On certain Cuntz-Pimsner algebras
Alex Kumjian
Let $A$ be a separable unital C*-algebra and let $\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a separable Hilbert space $\Hf$ such that $\pi(A) \cap \Kc(\Hf) = \{ 0 \}$. % and the natural embedding $A \emb \Oc_E$, where $\Oc_E$ is the % Cuntz-Pimsner algebra associated to $E$, induces a $KK$-equivalence. %(see \cite[4.5]{pm}). We show that $\Oc_E$, the Cuntz-Pimsner algebra associated to the Hilbert $A$-bimodule $E = \Hf \ot_{\C} A$, is simple and purely infinite. If $A$ is nuclear and belongs to the bootstrap class to which the UCT applies, then the same applies to $\Oc_E$. Hence by the Kirchberg-Phillips Theorem the isomorphism class of $\Oc_E$ only depends on the $K$-theory of $A$ and the class of the unit.
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\title[Cuntz-Pimsner algebras]{On certain Cuntz-Pimsner algebras}
\author[a.~kumjian]{Alex Kumjian}
\address{Department of Mathematics, University of Nevada, Reno NV
89557, USA}
\email{alex@unr.edu}
\thanks{Research partially supported by \textsc{nsf} grant DMS-9706982}
\subjclass{Primary 46L05; Secondary 46L55.}
\keywords{C*-algebra, Hilbert bimodule, %Cuntz-Pimsner algebra, 
simple, purely infinite} 
\date{28 August 2001}
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\begin{abstract} 
Let $A$ be a separable unital C*-algebra and let 
$\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a
separable Hilbert space $\Hf$ such that 
$\pi(A) \cap \Kc(\Hf) = \{ 0 \}$. 
% and the natural embedding $A \emb \Oc_E$, where $\Oc_E$ is the
% Cuntz-Pimsner algebra associated to $E$, induces a $KK$-equivalence.
%(see \cite[4.5]{pm}).
We show that $\Oc_E$, the Cuntz-Pimsner algebra associated to the Hilbert
$A$-bimodule $E = \Hf \ot_{\C} A$, is simple and purely infinite.  
If $A$ is nuclear and belongs to the bootstrap class to which the UCT
applies, then the same applies to $\Oc_E$.
Hence by the Kirchberg-Phillips Theorem the isomorphism class of
$\Oc_E$ only depends on the $K$-theory of $A$ and the class of the unit.
\end{abstract}
%% $(K_*(A), [1_A])$. does not depend on the choice of representation $\pi$.  
%%Moreover, if $A = C(X)$ with $X$ contractible, then $\Oc_E \cong
%%\Oc_\infty$. 
\maketitle

%\section{Introduction}\label{intro}

In his seminal paper \cite{pm}, Pimsner constructed a C*-algebra
$\Oc_E$ from a Hilbert bimodule over a C*-algebra $A$ as a quotient of
a concrete C*-algebra $\Tc_E$, an analogue of the Toeplitz algebra,
acting on the Fock space associated to $E$. 
There has recently been much interest in these Cuntz-Pimsner algebras
(or Cuntz-Krieger-Pimsner algebras), which generalize both crossed
products by $\Z$ and Cuntz-Krieger algebras, as well as the associated
Toeplitz algebras. %both in their own right, as 
%generalizations of Cuntz-Krieger algebras (or graph C*-algebras),
%and as a tool in the study of reduced free product C*-algebras.   
The structure of these C*-algebras is not yet fully understood,
although considerable progress has been made.  For example,  
Pimsner found a six-term exact sequence for the $K$-theory of $\Oc_E$
which generalizes the Pimsner-Voiculescu exact sequence 
(see \cite[Theorem 4.8]{pm});
conditions for simplicity were found in \cite{sc2,ms,kpw1,dpz}
and for pure infiniteness in \cite{z}.

% for the $K$-theory of the crossed product of a
%C*-algebra by an automorphism was given in Pimsner's original paper
%%gives two six-term exact sequences for $KK$-theory). 

The purpose of the present note is to analyze the structure of
Cuntz-Pimsner algebras associated to a certain class of Hilbert
bimodules.  Let $A$ be a separable unital C*-algebra and let 
$\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a
separable Hilbert space $\Hf$ such that 
$\pi(A) \cap \Kc(\Hf) = \{ 0 \}$.  Then $E = \Hf \ot_{\C} A$ is a
Hilbert bimodule over $A$ in a natural way. %  Since the left action of
% $A$ does not meet the compacts $\Kc(E)$, the Toeplitz algebra $\Tc_E$
% may be identified with the Cuntz-Pimsner algebra $\Oc_E$.  
% Such bimodules include
% those of the form, $L^2(X, \mu) \ot_{\C} C(X)$ where $A = C(X)$ with
% $X$ a second countable compact space and $\mu$ is a nonatomic Borel
% measure with full support ($\pi$ is given by left multiplication).  
%C*-algebra which is $KK$-equivalent to $A$.  
We show that $\Oc_E$ is separable, simple and purely infinite.
If $A$ is nuclear and in the bootstrap class, then the same holds for
$\Oc_E$ and thus by the Kirchberg-Phillips theorem the
isomorphism class of $\Oc_E$ is completely determined by the
$K$-theory of $A$ together with the class of the unit (since $\Oc_E$
is $KK$-equivalent to $A$). 

Many examples of Cuntz-Pimsner algebras found in the
literature arise from Hilbert bimodules which are finitely generated
and projective; in such cases the left action must consist entirely of
compact operators.  Our examples do not fall in this class; in fact,
the left action has trivial intersection with the compacts.  And this
has some interesting consequences:
$\Oc_E \cong \Tc_E$ (see \cite[Corollary 3.14]{pm}) and the natural
embedding $A \emb \Oc_E$ induces a $KK$-equivalence 
(see \cite[Corollary 4.5]{pm}). 

In \S \ref{pre} we review some basic facts concerning the
construction of $\Tc_E$ as operators on the Fock space of $E$
and the gauge action $\lm : \T \ra \Aut(\Tc_E)$.  
We assume that the left action of $A$ does not meet the compacts $\Kc(E)$
%a requirement easily satisfied in the case of interest, 
and identify $\Oc_E$ with $\Tc_E$. 
The fixed point algebra $\Fc_E$, the analogue of the AF-core of a
Cuntz-Krieger algebra, contains a canonical descending sequence of
essential ideals indexed by $\N$ with trivial intersection.  The crossed 
product $\Oc_E \rt_{\lm} \T$ has a similar collection of essential ideals
indexed by $\Z$ on which the dual group of automorphisms acts in a
natural way.  By Takesaki-Takai duality 
$$
\Oc_E \ot \Kc(L^2(\T)) \cong (\Oc_E \rt_{\lm} \T) \rt_{\wh\lm} \Z;
$$
hence, much of the structure of $\Oc_E$ is revealed through an
analysis of the double crossed product. 
%$(\Oc_E \rt_{\lm} \T) \rt_{\wh\lm} \Z$.% (see \cite{dpz}).   

In \S \ref{result} we show that if $E$ is the Hilbert bimodule over
$A$ associated to a representation as described above, then for every
nonzero positive element $d \in \Oc_E$ there is a $z \in \Oc_E$ so
that $z^*dz = 1$; it follows that $\Oc_E$ is simple and purely
infinite (see Theorem \ref{spi}).  The proof of this proceeds
through a sequence of lemmas and is patterned on the proof of
\cite[Theorem 2.1]{rr}, which is in turn based on a key lemma of Kishimoto 
(see \cite[Lemma 3.2]{ks}).  Our argument uses the version of this
lemma found in \cite[Lemma 7.1]{op3} and this requires that we show
that the Connes spectrum of the dual action is full (this is also an
ingredient in the proof of simplicity found in \cite{dpz}).  
We invoke a version of a key lemma of R\o rdam for crossed products by
$\Z$ which arise from automorphisms with full Connes spectrum. 
The fact that $\Oc_E$ embeds equivariantly into 
$(\Oc_E \rt_{\lm} \T) \rt_{\wh\lm} \Z$ allows us to apply this lemma
to $\Oc_E$.  In \S \ref{app} we use the Kirchberg-Phillips
theorem to collect some consequences of this theorem as indicated
above and discuss certain connections with reduced (amalgamated) free
products.  

We fix some notation and terminology.
Given a C*-algebra $B$ we let $\wh{B}$ denote its spectrum, that is,
the collection of irreducible representations modulo unitary
equivalence endowed with the Jacobson topology (see \cite[\S 4.1]{pd}).
If $I$ is an ideal in a C*-algebra $B$, then every irreducible
representation of $I$ extends uniquely to an irreducible
representation of $B$.  This allows one to identify $\wh{I}$ with an
open subset of $\wh{B}$, the complement of which consists of the 
classes of irreducible representations which vanish on $I$. 
     %%Then $B$ is said to be $\bt$-simple if $B$ has no proper
     %%nontrivial $\bt$-invariantideal.   
Given a *-automorphism $\bt$ of a C*-algebra $B$, let $\Gm(\bt)$
denote the Connes spectrum of $\bt$ (see \cite{o,co} or 
\cite[\S 8.8]{pd}); recall that 
$$
\Gm(\bt) = \bigcap_H {\rm Sp}\,(\bt|_H)
$$
where the intersection is taken over all $\bt$-invariant hereditary
subalgebras $H$.  A C*-algebra is said to be purely infinite if every
hereditary subalgebra contains an infinite projection.

I wish to thank D.~Shlyakhtenko for certain helpful remarks relating
to material in \S \ref{app}.
\section{Preliminaries}\label{pre}

We review some basic facts concerning Cuntz-Pimsner algebras; we shall
be mainly interested in those which arise from bimodules for which the
left action has trivial intersection with the compacts (see Remark
\ref{oe=te}). 
Let $A$ be a C*-algebra.  

\begin{df} {\rm (see \cite{ri,ka,l})}
Let $E$ be a right $A$-module.  Then $E$ is said to be a (right)
pre-Hilbert $A$-module if it is equipped with an $A$-valued inner
product $\lb \cdot, \cdot \rb_A$   
which satisfies the following conditions for all  $\xi, \et, \zt \in E$,  
$s, t \in \C$, and $a \in A$:
\begin{itemize}
\item[i.]  $\lb \xi, s \et + t\zt \rb_A  = 
         s\lb \xi,  \et \rb_A  + t\lb \xi,  \zt \rb_A$
\item[ii.] $ \lb \xi,  \eta a\rb_A  =  \lb \xi,  \et \rb_A  a $
\item[iii.] $ \lb \et,  \xi \rb_A  =  \lb \xi,  \et \rb_A^*$
\item[iv.] $\lb \xi, \xi \rb_A  \ge 0$ and  $\lb \xi, \xi \rb_A  = 0$ 
only if $\xi = 0$.
\end{itemize}
$E$ is said to be a (right) Hilbert $A$-module if it is complete in
the norm: $\|\xi\| = \| \lb \xi, \xi \rb_A \|^{1/2}$. 
\end{df}

Let $E$ be a Hilbert $A$-module.  Then $E$ is said to be {\em full} if
the span of the values of the inner product is dense.   
The collection of bounded adjointable operators on $E$, $\Lc(E)$, is a
C*-algebra. The closure of the span of operators of the form
$\tht_{\xi,\et}$ for $\xi, \et \in E$ 
(where $\tht_{\xi,\et}(\zt) = \xi\lb \et, \zt\rb_A$ for $\zt \in E$) 
forms an essential ideal in $\Lc(E)$ which is denoted $\Kc(E)$. A
Hilbert space is a Hilbert module over $\C$. 

\begin{df}
Let $E$ be a Hilbert $A$-module and $\ph : A \ra \Lc(E)$ be an
injective $*$-homomorphism.  Then the pair $(E, \ph)$ is said to be
Hilbert bimodule over $A$ (or Hilbert $A$-bimodule). 
\end{df}

Pimsner defines the Cuntz-Pimsner algebra $\Oc_E$ as a quotient of the
analogue of the Toeplitz algebra, $\Tc_E$, generated by creation
operators on the Fock space of $E$ (see \cite{pm}).  The injectivity
of $\ph$ is not really necessary (see \cite[Remark 1.2.1]{pm}).  We will
henceforth assume that $E$ is full (see \cite[Remark 1.2.3]{pm}).  
%% We review the construction.

The Fock space of $E$ is the Hilbert $A$-module
$$
\Ec_+ = \bigoplus_{n = 0}^\infty E^{\ot n}
$$
where $E^{\ot 0} = A$ and for $n > 0$, $E^{\ot n}$ is the $n$-fold
tensor product: %%of Hilbert $A$-module
$$
E^{\ot n} = E \ot_A \cdots \ot_A E.
$$
Note that $\Ec_+$ is also a Hilbert $A$-bimodule with left action defined
by $\ph_+(a)b = ab$ for $a, b \in A = E^{\ot 0}$ and
$$
\ph_+(a)(\xi_1 \ot \cdots \ot \xi_n) =
\ph(a)\xi_1 \ot \cdots \ot \xi_n
$$
for $a \in A$ and $\xi_1 \ot \cdots \ot \xi_n \in E^{\ot n}$.

Then $\Tc_E \sub \Lc(\Ec_+)$ is the C*-algebra generated by the creation
operators $T_\xi$ for $\xi \in E$ where $T_\xi(a) = \xi a$ and
$$
T_\xi(\xi_1 \ot \cdots \ot \xi_n) = 
\xi \ot \xi_1 \ot \cdots \ot \xi_n.
$$
Observe that ${T_\xi}^*T_\et = \ph_+(\lb \xi, \eta\rb_A)$
for $\xi, \et \in E$.
Since $E$ is full, $\ph_+(A) \sub \Tc_E$; let $\io : A \emb \Tc_E$
denote the embedding.  Note that one may define $T_\xi$ for 
$\xi \in E^{\ot n}$ in an analogous manner and that we have  
${T_\xi}^*T_\et = \io(\lb \xi, \et\rb_A)$ 
for $\xi, \et \in E^{\ot n}$.

There is an embedding $\io_n : \Kc(E^{\ot n}) \emb \Tc_E$ (identify
$\Kc(E^{\ot 0})$ with $A$), given for $n > 0$ by 
$\io_n(\tht_{\xi,\et}) = T_{\xi}{T_{\et}}^*$ 
for $\xi, \et \in E^{\ot n}$.
%%the image of which (for $n > 0$) is the C*-subalgebra generated by
%%operators of the form $T_{\xi}{T_{\et}}^*$
%%T_{\xi_1}\cdots T_{\xi_n} {T_{\et_1}}^*\cdots{T_{\et_n}}^*
Note that such operators preserve the grading of $\Ec_+$ and that there is
an embedding $\Kc(E^{\ot n}) \emb \Lc(E^{\ot m})$ for $m \ge n$. 
%% Let $\Fc_E$ denote the C*-subalgebra of $\Tc_E$ that is generated
%% by copies of $\Kc(E^{\ot n})$ for $n \ge 0$. 
Let $C_n$ denote the C*-subalgebra of $\Tc_E$ generated by operators
of the form $T_{\xi}{T_{\et}}^*$ for $\xi, \et \in E^{\ot k}$ with 
$k \le n$ (by convention $C_0 = \io(A)$).  Then the $C_n$ form an
ascending family of C*-subalgebras. 

\begin{rem}\label{oe=te}
Suppose $\ph(A) \cap \Kc(E) = \{ 0 \}$; then the natural map 
$C_n \ra \Lc(E^{\ot m})$ is an embedding for $m \ge n$.
By \cite[Corollary 3.14]{pm} $\Tc_E \cong \Oc_E$ and the inclusion $A
\emb \Oc_E$ induces a $KK$-equivalence (see \cite[Corollary
4.5]{pm}).  Under the isomorphism of $\Tc_E$ with $\Oc_E$,
$\ol{\cup_n C_n}$ is mapped to $\Fc_E$, the analog of the AF core of a
Cuntz-Krieger algebra.
\end{rem}

For the remainder of this section we shall tacitly assume that 
$\ph(A) \cap \Kc(E) = \{ 0 \}$ and identify $\Tc_E$ with $\Oc_E$.

\begin{pro}\label{jn}
For each $n \in \N$ the C*-subalgebra, $J_n$, generated by
$\io_k(\Kc(E^{\ot k}))$ for $k \ge n$ is an essential ideal in
$\Fc_E$. We obtain a descending sequence of ideals   
$$
J_0 \supset J_1 \supset J_2 \supset \cdots
$$
with $J_0 = \Fc_E$ and  %%If $\ph(A)\cap\Kc(E) =  \{ 0 \}$, then
$\cap_n J_n = \{ 0 \}$.  Furthermore, 
$J_n/J_{n+1} \cong \Kc(E^{\ot n})$ (thus $J_n/J_{n+1}$ is strong
Morita equivalent to $A$) and the restriction of the quotient map
yields an isomorphism $C_n \cong \Fc_E/J_{n + 1}$. 
\end{pro}
\begin{proof}
Given $n \in \N$ it is clear that $J_n$ is an ideal (see
\cite[Definition 2.1]{pm}).  To see that $J_n$ is essential it suffices to
show that for every $m$ and nonzero element $c \in C_m$ there is an element 
$d \in \Kc(E^{\ot k})$ for some $k \ge n$ such that $c\io_k(d) \ne 0$.
Let $k$ be an integer with $k \ge \max(m, n)$; since the map from $C_m$ to 
$\Lc(E^{\ot k})$ is an embedding for $k \ge m$, $c\xi \ne 0$ for
some $\xi \in E^{\ot k}$.  Then $cT_{\xi}{T_{\xi}}^* \ne 0$ and we take 
$d = \tht_{\xi,\xi}$.

The $J_n$ form a descending sequence of ideals by construction.
Since $\ph(A) \cap \Kc(E) = \{ 0 \}$, $C_m \cap J_n = \{ 0 \}$ 
for $m < n$.  Hence, $\cap_n J_n = \{ 0 \}$, for $\Fc_E$ is the
inductive limit of the $C_m$.  Further, for each $n$ we have 
$$
J_n = \io_n(\Kc(E^{\ot n})) + J_{n + 1}\quad\text{and}\quad
\io_n(\Kc(E^{\ot n})) \cap J_{n + 1} = \{ 0 \};
$$ 
it follows that $J_n/J_{n+1} \cong \Kc(E^{\ot n})$.  Finally, since
$$
\Fc_E = C_n + J_{n + 1}\quad\text{and}\quad
C_n \cap J_{n + 1} = \{ 0 \},
$$
we have $C_n \cong \Fc_E/J_{n + 1}$. 
\end{proof}

There is a strongly continuous action 
$$
\lm : \T \ra \Aut(\Oc_E)
$$
such that $\lm_t(T_\xi) = tT_\xi$.  The fixed point algebra under this
action is $\Fc_E$ and we have a faithful conditional expectation
$P_E: \Oc_E \ra \Fc_E$ given by
$$
P_E(x) = \int_\T \lm_t(x)\,dt.
$$
Consider the spectral subspaces of $\Oc_E$ under this action: 
for $n \in \Z$
$$
(\Oc_E)_n = %\doteq
\{ x \in \Oc_E : \lm_t(x) = t^nx \text{ for all }t \in \T \}.
$$

\begin{rem}\label{spec}
Note that $(\Oc_E)_n$ is the closure of the span of
elements of the form $T_{\xi}{T_{\et}}^*$ where $\xi \in E^{\ot k}$
and $\et \in E^{\ot l}$ with $n = k - l$. 
For $n \ge 0$ and $x \in (\Oc_E)_n$ we have $x^*x \in \Fc_E$ and 
$xx^* \in J_n$.  We may regard $(\Oc_E)_n$ as an 
$J_n$-$\Fc_E$-equivalence bimodule (see \cite{ri}); hence, $J_n$ is
strong Morita equivalent to $\Fc_E$ for each $n \ge 0$. If we regard 
$(\Oc_E)_1$ as a Hilbert $\Fc_E$-bimodule we have
(cf.\ \cite[\S 2]{pm} and \cite[\S 1.4]{sc2})
$$    %E_\infty = 
 (\Oc_E)_1  \cong E \ot_A \Fc_E,
$$
where the isomorphism is implemented by the map 
$\xi \ot a \mapsto T_\xi a$.
The crossed product $\Oc_E \rt_{\lm} \T$ may be identified with the
closure of the subalgebra of $\Oc_E \ot \Kc(\ell^2(\Z))$ consisting of
finite sums of the form
$$
\sum x_{ij} \ot e_{ij} 
$$
where $e_{ij}$ are the standard rank one partial isometries in 
$\Kc(\ell^2(\Z))$ and $x_{ij} \in (\Oc_E)_{j - i}$.
\end{rem}

Let $\wh{\lm}: \Z \ra \Aut(\Oc_E \rt_{\lm} \T)$ denote the dual
automorphism group. 

\begin{pro}\label{id}
%%Suppose $\ph(A) \cap \Kc(E) = \{ 0 \}$.
There is an embedding $\eps : \Fc_E \emb \Oc_E \rt_{\lm} \T$ onto a
corner and a collection of essential ideals $\{ I_n \}_{n \in \Z}$ 
in $\Oc_E \rt_{\lm} \T$ satisfying the following conditions:
\begin{itemize}
\item[i.] For all $n \in \Z$, $\Fc_E$ is strong Morita equivalent to
$I_n$ and $A$ is strong Morita equivalent to $I_n/I_{n+1}$.
\item[ii.]  For all $n \ge 0$, $\eps(J_n) = \eps(1)I_n\eps(1)$.
\item[iii.] $I_n \sub I_m$ if $m \le n$.
\item[iv.] $\cap_n I_n = \{ 0 \}$
\item[v.] $\ol{\cup_n I_n} = \Oc_E \rt_{\lm} \T$
\item[vi.] $\wh{\lm}_k(I_n) = I_{n+k}$
\end{itemize}
\end{pro}
\begin{proof}
We use the identification of $\Oc_E \rt_{\lm} \T$ with a C*-subalgebra of 
$\Oc_E \ot \Kc(\ell^2(\Z))$ given in Remark \ref{spec}.  For each $n$
let $I_n$ be the ideal generated by $p_n = 1 \ot e_{nn}$.  Since 
$\Fc_E = (\Oc_E)_0$, it follows that $\Fc_E$ is isormorphic to the
corner determined by $p_n$ and thus is strong Morita 
equivalent to $I_n$.  The desired embedding  
$\eps : \Fc_E \emb \Oc_E \rt_{\lm} \T$ is given by
$\eps(a) = a \ot e_{00}$.  

Given an element of the form $a_{mn} = x_{mn} \ot e_{mn}$ in 
$\Oc_E \rt_{\lm} \T$ with $m \le n$, we have 
$$
{a_{mn}}^*a_{mn} = {x_{mn}}^*x_{mn} \ot e_{nn}
\quad\text{ and }\quad
a_{mn}{a_{mn}}^* = x_{mn}{x_{mn}}^* \ot e_{mm}
$$ 
with $x_{mn}{x_{mn}}^* \in J_{n - m}$; since $p_n$ may be expressed as a
finite sum of elements of the form ${a_{mn}}^*a_{mn}$, it follows that 
$I_n \sub I_m$ and that $p_mI_np_m = J_{n - m} \ot e_{mm}$.  
Thus $\eps(J_n) = \eps(1)I_n\eps(1)$ for all $n \ge 0$. 
% Thus $I_n$ is strong Morita equivalent to $J_n$ which in turn is
% strong Morita equivalent to $\Fc_E$.
Assertion (vi) follows from the fact that 
$\wh{\lm}_k(p_n) = 1 \ot p_{n + k}$.  
The remaining assertions follow from Proposition \ref{jn}.
%%Only (iv) and (vi) require the assumption $\ph(A) \cap \Kc(E) = \{ 0 \}$.
\end{proof}


\section{$\Oc_E$ is simple and purely infinite}\label{result}
Let $A$ be a separable unital C*-algebra and let 
$\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a
separable Hilbert space $\Hf$.
%% such that $\pi(A) \cap \Kc(\Hf) = \{ 0 \}$. 

\begin{pro}\label{bimod}
With  $A$ and $\pi : A \ra \Lc(\Hf)$ as above,
$$
E = \Hf \ot_{\C} A
$$
is a full Hilbert bimodule over $A$ under the operations
$$
\lb \xi \ot a, \et \ot b \rb_A = \lb \xi, \et \rb a^*b,
\qquad
\ph(a)(\xi \ot b) = \pi(a)\xi \ot b
$$
for all $\xi, \et \in \Hf$ and $a, b \in A$.
Moreover, if  $\pi(A) \cap \Kc(\Hf) = \{ 0 \}$, then
$\ph(A) \cap \Kc(E) = \{ 0 \}$ and $\Oc_E \cong \Tc_E$.
\end{pro}
\begin{proof}
Note that $E = \Hf \ot_{\C} A$ is the tensor product of the Hilbert
$A$-$\C$-bimodule $\Hf$ and the Hilbert $\C$-$A$-bimodule $A$ 
as defined by Rieffel in \cite{ri} (see also \cite[Ch.\ 4]{l}).
The natural map from $\Lc(\Hf)$ to $\Lc(E) = \Lc(\Hf \ot_{\C} A)$ induces
an embedding $\Lc(\Hf)/\Kc(\Hf) \emb \Lc(E)/\Kc(E)$ (since $\Kc(\Hf)$
is mapped into $\Kc(E)$ and the Calkin algebra $\Lc(\Hf)/\Kc(\Hf)$ is
simple).  Hence, if $\pi(A) \cap \Kc(\Hf) = \{ 0 \}$, then  
$\ph(A) \cap \Kc(E) = \{ 0 \}$.  The last assertion, 
$\Oc_E \cong \Tc_E$, follows by \cite[Corollary 3.14]{pm}.
\end{proof}

Henceforth, we assume that $\pi(A) \cap \Kc(\Hf) = \{ 0 \}$ and
identify $\Oc_E$ with $\Tc_E$.
The aim of this section is to show that $\Oc_E$ is simple and purely
infinite.  Simplicity may be proven directly by invoking
\cite[Theorem 3.9]{sc2}: if $A$ is unital and $E$ is full, then $\Oc_E$ is
simple if and only if $E$ is minimal and nonperiodic.
Lemma \ref{hyp} would then be a consequence of \cite[Theorem 6.5]{op1}.
We follow a more indirect route patterned on the proof of 
\cite[Theorem 2.1]{rr}; this will also show that $\Oc_E$ is
purely infinite. 

\begin{rem}\label{rep}
With $E = \Hf \ot_{\C} A$ as above, we have 
$E^{\ot n} \cong \Hf^{\ot n} \ot_{\C} A$ via the map
$$
(\xi_1 \ot a_1) \ot (\xi_2 \ot a_2) \ot \cdots \ot (\xi_n \ot a_n) \mapsto 
(\xi_1 \ot \pi(a_1)\xi_2 \ot \cdots \ot \pi(a_{n-1})\xi_n) \ot a_n;
$$
similarly, if $\sg: A \ra \Lc(\Kf)$ is a representation of $A$ on a
Hilbert space $\Kf$, then  
$$
E^{\ot n}\ot_A \Kf \cong E^{\ot n-1} \ot_A \Hf \ot \Kf.
%%\cong \Hf^{\ot n} \ot \Kf
$$
Recall that the action of $\Fc_E$ on Fock space preserves the natural
grading.  Let $\wt\sg_n$ denote the representation of $\Fc_E$ on 
$E^{\ot n}\ot_A \Kf$ given by left action on $E^{\ot n}$.  Then the
restriction of $\wt\sg_n$ to $C_{n-1}$ is faithful: indeed, this
follows from the facts that the natural map  
$$
\Lc(E^{\ot n-1}) \ra \Lc(E^{\ot n-1} \ot_A \Hf \ot \Kf)
\cong \Lc(E^{\ot n}\ot_A \Kf)
$$
is an embedding (since $\pi$ is faithful) and that 
$\wt\sg_n|_{\Kc(E^{\ot n-1})}$ factors through $\Lc(E^{\ot n-1})$.
Note that $\wt\sg_n$ is equivalent to the representation of $\Fc_E$
obtained  from $\sg$ as follows: use the strong Morita 
equivalence between $A$ and $J_n/J_{n+1}$ % (see Proposition \ref{jn})
to obtain a representation of $J_n/J_{n+1}$ and extend this to a
representation of $\Fc_E$.  Since the restriction of
$\wt\sg_n$ to $C_{n-1}$ is faithful, $\ker\wt\sg_n \sub J_{n}$ (see
Proposition \ref{jn}).  It follows that the closure of a point in 
$\wh{J_n} - \wh{J_{n+1}}$ contains the complement of $\wh{J_n}$.  
A similar assertion holds for $\Oc_E \rt_{\lm} \T$: 
for any $n \in \Z$ the closure of a point in $\wh{I_n} - \wh{I_{n+1}}$
contains the complement of $\wh{I_n}$. 
%% it suffices to show this for $\Kc(E^{\ot n-1})$ and 
\end{rem}

\begin{lem}\label{hyp}
With $A$ and $E$ as above, $\Gm(\wh\lm_1) = \T$ where $\wh\lm$ is
the dual action of $\Z$ on $\Oc_E \rt_{\lm} \T$. 
\end{lem}
\begin{proof}
By \cite[Theorem 4.6]{op2} it suffices to find a dense invariant
subset of $(\Oc_E \rt_{\lm} \T)\sphat$\, on which ${\wh\lm_1}^*$ acts freely. 
That is, we must find an irreducible representation $\sg$ of 
$\Oc_E \rt_{\lm} \T$ such that, 
$\{ [\sg \circ \wh\lm_n] : n \in \Z \}$, the orbit of the
unitary equivalence class of $\sg$ under $\wh\lm^*$, is dense in 
$(\Oc_E \rt_{\lm} \T)\sphat$\, and 
$[\sg \circ \wh\lm_m] \ne [\sg \circ \wh\lm_n]$ if 
$m \ne n$.  Let $\sg_0$ be an irreducible representation of $A$ and
use the strong Morita equivalence between $A$ and $I_0/I_1$ to obtain
an irreducible representation $\sg'$ of $I_0/I_1$.  Then $\sg$, the
extension of $\sg'$ to $\Oc_E \rt_{\lm} \T$, is also irreducible. 
The classes $[\sg \circ \wh\lm_n]$ are distinct, for if $m < n$, 
$\sg \circ \wh\lm_m$ vanishes on $I_n$.
Moreover, for each $n \in \Z$ the closure of $[\sg\circ\wh\lm_n]$ in
$(\Oc_E \rt_{\lm} \T)\sphat$\, includes the classes of all irreducible
representations which vanish on $I_n$ (since
$[\sg\circ\wh\lm_n] \in \wh{I_n} - \wh{I_{n+1}}$,
% $I_{n+1} \sub \ker(\sg\circ\wh\lm_n) \sub I_n$, 
see Remark \ref{rep}).  
Hence, $\{ [\sg \circ \wh\lm_n] : n \in \Z \}$ is dense in 
$(\Oc_E \rt_{\lm} \T)\sphat$.  
\end{proof}

Using Takesaki-Takai duality we show below that a C*-algebra $D$ equipped
with an action $\al$ of $\T$ may be embedded equivariantly as a corner in
$(D \rt_{\al} \T) \rt_{\wh\al} \Z$.  This fact is related to
Rosenberg's observation that the fixed point algebra under a compact
group action embeds as a corner in the crossed product (see \cite{ro}).

\begin{pro}\label{duality}
Given a unital C*-algebra $D$ and a strongly continuous action 
$\al : \T \ra \Aut(D)$,
there is an isomorphism $\psi$ of $D$ onto a full corner of 
$(D \rt_{\al} \T) \rt_{\wh\al} \Z$ which is equivariant in the sense
that $\wh{\wh{\al}}_t\circ\psi = \psi\circ\al_t$ for all $t \in \T$.
Moreover, $\psi(1) \in D \rt_{\al} \T$.
\end{pro}
\begin{proof}
By Takesaki-Takai duality \cite[7.9.3]{pd} there is an isomorphism
$$
\gm: D \ot \Kc(L^2(\T)) \cong (D \rt_{\al} \T) \rt_{\wh\al} \Z,
$$ 
which is equivariant with respect to $\al \ot \Ad \rho$ and
$\wh{\wh{\al}}$ (where $\rho$ is the right regular representation of
$\T$ on $L^2(\T)$).  The desired embedding is obtained by finding an
$\Ad\rho$ invariant minimal projection $p$ in $\Kc(L^2(\T))$ 
(cf.\ \cite{ro}):  set $\psi(d) = \gm(d \ot p)$ for $d \in D$.
Since $\psi$ is equivariant, $\psi(1)$ is in the
fixed point algebra of $\wh{\wh{\al}}$; hence, 
$\psi(1) \in D \rt_{\al} \T$.
\end{proof}

The following lemma is adapted from \cite[Lemma 2.4]{rr}; the proof is
patterned on R\o rdam's but we substitute 
\cite[Lemma 7.1]{op3} for \cite[Lemma 3.2]{ks}. 

\begin{lem}\label{key}
Let $B$ be a C*-algebra and let $\bt$ be an automorphism of $B$ such
that $\Gm(\bt) = \T$ and let $P$ denote the canonical conditional
expectation from $B \rt_\bt \Z$ to $B$.   Then for every positive
element $y \in B \rt_\bt \Z$ and $\ep > 0$ there are  
positive elements $x, b \in B$ such that 
$$
\| b \| > \| P(y) \| - \ep, \quad \| x \| \le 1 \quad \text{and} \quad
\| xyx - b \| < \ep.
$$  
If $y$ is in the corner determined by a projection $p \in B$, then
$x, b$ may also be chosen to be in the corner.
\end{lem}
\begin{proof}
As in the proof of \cite[Lemma 2.4]{rr} we may assume (by perturbing
$y$ if necessary) that $y$ is of the form 
$$
y = 
y_{-n}u^{-n} + \cdots + y_{-1}u^{-1} + y_0 + y_1u + \cdots + y_nu^n
$$
for some $n$ where $y_j \in B$ and $u$ is the canonical unitary in 
$B \rt_\bt \Z$ implementing the automorphism $\bt$; note that 
$y_0 = P(y)$ is positive.  

By \cite[Theorem 10.4]{op3} $\bt^k$ is properly outer for all 
$k \ne 0$.  Hence, by \cite[Lemma 7.1]{op3} there is a positive
element $x$ with $\| x \| = 1$ such that
$$
\| xy_0x \| > \| y_0 \| - \ep, \quad \text{and} \quad
\| xy_ku^kx \| = \| xy_k\bt^k(x) \| < \ep/2n
$$
for $0 < | k | \le n$. Set $b = xy_0x$; then a straightforward
calculation yields $\| xyx - b \| < \ep$.  
We now verify the last assertion.
Suppose that $y$ is in the corner determined by a projection 
$p \in B$; we may again assume that $y$ is of the above form.
Since $P$ is a conditional expectation onto $B$, 
%%py_0p = pP(y)p = P(pyp) = P(y) = y_0,
$y_0 = P(y)$ is also in the corner determined by $p$. 
In the proof of \cite[Lemma 7.1]{op3} the positive element $x$ 
is constructed in the hereditary subalgebra determined by $y_0$;
hence we may assume that $x$ and therefore also $b = xy_0x$ lies
in the same corner. 
\end{proof}

Recall that $C_n$ is the C*-subalgebra of $\Fc_E$ generated by operators
of the form $T_{\xi}{T_{\et}}^*$ for $\xi, \et \in E^{\ot k}$ with 
$k \le n$ and that they form an ascending family of C*-subalgebras
with dense union. The subspace $E^{\ot n}$ is left invariant by $C_n$
and one has an embedding $C_n \emb \Lc(E^{\ot n})$.

\begin{lem}\label{easy}
Given a positive element $c \in C_n$ and $\ep > 0$, there is 
$\xi \in E^{\ot n}$ with $\| \xi \| = 1$ such that 
${T_\xi}^*cT_{\xi} \in C_0$ and 
$\| {T_\xi}^*cT_{\xi} \| > \| c \| - \ep$.
\end{lem}
\begin{proof}
The first assertion follows from a straightforward calculation: given  
$c \in C_n$ and $\xi \in E^{\ot n}$, then $c \xi \in E^{\ot n}$ and
$$
{T_\xi}^*cT_{\xi} = {T_\xi}^*T_{c\xi} = \io(\lb \xi, c\xi \rb_A) \in C_0.
$$
The second assertion follows from the embedding 
$C_n \emb \Lc(E^{\ot n})$ and the fact
$$
\| d \| = 
\sup\{ \| \lb \xi, d\xi \rb_A \| : \xi \in E^{\ot n}, \| \xi \| = 1 \} 
$$
for $d \in \Lc(E^{\ot n})$ positive.
\end{proof}

\begin{lem}\label{coup}
Given a positive element $a \in A$ and $\ep > 0$ with 
$\| a \| > \ep$, there is $\et \in E$ with 
$\| \et \| \le (\| a \| - \ep)^{-1/2}$ such that
${T_\et}^*\io(a)T_{\et} = 1$.
\end{lem}
\begin{proof}
Let $f$ be a continuous nonzero real-valued function supported on the
interval $[ \| a \| - \ep, \| a \| ]$ and choose a vector 
$\zt \in \pi(f(a))\Hf$ such that $\lb \zt, \pi(a)\zt \rb = 1$; 
we have 
$$
(\| a \| - \ep)\| \zt \|^2 \le \|\lb \zt, \pi(a)\zt \rb\| = 1.
$$
Then $\et = \zt \ot 1 \in E$ satisfies the desired conditions.
\end{proof}
It will now follow that $\Oc_E$ is simple and purely infinite (cf.\
proof of \cite[Theorem 2.1]{rr}). 

\begin{thm}\label{spi}
For every nonzero positive element $d \in \Oc_E$ there is a
$z \in \Oc_E$ so that $z^*dz = 1$.  Hence, $\Oc_E$ is simple and
purely infinite. 
\end{thm}
\begin{proof}
Let  $d \in \Oc_E$ be a nonzero positive element and choose $\ep$
so that $0 < \ep < \| P(d) \|/4$.  By Proposition \ref{duality} there
is a $\T$-equivariant isomorphism $\psi$ from $\Oc_E$ onto a corner of
$(\Oc_E \rt_{\lm} \T) \rt_{\wh\lm} \Z$ determined by a
projection $p \in \Oc_E \rt_{\lm} \T$.  We now apply Lemma \ref{key}
to the element $y = \psi(d)$ and the automorphism $\beta = \wh{\lm}_1$
(note $\Gm(\wh\lm_1) = \T$ by Lemma \ref{hyp}).  We identify
$\Oc_E$ with the  corner determined by $p$; note that under this
identification $\Fc_E$ is identified with $p(\Oc_E \rt_{\lm} \T)p$.
There are then positive elements $x, b \in \Fc_E$ so that 
$$
\| b \| > \| P(d) \| - \ep, \quad 
\| x \| \le 1 \quad \text{and} 
\quad \| xdx - b \| < \ep.
$$ 
Since $\cup_n C_n$ is dense in $\Fc_E$ we may
assume that $b \in C_n$ for some $n$. Hence, by Lemma \ref{easy} there
is $\xi \in E^{\ot n}$ with $\| \xi \| = 1$ such that 
$$
{T_\xi}^*bT_{\xi} \in C_0\quad \text{and} \quad 
\| {T_\xi}^*bT_{\xi} \| > \| b \| - \ep.
$$
Let $a$ denote the unique element of $A$ such that 
$\io(a) = {T_\xi}^*bT_{\xi}$; then $\| a \| > \| P(d) \| - 2\ep$ and 
$$
\| {T_\xi}^*xdxT_{\xi} - \io(a) \| =
 \| {T_\xi}^*(xdx - b)T_{\xi} \| < \ep.
$$
By Lemma \ref{coup} there is $\et \in E$
such that ${T_\et}^*\io(a)T_{\et} = 1$ and  
$$
\| \et \| \le (\| a \| - \ep)^{-1/2} 
< (\| P(d) \| - 3\ep)^{-1/2} < \ep^{-1/2}. 
$$
It follows that 
$$
\| {T_\et}^*{T_\xi}^*xdxT_{\xi}{T_\et} - 1 \| =
\| {T_\et}^*({T_\xi}^*xdxT_{\xi} - \io(a))T_\et \| \le
\| {T_\xi}^*xdxT_{\xi} - \io(a) \|(\ep^{-1/2})^2 < 1.
$$
Therefore, $c = {T_\et}^*{T_\xi}^*xdxT_{\xi}{T_\et}$ is an
invertible positive element and we take $z = xT_{\xi}{T_\et}c^{-1/2}$.
\end{proof}

\section{Applications and concluding remarks}\label{app}

We collect some applications of the above theorem and consider certain
connections with the theory of reduced (amalgamated) free product
C*-algebras. First we consider criteria under which the
Kirchberg-Phillips Theorem applies  
(see \cite[Theorem C]{kr}, \cite[Corollary 4.2.2]{ph}). 

\begin{thm}\label{main}
Let $A$ be a separable nuclear unital C*-algebra which belongs to the
bootstrap class to which the {\sc uct} applies (see \cite{rs}); let 
$\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a
separable Hilbert space $\Hf$ such that 
$\pi(A) \cap \Kc(\Hf) = \{ 0 \}$ 
and let $E$ denote the Hilbert $A$-bimodule $\Hf \ot_{\C} A$.
% Hilbert bimodule over $A$ constructed above (see Prop \ref{bimod}).  
Then $\Oc_E$ is a unital Kirchberg algebra (simple,
purely infinite, separable and nuclear) which belongs to the bootstrap
class.  
%% to which the {\sc uct} applies which is $KK$-equivalent to $A$.
Hence, the Kirchberg-Phillips Theorem applies and the isomorphism
class of $\Oc_E$ only depends on $(K_*(A), [1_A])$ and not on the
choice of representation $\pi$.  
\end{thm}
\begin{proof}
First note that $\Oc_E$ is simple and purely infinite by Theorem
\ref{spi}. If $A$ is nuclear, then the argument given in the proof of
\cite[Theorem 2.1]{ds} shows that $\Oc_E$ must also be nuclear
(alternatively, the nuclearity of $\Oc_E$ follows from the
structural results discussed in \S \ref{pre}).  Hence, $\Oc_E$ is a
unital Kirchberg algebra.  Recall that the inclusion $A \emb \Oc_E$
defines a $KK$-equivalence (see \cite[Corollary 4.5]{pm}) which induces
a unit-preserving isomorphism $K_*(A) \cong K_*(\Oc_E)$.
Hence, if $A$ is in the bootstrap class, then $\Oc_E$ is also. 
Therefore, the Kirchberg-Phillips Theorem applies and the isomorphism
class of $\Oc_E$ only depends on $(K_*(A), [1_A])$. 
% Given two faithful representations, $\pi$, $\pi'$, of $A$, the images
% of which contain no nontrivial compact operators, let $E$, $E'$ denote the
% corresponding Hilbert bimodules over $A$.  Since $\Oc_E$ and $\Oc_{E'}$
% are $KK$-equivalent (both are $KK$-equivalent to $A$) in a way which
% preserves the class of the unit, $\Oc_E \cong \Oc_{E'}$ by the 
% Kirchberg-Phillips Theorem (see \cite[Theorem C]{kr}, 
% \cite[Corollary 4.2.2]{ph}). 
\end{proof}

Let $X$ be a second countable compact space, let $\mu$ be a nonatomic
Borel measure with full support and let 
$$
\pi : C(X) \ra \Lc(L^2(X, \mu))
$$
be the representation given by multiplication of functions.  Then
$\pi$ is faithful and
$$
\pi(C(X)) \cap \Kc(L^2(X, \mu)) = \{ 0 \}.
$$
Hence, we may apply the
above theorem with $A = C(X)$ and $\Hf = L^2(X, \mu)$.

\begin{cor}\label{abel}
Let $X$ and $\mu$ be as above.  Then
$$
E = L^2(X, \mu) \ot_{\C} C(X)
$$ 
is a Hilbert bimodule over $C(X)$ and $\Oc_E$ is a unital Kirchberg algebra.
The embedding  $C(X) \emb \Oc_E$ induces a (unit preserving) 
$KK$-equivalence.  Hence, the isomorphism class of $\Oc_E$ only depends on
$(K_*(C(X)), [1_{C(X)}])$ (and not on $\mu$); moreover, if $X$ is
contractible, then $\Oc_E \cong \Oc_\infty$.
\end{cor}

The following proposition is Theorem 5.6 of \cite{l}
(see also \cite[Theorem 3]{ka}); Lance calls this the 
Kasparov-Stinespring-Gelfand-Naimark-Segal construction.

\begin{pro}
Let $B$ and $C$ be C*-algebras, let $F$ be a Hilbert $C$-module and
let $f : B \ra \Lc(F)$ be a completely positive map, then there
is a Hilbert $C$-module $E_f$, a $*$-homomorphism  
$\ph_f : B \ra \Lc(E_f)$ and an element $v_f \in \Lc(F, E_f)$ such
that $f(b) = {v_f}^*\ph_f(b)v_f$ and $\ph_f(B)v_fF$ is dense in
$E_f$. 
\end{pro}

I am grateful to D.~Shlyakhtenko for the following observation.  Let
$\Tc$ denote the ``usual'' Toeplitz algebra (i.e.\ $\Tc_E$ where $E$
is the $1$-dimensional Hilbert bimodule over $\C$) and let $g$ denote
the vacuum state on $\Tc$.

\begin{pro}
Let $A$ be a separable unital C*-algebra and let 
$\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a
separable Hilbert space $\Hf$ such that $\pi$ has a cyclic
vector $\xi \in \Hf$.  Let $f$ denote the vector state 
$\lb \xi, \pi(\cdot)\xi \rb$ and let $\tilde{f}$ denote the
corresponding completely positive map from $A$ to $\Lc(A)$ 
(given by $\tilde{f}(a) = f(a)1$).
Then $E = E_{\tilde{f}} \cong \Hf \ot A$ and $\Tc_{E}$ may be
realized as a reduced free product (see \cite{a,v}): 
$$
(\Tc_{E}, h) \cong (A, f) * (\Tc, g)
$$
for some state $h$ on $\Tc_{E}$.
\end{pro}
\begin{proof}
This follows from \cite[Theorem 2.3, Corollary 2.5]{sh}.
\end{proof}

As a result of this observation part (at least) of Corollary \ref{abel} follows
from the existing literature on reduced free products.  The simplicity
follows from a theorem of Dykema \cite[Theorem 2]{dy}.  Criteria for
when reduced free products are purely infinite have been found by
Choda, Dykema and R{\o}rdam in a series of papers \cite{dr1,dr2,dc};
but none seem to apply generally to the case considered in the corollary.

A theorem of Speicher (see \cite{sp}) on reduced amalgamated free
products (see \cite[\S 5]{v}) and Toeplitz algebras associated to Hilbert
bimodules yields a curious stability property of the algebras we have
been considering.  The following is the version given in 
\cite[Theorem 2.4]{bds}.
\begin{pro}
Suppose that $E_1$ and $E_2$ are full Hilbert bimodules over the
C*-algebra $A$.  Then
$$
\Tc_{E_1 \op E_2} = \Tc_{E_1} *_A \Tc_{E_2}.
$$
\end{pro}
We obtain the following corollary.  
\begin{cor}
Let $A$ be a separable nuclear unital C*-algebra which belongs to the
bootstrap class to which the {\sc uct} applies (see \cite{rs}) and let 
$\pi : A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a
separable Hilbert space $\Hf$ such that 
$\pi(A) \cap \Kc(\Hf) = \{ 0 \}$.  
Let $E$ be the Hilbert bimodule $\Hf \ot_\C A$.  Then
$$
\Oc_E \cong \Oc_E *_A \Oc_E.
$$
\end{cor}
\begin{proof}
Observe that $E \op E = (\Hf \op \Hf) \ot_{\C} A$.  Since 
$\pi \op \pi : A \ra \Lc(\Hf \op \Hf)$ is a faithful representation and
$
(\pi \op \pi)(A) \cap \Kc(\Hf \op \Hf) = \{ 0 \},
$
the result follows follows from Theorem \ref{main} and the above
proposition. 
\end{proof}

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%% conditions for simplicity include E is finitely genrated 
%% Connes spectrum of dual action = \T - see Th 4.7
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Math. Soc., to appear. %% Th. 2.1 for exactness of $\Tc_E$
\bibitem[E]{e}
George A. Elliott, {\em Some simple C*-algebras constructed as crossed
products with discrete outer automorphism groups},
Publ. Res. Inst. Math. Sci. \textbf{16} (1980) 299--311. 

\bibitem[FR]{fr}
N. J. Fowler and I. Raeburn, {\em The Toeplitz algebra of a
Hilbert bimodule}, Indiana Univ. Math. J. \textbf{48} (1999) 155--181. 
\bibitem[FRM]{frm}
N. J. Fowler, I. Raeburn and P. S. Muhly, 
{\em Representations of Cuntz-Pimsner C*-algebras}, preprint.
\bibitem[KPW1]{kpw1}
T. Kajiwara, C. Pinzari and Y. Watatani,
{\em Ideal structure and simplicity of the $C\sp *$-algebras generated
by Hilbert bimodules}, J. Funct. Anal. \textbf{159} (1998) 295--322.
\bibitem[KPW2]{kpw2}
%% conditions for simplicity when X finite projective
T. Kajiwara, C. Pinzari and Y. Watatani,
{\em Hilbert C*-bimodules and countably generated Cuntz-Krieger algebras}, 
J. Operator Theory \textbf{45} (2001) 3--18.
\bibitem[Ka]{ka}
Kasparov, G. G. 
{\em Hilbert C*-modules: theorems of Stinespring and Voiculescu},
J. Operator Theory \textbf{4} (1980) 133--150. 
%% Theorem 3 -- analog of Stinespring for stable modules.
\bibitem[Kr]{kr}
E. Kirchberg, {\em The classification of purely infinite C*-algebras
using Kasparov's theory}, Fields Institute Communications, to appear.
%%Theorem C
\bibitem[Ks]{ks} A. Kishimoto, {\em Outer automorphisms and reduced
crossed products of simple C*-algebras}, Comm. Math. Phys. \textbf{81}
(1981) 429--435. 
%% Lemma 3.2
\bibitem[L]{l}
E. C. Lance, {\em Hilbert C*-modules. A toolkit for operator
algebraists}, London Math. Soc. Lecture Note Series, \textbf{210}, 
Cambridge University Press, Cambridge, 1995.  
%% Th 5.6  strict is a weaker then nondegenerate
\bibitem[MS]{ms}
P. S. Muhly and B. Solel, 
{\em On the simplicity of some Cuntz-Pimsner algebras}, 
Math. Scand. \textbf{83} (1998) 53--73.
\bibitem[O]{o}
D. Olesen,  {\em Inner *-automorphisms of simple C*-algebras},
Comm. Math. Phys. \textbf{44} (1975) 175--190.
\bibitem[OP1]{op1}
D. Olesen and G. K. Pedersen,  
{\em Applications of the Connes Spectrum to C*-dynamical Systems}, 
J. Funct. Anal. \textbf{30} (1978) 179--197.
%Th 6.5 \Z-simple + \Gm = \T => A x \Z simple.
\bibitem[OP2]{op2}
D. Olesen and G. K. Pedersen,  
{\em Applications of the Connes Spectrum to C*-dynamical Systems, II}, 
J. Funct. Anal. \textbf{36} (1980) 18--32. %% Proposition 4.4.
\bibitem[OP3]{op3}
D. Olesen and G. K. Pedersen,  
{\em Applications of the Connes Spectrum to C*-dynamical Systems, III}, 
J. Funct. Anal. \textbf{45} (1982) 357--390.
%%Th 10.4  (i) \Gm(\al) = \T iff (v) \wh A has dense invariant set on which
%%\al acts freely iff (vi) for all n \ne 0 \al^n is properly outer.
\bibitem[Pa]{pa}
W. L. Paschke, 
{\em The crossed product of a C*-algebra by an endomorphism}, 
Proc. Amer. Math. Soc. \textbf{80} (1980) 113--118. %% see Th. 2
\bibitem[Pd]{pd}
G. K. Pedersen, {\em C*-algebras and their automorphism groups},
Academic Press, London, 1979. %% Takesaki-Takai duality Th 7.9.3
\bibitem[Ph]{ph}
N. C. Phillips, {\em A classification theorem for nuclear purely
infinite simple C*-algebras}, Doc. Math. \textbf{5} (2000) 49--114.
%% Cor. 4.2.2
\bibitem[Pm]{pm}  M. Pimsner, {\em A class of C*-algebras generalizing
both Cuntz-Krieger algebras and crossed products by $\Z$}, in ``Free
probability theory'' (D. Voiculescu, Ed.), 189-212, Fields Institute
Communications \textbf{12}, Amer. Math. Soc., Providence, 1997. 
\bibitem[PV]{pv}  M. Pimsner and D. Voiculescu,  
{\em Exact sequences for $K$-groups and Ext-groups of certain
cross-product C*-algebras},  
J. Operator Theory \textbf{4} (1980) 93--118. 
\bibitem[Pn]{pn}
C. Pinzari, {\em The ideal structure of Cuntz-Krieger-Pimsner algebras
and Cuntz-Krieger algebras over infinite matrices},  
in ``Operator Algebras and Quantum Field Theory (Accademia
Nazionale dei Lincei, Roma, July 1-6 1996)'' (S. Doplicher et al,
Eds.), 136--150, International Press 1997. 
%%R. Longo, J. Roberts and L. Zsido, Proceedings Rome Conference 

\bibitem[Ri]{ri}		
M. A. Rieffel, {\em Induced Representations of C*-algebras}, Advances Math. 
\textbf{13} (1974) 176-257.

\bibitem[R\o]{rr} 
M. R\o rdam, {\em Classification of certain infinite simple
C*-algebras, III}, in ``Operator algebras and their applications''
(P. Fillmore and J. Mingo, Eds.), 257--283, Fields Institute
Communications \textbf{13}, Amer. Math. Soc., Providence, 1997.
% Th 2.1 - Let A simp, sep, ex, ap div, fin, \al \in Aut A
% A x_\al Z sim pi iff A has no dd nonzero inv trace

\bibitem[Ro]{ro} 
J. Rosenberg, {\em Appendix to:\ ``Crossed products of UHF algebras by
product type actions,'' by O. Bratteli},
%% Duke Math. J. {textbf 46} (1979) 1--23  
Duke Math. J. \textbf{46} (1979) 25--26.
\bibitem[RS]{rs}
J. Rosenberg and C. Schochet, {\em The K\"unneth theorem and the
universal coefficient theorem for Kasparov's generalized $K$-functor},
Duke Math. J. \textbf{55} (1987) 431-474. 

\bibitem[Sc1]{sc1}
J. Schweizer, {\em Crossed products by C*-correspondences and Cuntz-Pimsner
algebras}, in ``C*-algebras, M\"unster, 1999'' (J. Cuntz and S. Echterhoff,
Eds.), 203--226, Springer Verlag, Berlin, 2000. 
\bibitem[Sc2]{sc2}
J. Schweizer, {\em Dilations of C*-correspondences and the simplicity of
Cuntz-Pimsner algebras}, J. Funct. Anal. \textbf{180} (2001) 404--425.
%% dilations seem to use a ``generalization'' of Takai duality?
%% Main Theorem 3.9
\bibitem[Sh]{sh}
D. Shlyakhtenko, {\em Some applications of freeness with
amalgamation}, J. Reine Angew. Math. \textbf{500} (1998), 191--212. 
\bibitem[Sp]{sp}
R. Speicher, {\em Combinatorial theory of the free product with
amalgamation and operator-valued free probability theory}, 
Mem. Amer. Math. Soc. \textbf{132}, no. 627 (1998).
\bibitem[T]{t}
H. Takai, {\em On a duality for crossed products of C*-algebras}, J.
Funct. Anal. \textbf{19} (1975), 25--39.
\bibitem[V]{v} 
D. Voiculescu, 
{\em Symmetries of some reduced free product C*-algebras}, 
in ``Operator algebras and their connections with topology and ergodic
theory, Bu\c{s}teni, 1983,'' (H. Araki et al, Eds.) 556--588, Lecture 
Notes in Math., 1132, Springer, Berlin, 1985. \bibitem[Z]{z}
J. Zacharias, {\em Quasi-free automorphisms of Cuntz-Krieger-Pimsner
algebras}, in ``C*-algebras, M\"unster 1999.'' (J. Cuntz and S. Echterhoff,
Eds.), 262--272, Springer Verlag, Berlin, 2000.
\end{thebibliography} 
\end{document}
Rosenberg, Jonathan Appendix to: ``Crossed products of UHF algebras by
product type actions'' [Duke Math. J. {46} (1979), no. 1, 1--23; MR
82a:46063 above] by O. Bratteli. Duke Math. J. 
46 (1979), no. 1, 25--26.
embedding of fixed point algebra as corner in crossed product.
\bibitem[]{}
\bibitem[]{}
\bibitem[Re]{re}
J. Renault, {\em A Groupoid approach to C*-algebras}, Lecture Notes in
Mathematics \textbf{793}, Springer-Verlag, Berlin, 1980. 
\bibitem[Ks1]{ks1} A. Kishimoto, 
{\em Simple crossed products by locally compact abelian groups}, 
Yokohama Math. J. \textbf{28} (1980) 69--85.
%% Prop 3.8, G discrete, A \al-simple => st Connes = Connes.

%\bibitem[KK]{kk} 
%A. Kishimoto and A. Kumjian, 
%{\em Crossed products of Cuntz algebras by quasi-free automorphisms}, 
%in ``Operator algebras and their applications''
%(P. Fillmore and J. Mingo, Eds.), 173--192, Fields Institute
%Communications \textbf{13}, Amer. Math. Soc., Providence, 1997.
%% Lemma 10

\begin{rem}\label{irr}
With $E$ as above and $I_n$ as in
Proposition \ref{id}, if $\pi$ is an irreducible representation of 
$\Oc_E \rt_{\lm} \T$ with $[\pi] \in \wh{I_n}$, then the closure of
$[\pi]$ contains the complement of $\wh{I_n}$.  It follows that if
$I$ is any ideal in $\Oc_E \rt_{\lm} \T$ then for all $n$, 
either $I \sub I_n$ or $I_n \sub I$.
\end{rem}
If $B$ is  $\bt$-simple and $\Gm(\bt) = \T$ then
the crossed product $B \rt \Z$ is simple 
(see \cite[Theorem 6.5]{op1}).
math0108194.extracted_bib
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D. Avitzour, {\em Free products of C*-algebras}, 
Trans. Amer. Math. Soc. \textbf{271} (1982) 423--435. 
\bibitem[BDS]{bds}
N. Brown, K. Dykema and D. Shlyakhtenko, 
{\em Topological Entropy of Free Product Automorphisms}, preprint. 
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A. Connes, {\em Une classification des facteurs de type III},
Ann. Sci. \'{E}cole Norm. Sup. \textbf{6} (1973) 133--252.
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J. Cuntz and W. Krieger, {\em A class of C*-algebras and
topological Markov  chains}, Invent. Math. \textbf{56} (1980) 251-268. 
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V. Deaconu, {\em Generalized solenoids and C*-algebras},
Pacific J. Math. \textbf{190} (1999),  247--260. 
%\bibitem[DKM]{dkm}
% V. Deaconu, A. Kumjian and P. Muhly, 
% {\em Cohomology of topological graphs and Cuntz-Pimsner algebras},
% J. Operator Theory, in press.
\bibitem[DPW]{dpz}
Doplicher, C. Pinzari, and R. Zuccante, {\em The C*-algebra of a
Hilbert bimodule}, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)
\textbf{1} (1998) 263--281.
%% conditions for simplicity include E is finitely genrated 
%% Connes spectrum of dual action = \T - see Th 4.7
\bibitem[Dy]{dy}
K. Dykema, {\em Simplicity and the stable rank of some free product
C*-algebras}, Trans. Amer. Math. Soc. \textbf{351} (1999) 1--40.
%%See Theorem 2: If states give faithful rep and centralizer of first
%%contains copy of C(X) on which state is atomless, then reduced free
%%product is simple (C*-algebras are unital).
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K. Dykema and M. Choda, {\em Purely infinite, simple C*-algebras
arising from free product constructions, III},  
Proc. Amer. Math. Soc. \textbf{128} (2000) 3269--3273.
\bibitem[DR1]{dr1}
K. Dykema and M. R\o rdam, {\em Purely infinite, simple C*-algebras
arising from free product constructions},   
Canad. J. Math. \textbf{50} (1998) 323--341.
\bibitem[DR2]{dr2}
K. Dykema and M. R\o rdam, {\em Purely infinite, simple C*-algebras
arising from free product constructions, II}, preprint.
\bibitem[DS]{ds}
K. Dykema and D. Shlyakhtenko, 
{\em Exactness of Cuntz-Pimsner C*-algebras}, Proc. Edinburgh
Math. Soc., to appear. %% Th. 2.1 for exactness of $\Tc_E$
\bibitem[E]{e}
George A. Elliott, {\em Some simple C*-algebras constructed as crossed
products with discrete outer automorphism groups},
Publ. Res. Inst. Math. Sci. \textbf{16} (1980) 299--311. 

\bibitem[FR]{fr}
N. J. Fowler and I. Raeburn, {\em The Toeplitz algebra of a
Hilbert bimodule}, Indiana Univ. Math. J. \textbf{48} (1999) 155--181. 
\bibitem[FRM]{frm}
N. J. Fowler, I. Raeburn and P. S. Muhly, 
{\em Representations of Cuntz-Pimsner C*-algebras}, preprint.
\bibitem[KPW1]{kpw1}
T. Kajiwara, C. Pinzari and Y. Watatani,
{\em Ideal structure and simplicity of the $C\sp *$-algebras generated
by Hilbert bimodules}, J. Funct. Anal. \textbf{159} (1998) 295--322.
\bibitem[KPW2]{kpw2}
%% conditions for simplicity when X finite projective
T. Kajiwara, C. Pinzari and Y. Watatani,
{\em Hilbert C*-bimodules and countably generated Cuntz-Krieger algebras}, 
J. Operator Theory \textbf{45} (2001) 3--18.
\bibitem[Ka]{ka}
Kasparov, G. G. 
{\em Hilbert C*-modules: theorems of Stinespring and Voiculescu},
J. Operator Theory \textbf{4} (1980) 133--150. 
%% Theorem 3 -- analog of Stinespring for stable modules.
\bibitem[Kr]{kr}
E. Kirchberg, {\em The classification of purely infinite C*-algebras
using Kasparov's theory}, Fields Institute Communications, to appear.
%%Theorem C
\bibitem[Ks]{ks} A. Kishimoto, {\em Outer automorphisms and reduced
crossed products of simple C*-algebras}, Comm. Math. Phys. \textbf{81}
(1981) 429--435. 
%% Lemma 3.2
\bibitem[L]{l}
E. C. Lance, {\em Hilbert C*-modules. A toolkit for operator
algebraists}, London Math. Soc. Lecture Note Series, \textbf{210}, 
Cambridge University Press, Cambridge, 1995.  
%% Th 5.6  strict is a weaker then nondegenerate
\bibitem[MS]{ms}
P. S. Muhly and B. Solel, 
{\em On the simplicity of some Cuntz-Pimsner algebras}, 
Math. Scand. \textbf{83} (1998) 53--73.
\bibitem[O]{o}
D. Olesen,  {\em Inner *-automorphisms of simple C*-algebras},
Comm. Math. Phys. \textbf{44} (1975) 175--190.
\bibitem[OP1]{op1}
D. Olesen and G. K. Pedersen,  
{\em Applications of the Connes Spectrum to C*-dynamical Systems}, 
J. Funct. Anal. \textbf{30} (1978) 179--197.
%Th 6.5 \Z-simple + \Gm = \T => A x \Z simple.
\bibitem[OP2]{op2}
D. Olesen and G. K. Pedersen,  
{\em Applications of the Connes Spectrum to C*-dynamical Systems, II}, 
J. Funct. Anal. \textbf{36} (1980) 18--32. %% Proposition 4.4.
\bibitem[OP3]{op3}
D. Olesen and G. K. Pedersen,  
{\em Applications of the Connes Spectrum to C*-dynamical Systems, III}, 
J. Funct. Anal. \textbf{45} (1982) 357--390.
%%Th 10.4  (i) \Gm(\al) = \T iff (v) \wh A has dense invariant set on which
%%\al acts freely iff (vi) for all n \ne 0 \al^n is properly outer.
\bibitem[Pa]{pa}
W. L. Paschke, 
{\em The crossed product of a C*-algebra by an endomorphism}, 
Proc. Amer. Math. Soc. \textbf{80} (1980) 113--118. %% see Th. 2
\bibitem[Pd]{pd}
G. K. Pedersen, {\em C*-algebras and their automorphism groups},
Academic Press, London, 1979. %% Takesaki-Takai duality Th 7.9.3
\bibitem[Ph]{ph}
N. C. Phillips, {\em A classification theorem for nuclear purely
infinite simple C*-algebras}, Doc. Math. \textbf{5} (2000) 49--114.
%% Cor. 4.2.2
\bibitem[Pm]{pm}  M. Pimsner, {\em A class of C*-algebras generalizing
both Cuntz-Krieger algebras and crossed products by $\Z$}, in ``Free
probability theory'' (D. Voiculescu, Ed.), 189-212, Fields Institute
Communications \textbf{12}, Amer. Math. Soc., Providence, 1997. 
\bibitem[PV]{pv}  M. Pimsner and D. Voiculescu,  
{\em Exact sequences for $K$-groups and Ext-groups of certain
cross-product C*-algebras},  
J. Operator Theory \textbf{4} (1980) 93--118. 
\bibitem[Pn]{pn}
C. Pinzari, {\em The ideal structure of Cuntz-Krieger-Pimsner algebras
and Cuntz-Krieger algebras over infinite matrices},  
in ``Operator Algebras and Quantum Field Theory (Accademia
Nazionale dei Lincei, Roma, July 1-6 1996)'' (S. Doplicher et al,
Eds.), 136--150, International Press 1997. 
%%R. Longo, J. Roberts and L. Zsido, Proceedings Rome Conference 

\bibitem[Ri]{ri}		
M. A. Rieffel, {\em Induced Representations of C*-algebras}, Advances Math. 
\textbf{13} (1974) 176-257.

\bibitem[R\o]{rr} 
M. R\o rdam, {\em Classification of certain infinite simple
C*-algebras, III}, in ``Operator algebras and their applications''
(P. Fillmore and J. Mingo, Eds.), 257--283, Fields Institute
Communications \textbf{13}, Amer. Math. Soc., Providence, 1997.
% Th 2.1 - Let A simp, sep, ex, ap div, fin, \al \in Aut A
% A x_\al Z sim pi iff A has no dd nonzero inv trace

\bibitem[Ro]{ro} 
J. Rosenberg, {\em Appendix to:\ ``Crossed products of UHF algebras by
product type actions,'' by O. Bratteli},
%% Duke Math. J. {textbf 46} (1979) 1--23  
Duke Math. J. \textbf{46} (1979) 25--26.
\bibitem[RS]{rs}
J. Rosenberg and C. Schochet, {\em The K\"unneth theorem and the
universal coefficient theorem for Kasparov's generalized $K$-functor},
Duke Math. J. \textbf{55} (1987) 431-474. 

\bibitem[Sc1]{sc1}
J. Schweizer, {\em Crossed products by C*-correspondences and Cuntz-Pimsner
algebras}, in ``C*-algebras, M\"unster, 1999'' (J. Cuntz and S. Echterhoff,
Eds.), 203--226, Springer Verlag, Berlin, 2000. 
\bibitem[Sc2]{sc2}
J. Schweizer, {\em Dilations of C*-correspondences and the simplicity of
Cuntz-Pimsner algebras}, J. Funct. Anal. \textbf{180} (2001) 404--425.
%% dilations seem to use a ``generalization'' of Takai duality?
%% Main Theorem 3.9
\bibitem[Sh]{sh}
D. Shlyakhtenko, {\em Some applications of freeness with
amalgamation}, J. Reine Angew. Math. \textbf{500} (1998), 191--212. 
\bibitem[Sp]{sp}
R. Speicher, {\em Combinatorial theory of the free product with
amalgamation and operator-valued free probability theory}, 
Mem. Amer. Math. Soc. \textbf{132}, no. 627 (1998).
\bibitem[T]{t}
H. Takai, {\em On a duality for crossed products of C*-algebras}, J.
Funct. Anal. \textbf{19} (1975), 25--39.
\bibitem[V]{v} 
D. Voiculescu, 
{\em Symmetries of some reduced free product C*-algebras}, 
in ``Operator algebras and their connections with topology and ergodic
theory, Bu\c{s}teni, 1983,'' (H. Araki et al, Eds.) 556--588, Lecture 
Notes in Math., 1132, Springer, Berlin, 1985. \bibitem[Z]{z}
J. Zacharias, {\em Quasi-free automorphisms of Cuntz-Krieger-Pimsner
algebras}, in ``C*-algebras, M\"unster 1999.'' (J. Cuntz and S. Echterhoff,
Eds.), 262--272, Springer Verlag, Berlin, 2000.
\end{thebibliography}