, \Delta \tilde{v}_j^*\Omega>\;\;\;(\;\mbox{as}\;\;v_i^*\Omega=\tilde{v}^*_i\Omega\;)$$
$$=<\clj v_i^*\Omega, \Delta \clj v_j^*\Omega>\;\;\;(\;\mbox{as}\;\; \clj v_i \clj = \tilde{v}_i\;)$$ 
$$=<\Omega, v_j \Delta^{-1}v^*_i\Omega>$$

\vsp
Now a simple computation shows that
$$||\Delta^{1 \over 2}v_k^*\Omega-\Delta^{-{1 \over 2}}v_k^*\Omega||^2 $$
$$=<v_k^*\Omega,\Delta v_k^*\Omega>+<v^*_k\Omega,\Delta^{-1}v_k^*\Omega>-2<\Omega,v_kv_k^*\Omega>$$
$$=0.$$
Hence by separating property of $\Omega$ for $\clm$ we
conclude that $\Delta v^*_k \Delta^{-1} =v^*_k$ for all $ 1 \le k \le d$. So $\Delta$
is affiliated to $\clm'$. As $\clj \Delta \clj =\Delta^{-1}$, $\Delta$ is
also affiliated to $\clm$. Hence $\Delta=I$ as $\clm$ is a factor. Thus $\phi_0$ is a tracial
state on $\clm$. That it is the unique trace follows as $\clm$ is either a type-I finite factor
or a type-II$_1$ factor.

\vsp
Now we will rule out the possibility for $\clm$ to be a type-II$_1$ factor. Since $v_k=v^*_k$, $\{v_I^*\Omega: |I| < \infty \}$
is total in $\clk$ and $\phi_0(\tau(y)x)=\phi_0(y\tau(x))$ for all $x,y \in \clm$. In particular for $x=v_Iv_J^*,\;|I|=|J|$ 
we have $\phi_0(\tau^n(x^*)\tau^n(x))=\phi_0(x^*\tau^{2n}(x))=\phi_0(v^*_I\tau^{2n}(x)v_J)
=\phi_0(v_{\tilde{I}}\tau^{2n}(x)v^*_{\tilde{J}})$ and so by R. T. Powers criteria [Po,Mo3] for factor property 
of $\omega$ we get 
$$\phi_0(\tau^n(x^*)\tau^n(x)) \raro |\phi_0(x)|^2$$ 
as $n \raro \infty$. In other words the sequence of self-adjoint contractive maps $T^n$ defined by $T:x\omega=\tau(x)\omega$ on 
the Hilbert space $\{v_Iv^*_J\Omega: |I|=|J| < \infty \}$ converges strongly to $|\Omega><\Omega|$. Now we appeal to
Corollary 4.4 in [Mo4] to conclude that $\cla_0$ can not be a type-II$_1$ factor.  This completes the proof of (c).

\vsp
For the last part of (b) we note that purity of $\omega$ ensures that the point spectrum of the contractive operator
$T$, defined by $Tx\Omega=\tau(x)\Omega$ on the KMS Hilbert space (see section 4), in the unit circle is trivial i.e.
$\{ z \in S^1: Tf=zf \mbox{for some non zero } f \in \clk \}$ is the trivial set $\{ 1 \}$. As $T$ is a contractive matrix
on a finite dimensional Hilbert space, the spectral radius of the contractive matrix $T-|\Omega><\Omega|$ is $\alpha$ for
some $\alpha < 1$. Now we use Proposition 3.1 and Corollary 3.5 (a) for any $X_l \in \cla_L $ and $X_r \in \cla_R$
to verify the following
$$e^{\delta k }|\omega(X_l \theta_k(X_r)) -\omega(X_l)\omega(X_r)| $$
$$= e^{\delta k}|\phi_0(\clj x_l \clj \tau_k(x_r))- \phi_0(x_l)\phi_0(x_r)| \raro 0$$
as $k \raro \infty $ for any $\delta >0$ so that $e^{\delta}\alpha < 1$ where $\clj x_l \clj = PX_lP$ and $x_r=PX_rP$ for
some $x_l,x_r \in \clm$. As $\alpha < 1$ such a $\delta >0$ exists. This completes the proof for (d).

\vsp
$\clm$ being a type-I factor and $G$ being a simply connected group, by a general result [Ki] any continuous action of the $G$ 
is implemented by an inner conjugation, i.e. there exists an unitary operator $g \raro u'(g) \in \clm$ so that 
$u(g)'x{u(g)'}^*=u(g)xu(g)^*$ for all $g \in G$. Thus we have $u'(g)v_ku'(g)^* = \sum_j v^k_j(g)v_j$ for all $1 \le k \le d$. 
Now for uniqueness let $g \raro u''(g) \in \clm$ be another such representation. Then $u''(g)u'(g)^* \in \clm \bigcap \clm'$ and 
$\clm$ being a factor, $\lambda(g)=u''(g)u'(g)^*$ is a scaler and as $u'(g) = \lambda(g) u''(g)$ and each one being a 
representation we also get $g \raro \lambda(g)$ is a representation and so $\lambda(g)=1$ as $G$ is simply connected. 
Hence uniqueness follows. 

\vsp
That irreducibility $g \raro u'(g)$ is equivalent to $G$-ergodicity of $(\clm,\alpha_g,\phi_0)$ follows from a more general result 
[BR], however here one can verify easily as $\clm$ is a type-I factor. For the non-trivial part of last statement we will use once 
more the property (b) i.e. self-adjointness of the Popescu elements $\{v_k,\;1 \le k \le d \}$. To that end let $E$ be a $G$-
irreducible projection in $\clm$ and set $E'= \clj E \clj $. We set von-Neumann algebra $\clm_{E}=E'\clm E'$. 
$\clm_{E}$ is a type-I finite factor as $\clm$ is so and $u(g) \clm_E u(g)^* = \clm_E$ for all $g$ as $E'$ is $G$-invariant. 
Further $\clm_{E}$ is also $G$-irreducible as $E$ is an $G$-irreducible projection in $\clm$. The vector state $\phi_{E}(X) = 
<\Omega,X\Omega>$ on $\clm_E$ being $G$-invariant and irreducible, we conclude that $\phi_{E}$ is the unique normalized trace 
on $\clm_{E}$ ( see [HLS] for details ). That the state $\psi_{E}$ on $\clo_d$ defined by 
$$\psi_{E}(s_Is_J^*)=<\Omega,E'v_Iv^*_JE'\Omega>\;\;|I|,|J| < \infty $$ 
is $\lambda$-invariant follows from the tracial property of $\phi_E$ on $\clm_{E}$ and 
self-adjoint property of the family $\{v_k: 1 \le k \le d \}$ ensuring that 
$\sum_k E'v^*_kE'v_kE'=\sum_k E'v_kv_k^*E'=E'$. Thus $\lambda$-invariance of $\psi_{E}$ ensures that
$$<\Omega,E'xE'\Omega>=<\Omega,E'\tau(x)E'\Omega>$$
for all $x \in \clm$. Tomita's modular operator being trivial we also have $E'\Omega=E\Omega$ and so 
by duality relation (4.1) we have
$$<\Omega,Ex\Omega> = <\Omega,\tau(E)x\Omega>$$ 
for all $x \in \clm$. Thus by cyclic and separating property of $\Omega$ for $\clm$, we get $\tau(E)=E$. Now 
by ergodic property of $(\clm,\tau,\phi_0)$, we conclude that $E$ is either $0$ or $1$. This completes the 
proof for (e). \qed 


\vsp
Before we end this section here we discuss our prime examples where $G=SU(2)$ which is a simply connected Lie group and by 
Clebsch-Gordon theory [Ha] assumption that invariant vectors of the representation $g \raro v(g) \otimes v(g)$ of $G$ 
is one dimensional, is satisfied. In [Mo2] we have proved that there is no translation, $SU(2)$ invariant pure real lattice
symmetric state on $\otimes_{\IZ}M_d$ where $d=2s+1$ and $s=\{ {1 \over 2}, {3 \over 2},..\}$. Here we aim to describe the
case for $d=2s+1$, where $s=\{1,2,..,\}$.


\vsp
\NI {\bf THEOREM 5.4: } Let $G=SU(2)$ in Theorem 5.3. Then $d \ge 3$ is an odd integer and all the statements in 
Theorem 5.3 are valid. Furthermore Popescu elements $(\clk,\clm,v_k,\;1 \le k \le d,\Omega)$ and the 
representation $g \raro u'(g)$ of $G$ satisfying the covariance relation in Theorem 5.3 (e) is uniquely 
determined up to unitary isomorphism and modulo a phase factor. 

\vsp
\NI {\bf PROOF: } Let $\cls$ be the set of non-negative half odd integers and integers. For $s \in \cls$ there exists a unique
up to unitary conjugacy $2s+1$ dimensional irreducible representation $g \raro v_s(g)$ of $SU(2)$ and any irreducible
representation is unitary conjugate with the representation $g \raro v_s(g)$ for some $s \in \cls$. By Clebsch-Gordon
theorem for any $s \le t \in \cls$ we have
$$v_s(g) \otimes v_t(g) \equiv v_{|t-s|}(g) \oplus v_{|t-s|+1}(g) \oplus ..\oplus v_{t+s}(g)$$
In particular the above decomposition shows that if a non-zero intertwining operator exists between the representation
$g \raro v_s(g) \otimes v_t(g)$ and $g \raro v_r(g)$ then it is uniquely determined modulo scaler. Thus 
the intertwining operator $V^*=(v_1^*,v_2^*,..,v_d^*)^{tr}$ is determined uniquely modulo a phase factor $\lambda$. The 
elements $\{v_k: 1 \le k \le d \}$ being self-adjoint, $\lambda$ is either $1$ or $-1$.

\vsp
By Clebsch-Gordon theory invariant vectors in $\!C^d \otimes \!C^d$ of the representation
$g \raro \bar{v(g)} \otimes v(g)$ of $SU(2)$ forms a one dimensional subspace and thus condition for
Proposition 5.3 holds good for $G=SU(2)$. If $d$ is an even integer, there is no orthonormal basis 
$(e_i)$ for $\IC^d$ such that the matrix elements $g \raro v^i_j(g)$ of the irreducible representation 
are real [see Mo2 for a proof ]. Thus $d$ is an odd integer.  \qed


\newsection{ Ground states of a Hamiltonian and detailed balance: }

\vsp
In this section we consider translation invariant Hamiltonian those are in detailed balance and look for possible 
application to ground states. Since these states are constructed as infinite volume ground states of spin models, 
we begin with explaining the mathematical definition of ground states (for more details, see [BR2]).    

\vsp
We now present in the following a standard criteria for definition of a ground states. To that end we consider a 
translation invariant Hamiltonian with finite range interaction. For simplicity we assume that $h_0=h^*_0 
\in \cla_{loc}$ and consider the finite volume Hamiltonian 
$$H_{[m,n]}=\sum_{n-1 \le j \le m} \theta_j(h_0).$$ 
The formal infinite volume limit of these Hamiltonian is denoted by
\be
H=\sum_{j \in \IZ} \theta_j(h_0)
\ee
The time evolution of $\alpha_t(Q)$ of $Q \in \cla$ is obtained via the thermodynamic limit 
$$\alpha_t(Q)=\mbox{lim}_{\Lambda \uparrow \IZ}e^{itH_{\Lambda}}Qe^{-itH_{\Lambda}}.$$
For more details we refer readers to any standard text [Ru,BR]. 

\vsp
A state $\omega$ is called invariant by $(\alpha_t)$ if $\omega (\alpha_t(Q))=\omega(Q)$ for all $t \in \IR$ and
$Q \in \cla$. We recall now the following standard criteria as definition of a ground state.

\vsp
\NI{ \bf DEFINITION 6.1:} Let $\omega$ be a state on $\cla$. We say $\omega$ is a {\it ground state} for the 
Hamiltonian $H$ ( formally given by (6.1) ) if and only if 
\be
\omega(Q^*[H,Q]) \ge 0 \mbox{ for any }\; Q \in \cla_{loc}
\ee

\vsp
In case $\omega$ is translation invariant, $\omega$ is a ground state if and only if $\omega$ minimizes the 
mean energy i.e.
\be
\omega(h_0)=\mbox{inf}\psi(h_0) 
\ee
where infrimum is taken over all translation invariant states $\psi$ on $\cla$. The set of ground states
are weakly compact non-empty convex set in the state space of $\cla$ and extremal points are pure states. 

\vsp
\NI{ \bf DEFINITION 6.2:} We say a Hamiltonian $H$ is {\it lattice symmetric } if and only if $\tilde{H}=H$. $H$ is
called {\it real } if $H^t=H$ ( real transpose ). It is in {\it quantum detailed balance } if it is lattice symmetric and
real. $H$ is $G$ invariant if $\gamma_g(H)=H$ for all $g \in G$.

\vsp
Let $\omega_0$ be a translation invariant ground state. Then $\omega = \int_G \omega_0 \gamma_g dg$ is a $G$ and 
translation invariant ground state. Let $\omega = \int_r \omega_r d\mu(r)$ be the factor decomposition of $\omega$. 
Then for almost all $r$ with respect to $\mu$, $\omega_r$ are translation and $G$ invariant. Ground states being a face 
in the convex set of all states on $\cla$, for almost all $r$ with respect to $\mu$, $\omega_r$ are ground states for 
$H$. Thus for a translation and $G$ invariant Hamiltonian there exists a translation and $G$ invariant ground state which 
is also a factor. However unless the ground state is unique, such a factor state in general need not be pure.  
  
\vsp
Note that $\tilde{\omega}$ is ground state for $\tilde{H}$ whenever $\omega$ is a ground state for $H$, where
$\tilde{H}$ is the operator associated with $\tilde{h}_0 \in \cla_{loc}$ (see section 3). A similar statement
is also valid for $H^t$ where $H^t$ is the Hamiltonian associated with $h^t_0 \in \cla_{loc}$ (see section 3).
Same hold for $\omega_g(x)=\omega(\gamma_g(x))$.

\vsp
\NI{ \bf PROPOSITION 6.3:} The set of ground states for $H$ is non-empty. Moreover 

\NI (a) the map $\omega \raro \tilde{\omega}$ determines an affine map on the set of ground states of 
a lattice symmetric $H$;     

\NI (b) the map $\omega \raro \bar{\omega}$ determines an affine map on the set of ground states. 

\NI (c) Let $H$ be in detailed balance. Then there exists a detailed balance ground state for $H$ and 
such states form a weakly compact convex subset of all ground states.    

\vsp
\NI {\bf PROOF:} $\omega$ being a ground state we have $\omega(Q^*[H,Q]) \ge 0$ for all $Q \in \cla_{loc}$.
Thus for $\tilde{H}=H$, we have $\tilde{\omega}(Q^*[H,Q]) = \omega(\tilde{Q}^*[\tilde{H},\tilde{Q}]) 
\ge 0$ for all $Q \in \cla_{loc}$. Hence $\tilde{\omega}$ is a ground state for for lattice symmetric 
$H$. That $\omega \raro \tilde{\omega}$ is an affine map follows by a simple application of the 
criteria (4.1). Since the set of ground states are weakly compact convex set, by Kakutani fixed 
point theorem we conclude that there exists a lattice symmetric state. That (b) is also true follows 
essentially along the same line.  

\vsp
The proof for (c) goes along the same line, since the set of lattice symmetric ground states are weakly compact 
( being a closed subset of all ground states ) and $\bar{\omega}$ is lattice symmetric whenever $\omega$ is lattice 
symmetric. Thus once more by a simple application of Kakutani fixed point theorem, (c) is true.  This completes 
the proof. \qed

\vsp
In the following we state our main results on ground states. 

\vsp
\NI {\bf THEOREM 6.4: } Let $H$ be in detailed balance with a unique ground state $\omega$ then $\omega$ is pure
and in detailed balance. If the correlation functions of $\omega$ decay exponentially then $\omega$ is a split 
state.

\vsp
\NI {\bf PROOF: } Uniqueness of the ground state will ensure that the ground state is translation invariant pure detailed 
balanced state on $\cla$. Thus the result is a simple application of Theorem 4.4. \qed

\vsp
\NI {\bf THEOREM 6.5: } Let $G$ be a simply connected Lie group and $g \raro v^i_j(g)$ be an irreducible representation 
on $\!C^d$ such that invariant vectors of the representation $g \raro v(g) \otimes v(g)$ is one dimensional and 
$H$ be a $G$-invariant real lattice symmetric Hamiltonian on $\cla= \otimes_{\IZ}M_d$. If the ground state of $H$ 
is unique say $\omega$ on $\cla$ then 

\NI (a) the representation $g \raro v^i_j(g)$ are with real entries;

\NI (b) $\omega$ is a split state; 

\NI (c) The spacial correlation function of $\omega$ decays exponentially;

\NI (d) $\omega$ is a quantum Markov state ( finitely correlated state ) and associated Popescu elements 
$(\clk,\clm,v_k,\phi_0)$ satisfies the following:

\NI (i) $\clk$ is a finite dimensional Hilbert space;

\NI (ii) $\clm=\{v_k,v_k^*,\;1 \le k \le d \}''=\clb(\clk)$;

\NI (iii) $\phi_0$ is the unique tracial state on $\clm$; 

\NI (iv) There exists a unique representation $g \raro u'(g) \in \clm$ so that 
$u'(g)v_ku'(g)^*=\beta_{v(g)}(v_k)$ for all $1 \le k \le d$; 

\NI (v) The representation $g \raro u'(g)$ is irreducible.

\vsp
\NI {\bf PROOF:} By uniqueness of the ground state, $\omega$ is a translation invariant, pure and $G$ invariant
state on $\cla$. We fix an extremal element $\psi \in K_{\omega'}$ so that $\psi=\bar{\psi}=\tilde{\psi}$ and 
consider $G$-covariant Popescu elements $(\clk,\clm,v_k,u'(g)\;1 \le k \le d,\Omega)$ defined as in Theorem 5.3.  
Thus the result follows by Theorem 5.3. \qed

\vsp 
Theorem 6.4 (a) gives an easy criteria to determine phase transition in the ground state for
a  $\gamma_g$-invariant real lattice symmetric Hamiltonian $H$ on $\cla=\otimes M_d$. In case 
$G=SU(2)$ and $d=2s+1$, $s={1 \over 2},{3 \over 2},... $, there exists [Mo2] no real representation 
$g \raro v^i_j(g)$ in $\!C^d$. Thus the ground state for such an $H$ is not unique. On the other side 
of the theory, [AKLT] model is $\gamma_g$-invariant real lattice symmetric and the model 
is known to have unique ground state and exponentially decaying spacial correlation function. So Theorem 
6.4 (c) says that such a phenomena is rather general once the symmetry group $G$ admits the hypothesis 
of Theorem 6.5. Further property (iv) and (v) in Theorem 6.5 (d) essentially brings down the problem 
of finding the solution for the ground state to a well known now a tangible problem of finding out 
Clebsch-Gordon coefficient of the symmetry group $G$.  
 
\vsp
We are left to discuss few motivating examples. 

\vsp
\NI {\bf ISING model:} The simplest exactly solvable model Ising model Hamiltonian is given by
$$H_{I}=\sum_{j \in \IZ } \sigma_z^{(j)}\sigma_z^{(j+1)}$$ 
where $\sigma_z$ is the Pauli matrix ... It is well known that there are translation and as well as 
non-translational invariant ground states for $H_{I}$ [BR]. Among the translation invariant ground states, 
there are two extremal points. The pure states with all spins up and all spins down are two pure ground state. 
None of the extremal points are in detailed balance. Nevertheless $H_I$ is in detailed balance and the mixed state 
with equal probability of those two extremal points are the unique detailed balance state. If the Gibbs state at positive 
temperature is unique, we also note that the state is in detailed balance. As detailed balance property is well preserved 
in the weak$^*$ limit, the detailed balance symmetry is preserved when we arrive at the ground state by taking limit from 
temperature states. This explains why in the limiting procedure of Onsager we only get the mixed state with all spin 
up or all spin down with equal probability as ground state. \qed

\vsp
\NI {\bf XY model: } We consider the exactly solvable XY model. The Hamiltonian $H_{XY}$ of the XY model is
determined by the following prescription:

$$H_{XY}=- \sum_{j \in \IZ} \{ \sigma_x^{(j)}\sigma_x^{(j+1)}+\sigma_y^{(j)} \sigma_y^{(j+1)} \} - 
2 \lambda \sum_{j \in \IZ} \sigma_z^{(j)},$$
where $\lambda$ is a real parameter stand for external magnetic field, $\sigma_x^{(j)},\sigma_y^{(j)}$ and
$\sigma_z^{(j)}$ are Pauli spin matrices at site $j$. It is well known [AMa] that ground state exists and unique. 
It is simple to verify that $\tilde{H}=H$ since we can rewrite $H_{XY}$ as sum over element of 
the form $\sigma_x^{(j-1)} \sigma_x^{(j)} + \sigma_y^{(j-1)} \sigma_y^{(j)}$. Since the transpose 
of $\sigma_x$ is itself, transpose of $\sigma_y$ is $-\sigma_y$ and transpose of $\sigma_z$ is itself, 
we also verify that $H_{XY}^t=H_{XY}$. Hence $H_{XY}$ is in detailed balance. A simple application of 
Theorem 6.5 says now that the correlation functions decay exponentially if and only if the ground state 
is split. It is also well known that for $|\lambda| \ge 1$ the unique ground state is a product state thus 
split state. On the other hand for $|\lambda| < 1$ the unique ground state is not a split state [Ma2, 
Theorem 4.3]. Hence spacial correlation function of the ground state decays exponentially if and only if 
$|\lambda| \ge 1$. \qed       

\vsp
\NI {\bf The Ghosh Majumdar model:} The model was already completely analyzed in [GM,AKLT]. The Ghosh Majumdar 
is a spin $1 \over 2$ chain with a next next neibour interaction given by:
$$h_0 = P^{3 \over 2}_{012}= {2 \over 3}(S^{\uparrow}_1.S^{\uparrow}_2 + S^{\uparrow}_2.S^{\uparrow}_3 + S^{\uparrow}_1.S^{\uparrow}_3) 
+ 1,$$
where $P^{3 \over 2}_{012}$ is the orthogonal projection onto the subspace with total spin equal
to $3 \over 2$. It is simple to check that $h_0$ is also real and lattice symmetric. However the irreducible 
representation $g \raro u(g)$ of $SU(2)$ do not admit a real representation with respect to any orthogonal basis 
in $\IC^2$. Thus phase transition takes place at $T=0$. At $T=0$ this model breaks the translation invariance of the 
chain and has two pure infinite volume ground states $\omega_1,\omega_2$, where $\omega_1$ is product of singlet states 
on next neibour pairs of the form $2i, 2i+1$ and $\omega_2$ is obtained from $\omega_1$ by translation over one 
lattice spacing. Both the pure ground states are $SU(2)$ invariant. On other hand the unique translational invariant 
ground state ${1 \over 2}(\omega_1+\omega_2)$ is not a pure state. \qed

\vsp
\NI {\bf XXX MODEL: } Here we consider the prime example where very little exact results were known. 
The Hamiltonian $H_{XXX}$ of the spin $s$ anti-ferromagnetic chain i.e. the Heisenberg's XXX model is 
determined by the following formula:
$$H_{XXX} = J \sum_{j \in \IZ} \{ S_x^{(j)}S_x^{(j+1)}+S_y^{(j)}S_y^{(j+1)} + 
S_z^{(j)} S_z^{(j+1)} \}$$
where $S_x^{(j)},S_y^{(j)}$ and $S_z^{(j)}$ are representation in $2s+1$ dimensional of Pauli spin matrices 
$\sigma_x,\sigma_y$ and $\sigma_z$ respectively at site $j$. Existence of ground 
state for XXX model follows from more general theory [BR vol-2]. If $s$ is an half-odd integer and $J > 0$ (anti-ferromagnet), 
in [Mo3] we have shown that ground states are not unique. 

\vsp
Here we discuss now for integer spin $s$ assuming that the ground state is unique. 

Since $H_{XXX}$ can be rewritten as sum of elements of the form  $$\{ S_x^{(j-1)}S_x^{(j)}+S_y^{(j-1)}S_y^{(j)} +  
S_z^{(j-1)} S_z^{(j)}\}$$, it is simple to check that $\bar{H}_{XXX}=H_{XXX}$. We also claim that 
$H_{XXX}^t=H_{XXX}$. To that end we consider the space $V_d$ of homogeneous polynomials in two
complex variable with degree $m,\;m \ge 0$ i.e. $V_d$ is the space of functions of the form
$$f(z_1,z_2)=a_0z^d_1+a_1z_1^{d-1}z_2+...+a_dz_2^d $$ 
with $z_1,z_2 \in \IC$ and $a_i's $ are arbitrary complex constants. Thus $V_d$ is a $d$-dimensional complex
vector space. The $d-$dimensional irreducible representation 
$\pi_d$ of the Lie-algebra $su(2)$ is given by 
$$\pi_d(X)f= - { \partial f \over \partial z_1 } ( X_{11}z_1+X_{12}z_2 ) +  
{ \partial f \over \partial z_2 } ( X_{21}z_1+X_{22}z_2 )$$  
where $X$ in any element in Lie-algebra $su(2)$. It is simple to verify that the transpose of 
$S_x=\pi_d(\sigma_x)$ is itself, transpose of $S_y=\pi_d(\sigma_y)$ is $-S_y$ and transpose of 
$S_z=\pi_d(\sigma_z)$ is itself. Thus $H_{XXX}^t=H_{XXX}$ for any $d$. So if the ground state 
for $H_{XXX}$ is unique, then the ground state is a pure translation invariant detailed balance and 
$SU(2)$ invariant state. Hence by Theorem 5.4 two point spacial correlation functions of the ground 
state of an integer spin $H_{XXX}$ (i.e. $d=2s+1$ for some integer $s \ge 1$ ) 
decays exponentially provided the ground state is unique. We recall that Theorem 5.4 gives a complete characterization of 
a pure $SU(2)$ and translation invariant state on $\cla=\otimes M_d$ where $d=2s+1$ modulo the dimension $n=2t+1$ of the 
irreducible representation $g \raro u'(g)$ for all $g \in SU(2)$. If the ground state for a $SU(2)$ and translation invariant 
Hamiltonian is unique, then the ground state is completely determined by the unique modulo unitary equivalence 
Popescu systems $(\clk,v_k,\;1 \le k \le d)$ given in Theorem 5.4. In such a case one natural question that 
arises here what is value of $n$ i.e. the dimension of the irreducible representation $g \raro u'(g)$? 

\vsp
We fix any integer $s$ and set $d=2s+1$ as before. Let $V^*=(v_1^*,..,v_d^*)^{tr}$ be the unique ( modulo a phase ) factor 
isometry intertwining the representation
$g \raro u(g)$ and $g \raro u(g) \otimes v(g)$ of $G=SU(2)$ ( Note that such an intertwining operator exists as
$s$ is an integer and Clebsch-Gordon decomposition of $v_s(g) \otimes v_t(g) \equiv v_{|t-s|} \oplus v_{|t-s|+1} \oplus ..\oplus
v_{t+s}$ admits a solution irrespective the value for $t \ge s$ as equality hold $t=|t-s|+k$ for $k=s$ which is an integer. 
However in case $t < s$ we should have $t=s-t+k$ for some integer $k$ i.e. $k=2t-s$. This shows also the range of $t$ as 
$2t-s = k < t < s$. In case $s=1$, for $t < s$, either $t=0$ or $t={1 \over 2}$. $t=0$ is not a feasible solution as $k=-1$ 
is not a valid value. However $t={1 \over 2}$ has a solution with $k=0$. For any integer $s$,    

\vsp
We set $s=1$ and we use variation formula (6.3) in order to find the value of $t$ which minimises the mean energy. 




To that end 
we use the isomorphism $\cla$ with $\mbox{UHF}_d \otimes \tilde{\mbox{UHF}}_d$ and compute the following: We consider 
the following standard representation of Lie algebra $SO(3):$

\ben
l_x=2^{1 \over 2} \left (\begin{array}{llll} 0&,&\;\; 1,\;\; 0 \\ 1&,&\;\;0,\;\;1 \\ 0&,&\;\;1,\;\;0
\end{array} \right ),
\een

\ben
l_y =2^{1 \over 2} \left (\begin{array}{llll} 0&,&\;\; -i,\;\; 0 \\ i&,&\;\;0,\;\;-i \\ 0&,&\;\;i,\;\;0
\end{array} \right ),
\een

\ben
l_z= \left (\begin{array}{llll} 1&,&\;\; 0,\;\; 0 \\ 0&,&\;\;0,\;\;0 \\ 0&,&\;\;0,\;\;-1
\end{array} \right ).
\een




\vsp
Taking $J=1$ for the time being, we compute 
$$h_0 = {1 \over 2} \{(e^1_2 +e^2_1 + e^2_3 + e^3_2 ) \otimes 
(e^1_2 + e^2_1 + e^2_3 + e^3_2) - (e^1_2 - e^2_1 + e^2_3 - e^3_2) \otimes (e^1_2 - e^2_1 
+ e^2_3 - e^3_2) \} $$ 
$$ + ( e^1_1 - e^3_3 ) \otimes ( e^1_1 - e^3_3 )$$
$$=(e^1_2 + e^2_3) \otimes (e^2_1 + e^3_2)  + (e^2_1 + e^3_2) \otimes (e^1_2+ e^2_3)$$
$$ + ( e^1_1 - e^3_3 ) \otimes ( e^1_1 - e^3_3 )$$

$$P\pi(h_0)P = v_1(v_2v^*_1 + v_3v^*_2)v^*_2 + v_2(v_2v^*_1 + v_3v^*_2 )v^*_3$$
$$ + \mbox{ complex conjugation of the previous term } $$ 
$$ + v_1(v_1v_1^* - v_3v^*_3 )v^*_1 - v_3(v_1v_1^*-v_3v^*_3)v^*_3$$

\vsp
At this point we note that as $h_0$ is $\gamma_g$ invariant, the operator $P\pi(h_0)P$ commutes with $n$ dimensional
irreducible representation $g \raro v_t(g)$ and thus it is a scaler multiple of $P$. Thus mean energy of the translation 
invariant state $\omega_t$ which depends on the value of $t$   
$$\omega_t(h_0) = \phi_0((v_2v^*_1+v_3v^*_2) \clj (v_2v_1^* + v_3v^*_2)^*\clj) + \mbox{ complex conjugation of the previous term } $$
$$+ \phi_0((v_1v_1^* - v_3v^*_3 ) \clj (v_1v^*_1-v_3v^*_3) \clj)$$
$$\omega_t(h_0) = \phi_0(\;(v_2v^*_1+v_3v^*_2)^2 \;+\;(v_1v^*_2+v_2v^*_3)^2\;+\;(v_1v_1^* - v_3v^*_3 )^2)$$

\vsp
Thus the problem is now boiled down to a feasible problem of computation of the map 
$t \raro \omega_t(h_0)$ on $\cls_*=\{ {1 \over 2},1,{3 \over 2},2,...\}$. $h_0$ being a bounded operator, 
the map $t \raro \omega_t(h_0)$ is bounded. Further uniqueness of the ground state says that there 
exists a unique $t_0 \in \cls_* $ where the mean energy map $t \raro \omega_t(h_0)$ will have its minimum. 

\vsp 
For any $t \in \cls_*$ explicit solution for $\{v_k: 1 \le k \le d \}$ satisfying the covariant relation 
are well known Clebsch-Gordon coefficients, thus in principle computation of $\omega_t(h_0)$ is possible at 
least numerically. Thus our main result gave a general constructive algorithm which finds a candidate for 
the unique ground state provided there is no phase transition at absolute zero temperature. Our analysis 
directly says very little about the uniqueness of the ground state which remains an open problem for integer spin 
Heisenberg anti-ferromagnet. However using the variational principle (6.3) we also have a criteria to test weather 
there exhibits a phase transition in the ground state for integer spin anti-ferromagnet. This scheme, though discussed 
in this particular model, holds valid for any translation invariant lattice symmetric real Hamiltonian. 


\bigskip
{\centerline {\bf REFERENCES}}

\begin{itemize} 
\bigskip
\item{[Ac]} A non-commutative Markov property, (in Russian), Functional.  anal. i Prilozen 9, 1-8 (1975).

\item{[AM]} Accardi, L., Mohari, A.: Time reflected
Markov processes. Infin. Dimens. Anal. Quantum Probab. Relat. Top., vol-2
,no-3, 397-425 (1999).

\item{[AL]} Affleck, L., Lieb, E.H.: A Proof of Part of Haldane's Conjecture on Spin Chains, Lett. Math. Phys, 12,
57-69 (1986). 

\item{[AMa]} Araki, H., Matsui, T.: Ground states of the XY model, Commun. Math. Phys. 101, 213-245 (1985).

\item {[Ar]} Arveson, W.: Pure $E_0$-semigroups and absorbing states. 
Comm. Math. Phys 187 , no.1, 19-43, (1997)

\item {[BR]} Bratteli, Ola., Robinson, D.W. : Operator algebras
and quantum statistical mechanics, I,II, Springer 1981.

\item {[BJP]} Bratteli, Ola., Jorgensen, Palle E.T. and Price, G.L.: 
Endomorphism of $\clb(\clh)$, Quantization, nonlinear partial differential 
equations, Operator algebras, ( Cambridge, MA, 1994), 93-138, Proc. Sympos.
Pure Math 59, Amer. Math. Soc. Providence, RT 1996.

\item {[BJKW]} Bratteli, Ola., Jorgensen, Palle E.T., Kishimoto, Akitaka and
Werner Reinhard F.: Pure states on $\clo_d$, J.Operator Theory 43 (2000),
no-1, 97-143.    

\item {[BJ]} Bratteli, Ola; Jorgensen, Palle E.T. Endomorphism of $\clb(\clh)$, II, 
Finitely correlated states on $\clo_N$, J. Functional Analysis 145, 323-373 (1997). 

\item {[DR]} Dagotto, E., Rice. T.M.: Surprise on the way from one-two-dimensional quantum magnets: The 
ladder materials. Science 271, 618-623 (1996).

\item {[FNW1]} Fannes, M., Nachtergaele, B., Werner, R.: Finitely correlated states on quantum spin chains,
Commun. Math. Phys. 144, 443-490(1992).

\item {[FNW2]} Fannes, M., Nachtergaele, B., Werner, R.: Finitely correlated pure states, J. Funct. Anal. 120, 511-
534 (1994). 

\item {[Fr]} Frigerio, A.: Stationary states of quantum dynamical semigroups, Comm. Math. Phys. 63 (1978) 269-276.

\item {[Ha]} Hall, B.C. : Lie groups, Lie algebras and representations: an elementary introduction, Springer 2003. 

\item{[HLS]} Hoegh-Krohn, R., Landstad, M.B., Stormer, E.: Compact ergodic groups of automorphisms, Ann. Math. (2)114:1(1981), 
75-86.

\item {[Ki]} Kirillov, A.: Elements of representation theory, Berlin-Heidelberg-New York:Springer-Verlag, 1976.

\item{[Mo1]} Mohari, A.: Markov shift in non-commutative probability, J. Funct. Anal. vol- 199 
, no-1, 190-210 (2003) Elsevier Sciences. 

\item{[Mo2]} Mohari, A.: $SU(2)$ symmetry breaking in quantum spin chain, http://arxiv.org/abs/math-ph/0509049

\item{[Mo3]} Mohari, A.: Pure inductive limit state and Kolmogorov's property,  http://arxiv.org/abs/0704.1987, 
accepted for publication by J. Funct. Anal, 2007, available online 17th Sep 2007, Elsevier Sciences.  

\item{[Mo4]} Mohari, A.: Jones index of a quantum dynamical semigroup,   http://arxiv.org/abs/0704.1989, Banff 
conference proceeding of BIRS conference; Operator methods in fractal analysis, wavelets and dynamical systems, 
December 2-7, Springer (to appear).  

\item{[Ma1]} Matsui, T.: A characterization of pure finitely correlated states. 
Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, no. 4, 647--661 (1998).

\item{[Ma2]} Matsui, T.: The split property and the symmetry breaking of the quantum spin chain, Comm. 
Maths. Phys vol-218, 293-416 (2001) 

\item{[Po]} Popescu, Gelu: Isometric dilations for infinite sequences of non-commutating operators, Trans. Amer. Math.
Soc. 316 no-2, 523-536 (1989)

\item{[Pow1]} Powers, Robert T.: Anals of Mathematics 1967. 

\item{[Pow2]} Powers, Robert T.: An index theory for semigroups of $*$-endomorphisms of
$\clb(\clh)$ and type II$_1$ factors.  Canad. J. Math. 40 (1988), no. 1, 86--114.


\end{itemize}

\end{document}




