algebraic formulation of quantum field theory [Haa96]. Typically, the
splitting property holds in algebraic quantum field theories because certain
"partition function" is finite for all "temperature" [BDL90a, BDL90b]. In the
case of Poisson algebras, the analogs of these "partition functions" are
rarely finite, but splitting property hold in general when the algebras are
hyperfinite. This is due to the special properties of Poisson algebras.

Lastly, we calculate the quantum relative entropy [Ara76, Ara77] of some
states on the Poisson algebra. When the algebra is type III, it is generally
impossible to calculate the relative entropy for a pair of arbitrary states.
Here our calculation holds when one state is the Poisson state and the other
is a bounded perturbation of the Poisson state. The formula relates the
(Araki) relative entropy of the Poisson algebra $\mathbb{P}_{\omega}N$ to the
Lindblad relative entropy [Lin74] of the original von Neumann algebra $N$.
Since the original von Neumann algebra $N$ is typically semi-finite in actual
applications, the Lindblad entropy on $N$ can be easily calculated using well-
known formula. This result will be extended and studied in depth in a
companion paper.

###  4.1. Factoriality and Type of Poisson Algebras

In this subsection, we determine the type of some Poisson algebras. The key
observation is the following proposition.

######  Proposition 4.1.

The Poisson algebra $\mathbb{P}_{\operatorname{tr}}\mathbb{B}(\ell_{2})$ is
the hyperfinite $II_{1}$ factor where $\ell_{2}$ denotes the separably
infinite dimensional Hilbert space $\ell_{2}(\mathbb{Z})$.

###### Proof.

From straight-forward calculations, it is clear that the Poisson state
$\phi_{\operatorname{tr}}$ is a tracial state on
$\mathbb{P}_{\operatorname{tr}}\mathbb{B}(\ell_{2})$. Therefore we only need
to show that the algebra is a factor.

To see this we need to consider the following sequence of subalgebras:
$\\{\mathbb{P}_{\operatorname{tr}_{k}}\mathbb{B}(\ell_{2}^{k})\\}_{k\geq 1}$
where $\ell_{2}^{k}\\!\\!\subset\\!\\!\ell_{2}$ is the $k$-dimensional
subspace of $\ell_{2}$ and $\operatorname{tr}_{k}$ is the canonical trace on
the subalgebra $\mathbb{B}(\ell^{k}_{2})$. Let
$\\{e_{k}\in\mathbb{B}(\ell_{2})\\}_{k}$ be the increasing family of
orthogonal projections such that $e_{k}\ell_{2}=\ell^{k}_{2}$. Then for each
pair $k<n$, there exists a normal $*$-homomorphism:

(4.1)
$\pi_{n,k}:\mathbb{P}_{\operatorname{tr}_{k}}\mathbb{B}(\ell^{k}_{2})\rightarrow\mathbb{P}_{\operatorname{tr}_{n}}\mathbb{B}(\ell_{2}^{n}):\Gamma(x)\mapsto\Gamma(x+e_{n}-e_{k})\hskip
2.84544pt.$

In addition,
$\phi_{\operatorname{tr}_{n}}\circ\pi_{n,k}=\phi_{\operatorname{tr}_{k}}$:

(4.2)
$\phi_{\operatorname{tr}_{n}}(\pi_{n,k}(\Gamma(x)))=\phi_{\operatorname{tr}_{n}}(\Gamma(x+e_{n}-e_{k}))=e^{\operatorname{tr}_{n}(x+e_{n}-e_{k}-e_{n})}=e^{\operatorname{tr}_{k}(x-e_{k})}=\phi_{\operatorname{tr}_{k}}(\Gamma(x))\hskip
2.84544pt.$

Hence the Poisson algebra $\mathbb{P}_{\operatorname{tr}}\mathbb{B}(\ell_{2})$
is isomorphic to the inductive limit of
$\\{(\mathbb{P}_{\operatorname{tr}_{k}}\mathbb{B}(\ell^{k}_{2}),\phi_{\operatorname{tr}_{k}})\\}_{k\geq
1}$.

On $\mathbb{B}(\ell_{2})$, we fix a sequence of unitaries weakly converging to
0: $\\{u_{2n}:=\rm{id}_{n}\otimes\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\in\mathbb{B}(\ell^{2n}_{2})\\}_{n\geq 1}$. Then on the weakly
dense $*$-subalgebra $\cup_{k\geq
1}\mathbb{P}_{\operatorname{tr}_{k}}\mathbb{B}(\ell^{k}_{2})\\!\\!\subset\\!\\!\mathbb{P}_{\operatorname{tr}}\mathbb{B}(\ell_{2})$,
we consider the following sesquilinear form:

(4.3)
$\big{(}\Gamma(x),\Gamma(y))\big{)}_{u}:=\lim_{n\rightarrow\infty}\phi_{\operatorname{tr}_{2n}}(\Gamma(x^{*})\Gamma(u^{*}_{2n})\Gamma(y)\Gamma(u_{2n}))\hskip
2.84544pt.$

For any pair of $\Gamma(x),\Gamma(y)$ in the dense subalgebra, by the upwards
closedness of the inductive sequence, there exists $m\in\mathbb{N}$ such that
$\Gamma(x),\Gamma(y)\in\mathbb{P}_{\operatorname{tr}_{m}}\mathbb{B}(\ell_{2}^{m})$.
Then for all $n\in\mathbb{N}$ such that $n>m$, we have the following
factorization:

(4.4)
$\displaystyle\begin{split}\phi&{}_{\operatorname{tr}_{2n}}(\Gamma(x^{*})\Gamma(u^{*}_{2n})\Gamma(y)\Gamma(u_{2n}))=\exp\operatorname{tr}_{2n}\big{(}(x^{*}+e_{2n}-e_{m})u^{*}_{2n}(y+e_{2n}-e_{m})u_{2n}-e_{2n}\big{)}\\\
&=\exp\operatorname{tr}_{2n}\big{(}\begin{pmatrix}x^{*}+e_{n}-e_{m}&0\\\
0&e_{n}\end{pmatrix}\begin{pmatrix}0&e_{n}\\\
e_{n}&0\end{pmatrix}\begin{pmatrix}y+e_{n}-e_{m}&0\\\
0&e_{n}\end{pmatrix}\begin{pmatrix}0&e_{n}\\\
e_{n}&0\end{pmatrix}-e_{2n}\big{)}\\\
&=\exp\big{(}\operatorname{tr}_{m}(x^{*}-e_{m})\big{)}\exp\big{(}\operatorname{tr}_{m}(y-e_{m})\big{)}=\phi_{\operatorname{tr}_{m}}(\Gamma(x^{*}))\phi_{\operatorname{tr}_{m}}(\Gamma(y))=\phi_{\operatorname{tr}}(\Gamma(x^{*}))\phi_{\operatorname{tr}}(\Gamma(y))\end{split}$

where the last equation follows from the state-preserving property of the maps
$\\{\pi_{n,m}\\}_{n\in\mathbb{N}}$. Hence on the dense subalgebra $\cup_{k\geq
1}\mathbb{P}_{\operatorname{tr}_{k}}\mathbb{B}(\ell^{k}_{2})$, the
sesquilinear form factorizes:

(4.5)
$\big{(}\Gamma(x),\Gamma(y)\big{)}_{u}=\phi_{\operatorname{tr}}(\Gamma(x^{*}))\phi_{\operatorname{tr}}(\Gamma(y))\hskip
2.84544pt.$

Now suppose $\Gamma(y)$ is in the center of the Poisson algebra
$\mathbb{P}_{\operatorname{tr}}\mathbb{B}(\ell_{2})$, we can calculate the
sesquilinear form in another way. For each $n\in\mathbb{N}$, we have:

(4.6)
$\phi_{\operatorname{tr}_{2n}}(\Gamma(x^{*})\Gamma(u_{2n}^{*})\Gamma(y)\Gamma(u_{2n}))=\phi_{\operatorname{tr}_{2n}}(\Gamma(x^{*})\Gamma(y)\Gamma(u^{*}_{2n})\Gamma(u_{2n}))=\phi_{\operatorname{tr}_{2n}}(\Gamma(x^{*})\Gamma(y))\hskip
2.84544pt.$

Again by the state-preserving property of the embeddings
$\\{\pi_{n,m}\\}_{n,m}$, we have:

(4.7)
$\big{(}\Gamma(x),\Gamma(y)\big{)}_{u}=\phi_{\operatorname{tr}}(\Gamma(x^{*})\Gamma(y))\hskip
2.84544pt.$

Since the equation
$\phi_{\operatorname{tr}}(\Gamma(x^{*})\Gamma(y))=\phi_{\operatorname{tr}}(\Gamma(x^{*}))\phi_{\operatorname{tr}}(\Gamma(y))$
holds for all $\Gamma(x^{*})$ in the dense subalgebra,
$\Gamma(y)=\phi_{\operatorname{tr}}(\Gamma(y))1$. Hence the center of the
Poisson algebra is trivial. ∎

######  Corollary 4.1.

Using the same notation as Proposition 4.1 and let $\omega$ be a n.s.f. weight
on a von Neumann algebra $N$. Then the Poisson algebra
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))$
is a factor and its centralizer is also a factor
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N_{\omega}\overline{\otimes}\mathbb{B}(\ell_{2}))$.
Consequently, the type of the Poisson algebra
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))$
is determined by the Arveson spectrum $Sp(\Delta_{\phi_{\omega}})$.

###### Proof.

Using the same construction as in Proposition 4.1 and replacing $\\{u_{2n}\\}$
by $\\{1\otimes u_{2n}\\}$, we have that
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))$
is a factor. The centralizer of the Poisson algebra is a von Neumann algebra
generated by
$\\{\Gamma(e^{ix}):e^{ix}\in(N\overline{\otimes}\mathbb{B}(\ell_{2}))_{\omega\otimes\operatorname{tr}},e^{ix}\in\mathcal{U}^{\omega\otimes\operatorname{tr}}_{ana}\\}$
where $e^{ix}$ is a unitary in the centralizer
$(N\overline{\otimes}\mathbb{B}(\ell_{2}))_{\omega\otimes\operatorname{tr}}\cong
N_{\omega}\overline{\otimes}\mathbb{B}(\ell_{2})$. Hence it is isomorphic to
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N_{\omega}\overline{\otimes}\mathbb{B}(\ell_{2}))$
and it is a factor.

It is well-known that the Arveson spectrum
$Sp(\Delta_{\phi_{\omega\otimes\operatorname{tr}}})$ is minimal if the
centralizer is a factor [Tak03]. Hence the Connes invariant of the Poisson
algebra is given by the group generated by the Arveson spectrum of the Poisson
state:

$S(\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2})))=\Gamma\big{(}Sp(\Delta_{\phi_{\omega\otimes\operatorname{tr}}})-\\{0\\}\big{)}\hskip
2.84544pt.$

Since the modular automorphism group of the Poisson state
$\phi_{\omega\otimes\operatorname{tr}}$ is a tensor product:
$\sigma^{\phi_{\omega}}_{t}\otimes id=\Gamma(\sigma^{\omega}_{t})\otimes id$,
the Arveson spectrum equals to its restriction to the first component:
$Sp(\Delta_{\phi_{\omega}})$. Lastly, since the $L_{2}$-space of the Poisson
algebra is isomorphic to the symmetric Fock space of the $L_{2}$-space of the
original algebra, the multiplicative subgroup generated by
$Sp(\Delta_{\phi_{\omega}})$ equals to the multiplicative subgroup generated
by $Sp(\Delta_{\omega})$. Therefore the type of the Poisson algebra
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))$
is determined by the spectrum $Sp(\Delta_{\omega})$. ∎

###  4.2. Type III Algebras from Principal Series Representation

Using the result on the type of Poisson algebra, in this section we construct
examples of type IIIλ Poisson factors from unitary principal series
representations. In addition, we will present a framework to construct
algebraic quantum field theories using unitary principal series
representations and Poissonization. The examples from this construction
satisfy certain axioms in AQFT [Haa96]. A dedicated study of Poissonization
and AQFT with more constructions will come in a separate paper.

####  4.2.1. Unitary Principal Series Representations and Parabolic Induction

We first recall the construction of unitary principal series representation as
an induced representation [Kna86, Ber92, CH15, Rep00]. Let $G$ be a real
semisimple Lie group and let $G\cong KAN$ be the Iwasawa decomposition of $G$.
Let $M$ be the centralizer of $A$ in $K$, then $P=MAN$ is a minimal parabolic
subgroup of $G$. In addition, let $\mathfrak{a}$ be the Lie algebra of $A$
such that $A=\exp(\mathfrak{a})$. Let $\Sigma^{+}$ be the set of positive
roots of $\mathfrak{a}$ in the Lie algebra $\mathfrak{g}$. Fix a purely
imaginary root $\nu\in\mathfrak{a}^{*}$ and fix a unitary irreducible
representation $\sigma:M\rightarrow U(H_{\nu})$ (since $K$ is compact, $M$ is
compact and $H_{\nu}$ is a finite dimensional Hilbert space), then there
exists a unitary representation of the parabolic group $P$ on $H_{\nu}$:

(4.8) $\mu_{P}:=\sigma\otimes e^{\nu}\otimes id:P=MAN\rightarrow
U(H_{\nu}):man\mapsto\sigma(m)e^{\nu(\log a)}$

where $\log a\in\mathfrak{a}$ is the generator of $a\in A$. Let
$\rho_{G}:=\frac{1}{2}\sum_{\alpha\in\Sigma^{+}}\alpha$ be the half-sum of the
roots in $\Sigma^{+}$. Then the induced unitary principal series
representation of $G$ acts on the Hilbert space of half-densities over $G/P$
with value in the Hilbert space $H_{\nu}$:

(4.9) $\Gamma^{2}(G/P,H_{\nu})\cong\\{F:G\rightarrow
H_{\nu}:F(gman)=\sigma(m^{-1})e^{-(\nu+\rho_{G})(\log
a)}F(g),\int_{G/P}|F|^{2}_{H_{\nu}}<\infty\\}$

where $G/P\cong K/M\cap K$ is a compact manifold and $|\cdot|_{H_{\nu}}$ is
the $P$-invariant inner-product on $H_{\nu}$. The integration on $G/P$ is over
the density $|F|^{2}_{H_{\nu}}$. The group acts on the half-density sections
by left-multiplication: $\mathcal{P}^{\sigma,\nu}(g)F(h):=F(g^{-1}h)$. If the
representation $\sigma$ is irreducible, then $\mathcal{P}^{\sigma,\nu}$ is
also irreducible. This procedure of constructing irreducible unitary
representations is called parabolic induction.

####  4.2.2. Type III Poisson Factors from Parabolic Induction

Given a irreducible unitary principal series representation
$\mathcal{P}^{\sigma,\nu}$ of a real semisimple Lie group $G$, we construct a
type IIIλ Poisson factor with strongly continuous $G$-action. Let
$\mathcal{P}^{\sigma,\nu}$ be an irreducible unitary principal series
representation of $G$ induced by the finite dimensional unitary
representation: $\mu_{P}:=\sigma\otimes e^{\nu}\otimes id$ (i.e.
$\mathcal{P}^{\sigma,\nu}=Ind^{G}_{P}(\mu_{P})$). Fix an element $a\in A$ and
let $\log\delta_{\nu}$ be the generator of the unitary $\mu_{P}(a)$ (i.e.
$\delta_{\nu}^{it}=\mu_{P}(e^{t\log a})$ and hence $\log\delta_{\nu}=|\nu(\log
a)|$). And consider the weight $\omega_{\nu}=\operatorname{tr}(\delta\cdot)$
on $\mathbb{B}(H_{\nu})$. Then we have:

######  Lemma 4.2.

The following statements are true:

  1. (1)

$\mathbb{P}_{\omega_{\nu}\otimes\operatorname{tr}}\mathbb{B}(H_{\nu}\otimes\ell_{2})$
is the hyperfinite II1 factor;

  2. (2)

Fix a unitary irreducible representation $\sigma:M\rightarrow U(H_{\sigma})$.
For two different purely imaginary roots $\nu,\mu\in\mathfrak{a}^{*}$,
$\mathbb{B}(H_{\nu}\oplus H_{\mu})=\mathbb{B}(H_{\sigma})\otimes
M_{2}(\mathbb{C})$ and the normalization of the weight
$\operatorname{tr}(\delta_{\nu}\oplus\delta_{\mu}\cdot)$ is given by
$\operatorname{tr}\otimes\operatorname{tr}_{2}(\begin{pmatrix}\frac{e^{|\nu(\log
a)|}}{e^{|\nu(\log a)|}+e^{|\mu(\log a)|}}&0\\\ 0&\frac{e^{|\mu(\log
a)|}}{e^{|\nu(\log a)|}+e^{|\mu(\log a)|}}\end{pmatrix}\cdot)$. Denote this
weight $\omega_{\nu,\mu}$. Then if $\lambda=\min\\{e^{|\nu\log a|-|\mu\log
a|},e^{|\mu\log a|-|\nu\log a|}\\}\in(0,1)$, the Poisson algebra
$\mathbb{P}_{\omega_{\nu,\mu}\otimes\operatorname{tr}}\mathbb{B}\big{(}(H_{\nu}\oplus
H_{\mu})\otimes\ell_{2}\big{)}$ is a type IIIλ factor;

  3. (3)

For three different purely imaginary roots $\nu,\mu,\eta\in\mathfrak{a}^{*}$,
$\mathbb{B}(H_{\nu}\oplus H_{\mu}\oplus
H_{\eta})=\mathbb{B}(H_{\sigma})\otimes M_{3}(\mathbb{C})$ and the
normalization of the weight
$\operatorname{tr}(\delta_{\nu}\oplus\delta_{\mu}\oplus\delta_{\eta}\cdot)$ is
given by

$\operatorname{tr}\otimes\operatorname{tr}_{3}(\begin{pmatrix}\frac{e^{|\nu\log
a|}}{e^{|\nu\log a|}+e^{|\mu\log a|}+e^{|\eta\log a|}}&0&0\\\
0&\frac{e^{|\mu\log a|}}{e^{|\nu\log a|}+e^{|\mu\log a|}+e^{|\eta\log
a|}}&0\\\ 0&0&\frac{e^{|\eta\log a|}}{e^{|\nu\log a|}+e^{|\mu\log
a|}+e^{|\eta\log a|}}\end{pmatrix}\cdot)\hskip 2.84544pt.$

And the Poisson algebra
$\mathbb{P}_{\omega_{\nu,\mu,\eta}\otimes\operatorname{tr}}\mathbb{B}\big{(}(H_{\nu}\oplus
H_{\mu}\oplus H_{\eta})\otimes\ell_{2}\big{)}$ is of type III1 if the triplets
$\lambda_{\nu,\mu}:=\min\\{e^{|\nu\log a|-|\mu\log a|},e^{|\mu\log a|-|\nu\log
a|}\\}$, $\lambda_{\nu,\eta}:=\min\\{e^{|\nu\log a|-|\eta\log a|},e^{|\eta\log
a|-|\nu\log a|}\\}$ and $\lambda_{\mu,\eta}:=\min\\{e^{|\eta\log a|-|\mu\log
a|},e^{|\mu\log a|-|\eta\log a|}\\}$ generate the multiplicative group
$\mathbb{R}_{+}$.

######  Remark 4.3.

Therefore for generic pairs of irreducible unitary principal series
representations, the induced Poisson algebras are of type IIIλ. And for
generic multiplets of irreducible unitary principal series representations,
the induced Poisson algebras are of type III1.

The construction we gave here is nothing but an adaptation of the Araki-Woods
factor to the Poissonization functor.

###### Proof.

The key observation is that the weight $\omega_{\nu}$ is invariant under
$P$-action. Using the Levi decomposition of $P=LN$, $L$-action preserves
$\omega_{\nu}$ since $A$ is Abelian and $M$ commutes with $A$. Since $N$ acts
trivially on $H_{\nu}$, $\omega_{\nu}$ is preserved by the entire $P$-action.
Therefore the automorphism on $\mathbb{B}(H_{\nu}\otimes\ell_{2})$ given by
the adjoint action $\mu_{P}(g)\cdot\mu_{P}(g)^{*}$ (where $g\in P$) preserves
the weight $\omega_{\nu}$. Hence by Proposition 3.6, there exists a strongly
continuous $P$-action on the Poisson algebra
$\mathbb{P}_{\omega_{\nu}\otimes\operatorname{tr}}\mathbb{B}(H_{\nu}\otimes\ell_{2})$.
The unitary implementation of this automorphism gives a unitary representation
of $P$ on the Haagerup $L_{2}$-space of the Poisson algebra:

(4.10) $U_{\Gamma(\mu_{P})}:P\rightarrow
U(L_{2}(\mathbb{P}_{\omega_{\nu}\otimes\operatorname{tr}}\mathbb{B}(H_{\nu}\otimes\ell_{2}),\phi_{\omega_{\nu}\otimes\operatorname{tr}}))\hskip
2.84544pt.$

In addition, by Corollary 4.1 the Poisson algebra is a factor. Then all three
statements follow directly from a simple calculation of the corresponding
Arveson spectrums. ∎

######  Corollary 4.4.

The unitary representation $U_{\Gamma(\mu_{P})}$ constructed in the proof of
Lemma 4.2 induces a unitary representation of $G$ via parabolic induction. In
addition, this group action induces a strongly continuous automorphism on the
Poisson algebra:

(4.11)
$\mathbb{B}(L^{2}(G/P,\Omega_{P}^{1/2}))\overline{\otimes}\mathbb{P}_{\omega_{\nu,\mu}\otimes\operatorname{tr}}\mathbb{B}((H_{\nu}\oplus
H_{\mu})\otimes\ell_{2})\cong\mathbb{P}_{\omega_{\nu,\mu}\otimes\operatorname{tr}}\mathbb{B}((H_{\nu}\oplus
H_{\mu})\otimes\ell_{2})$

where the Hilbert space
$L^{2}(G/P,\Omega_{P}^{1/2}):=\\{f:G\rightarrow\mathbb{C}:f(gman)=e^{-\rho_{G}(\log
a)}f(g),\int_{G/P}|f|^{2}<\infty\\}$ is the space of half-densities over
$G/P$. The isomorphism of the von Neumann algebras follows from the absorption
property of type III factors.

###### Proof.

To see that we have an induced representation, we need to calculate that the
unitary representation $U_{\Gamma(\mu_{P})}$:

(4.12)
$\displaystyle\begin{split}U_{\Gamma(\mu_{P})}(man)\lambda_{\emptyset\emptyset}(x_{1},...,x_{n})\phi_{\omega_{\nu}\otimes\operatorname{tr}}^{1/2}&=\lambda_{\emptyset\emptyset}(Ad_{\mu_{P}(man)}x_{1},...,Ad_{\mu_{P}(man)}x_{n})\phi_{\omega_{\nu}\otimes\operatorname{tr}}^{1/2}\\\
&=\lambda_{\emptyset\emptyset}(Ad_{\sigma(m)e^{\nu(\log
a)}}x_{1},...,Ad_{\sigma(m)e^{\nu(\log
a)}}x_{n})\phi_{\omega_{\nu}\otimes\operatorname{tr}}^{1/2}\\\
&=U_{\Gamma(\mu_{P})}(ma)\lambda_{\emptyset\emptyset}(x_{1},...,x_{n})\phi_{\omega_{\nu}\otimes\operatorname{tr}}^{1/2}\hskip
2.84544pt.\end{split}$

Hence the nilpotent subgroup $N$ does not act on the Haagerup $L_{2}$-space of
the Poisson algebra. Denote this Hilbert space as $\mathcal{H}_{\mathbb{P}}$,
we consider the space of half-densities on $G/P$ with value in
$\mathcal{H}_{\mathbb{P}}$:

(4.13)
$\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}}):=\\{F:G\rightarrow\mathcal{H}_{\mathbb{P}}:F(gman)=U_{\Gamma(\mu_{P})}(ma)^{*}e^{-\rho_{G}(\log
a)}F(g),\int_{G/P}|F|^{2}_{\mathcal{H}_{\mathbb{P}}}<\infty\\}\hskip
2.84544pt.$

The type I factor $\mathbb{B}(L^{2}(G/P,\Omega^{1/2}_{P}))$ acts on
$\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}})$ be left regular action, while the
Poisson factor acts on $\mathcal{H}_{\mathbb{P}}$ by standard representation.
Finally, the unitary representation of $G$ on
$\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}})$ induces a strongly continuous
automorphism on
$\mathbb{B}(L^{2}(G/P,\Omega^{1/2}_{P}))\overline{\otimes}\mathbb{P}_{\omega_{\nu}\otimes\operatorname{tr}}\mathbb{B}(H_{\nu}\otimes\ell_{2})$.
∎

######  Example 4.5.

Let $G=SO_{0}(n+1,1)$ be the (connected) conformal group on $\mathbb{R}^{n}$.
Then the Iwasawa decomposition of $G$ is given by $K=SO(n)$ (the rotation),
$A=SO_{0}(1,1)\cong\\{\begin{pmatrix}\cosh\theta&\sinh\theta\\\
\sinh\theta&\cosh\theta\end{pmatrix}:\theta\in\mathbb{R}\\}$ (the boost or
dilation), and $N=\mathbb{R}^{n}$ (the translation). The centralizer of $A$ in
$K$ is given by $M=SO(n-1)$. Since the real Lie algebra of $A$ is
$\mathfrak{a}_{\mathbb{R}}=\mathbb{R}\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$,
a purely imaginary root on $\mathfrak{a}$ is given by an imaginary number
$\nu=it_{\nu}\in i\mathbb{R}$. And for an arbitrary element
$a=\begin{pmatrix}\cosh\theta&\sinh\theta\\\
\sinh\theta&\cosh\theta\end{pmatrix}\in SO_{0}(1,1)\cong A$, the pairing can
be calculated explicity: $\nu(\log a)=it_{\nu}\theta$. In this example, we fix
the boost: $\theta=2\pi$. Then the key parameter $\nu(\log a)=2\pi it_{\nu}$
where $t_{\nu}\in\mathbb{R}$. The quotient space is the sphere $G/P=K/M\cap
K=SO(n)/SO(n-1)\cong\mathbb{S}^{n}$. Then for any unitary irreducible
representation $U_{\nu}:SO(n-1)\rightarrow U(H_{\nu})$, there exists a Poisson
algebra
$\mathbb{B}(L^{2}(\mathbb{S}^{n},\Omega^{1/2}_{P}))\overline{\otimes}\mathbb{P}_{\omega_{\nu}\otimes\operatorname{tr}}\mathbb{B}(H_{\nu}\otimes\ell_{2})$
where the weight on $\mathbb{B}(H_{\nu})$ is given by
$\omega_{\nu}=\operatorname{tr}(e^{2\pi t_{\nu}}\cdot)$. For a pair
$t_{\nu},t_{\mu}\in\mathbb{R}$ such that
$\lambda_{\nu,\mu}=\min\\{e^{2\pi(|t_{\nu}|-|t_{\mu}|)},e^{2\pi(|t_{\mu}|-|t_{\nu}|)}\\}\in(0,1)$,
then
$\mathbb{P}_{\omega_{\nu,\mu}\otimes\operatorname{tr}}\mathbb{B}((H_{\nu}\oplus
H_{\mu})\otimes\ell_{2})$ is a type IIIλ factor. Similar results hold for
general multiplets of $\\{t_{\nu}\\}$’s.

####  4.2.3. Nets of Poisson Algebras from Unitary Principal Series
Representations

As an application of the constructions so far, we briefly discuss algebraic
quantum field theories (AQFT) [Haa96]. The AQFT approach studies the local
algebras of observables in a quantum field theory. It is an axiomatic
framework that emphasizes on the necessary properties these algebras must
satisfy. Briefly these properties are isotony, covariance under a (strongly-
continuous) group action, the existence of a group invariant state, and
locality. The axioms are often modified to accommodate different physical
situations. For example, one would often require there to be a unique group-
invariant state in order to study the so-called vacuum sector of the
underlying AQFT. As another example, one would often require the group
representation to satisfy certain spectral conditions (i.e. positive energy
condition) in order to ensure the existence of well-defined "Hamiltonian".
333Note however, there are natural examples where the positive energy
condition is not satisfied [BDH15, KLM01] even in the case of conformal nets
(i.e. $G=SU(1,1)$). The positive energy condition requires the group action on
the local algebras to come from discrete series representation. In our
examples, although the group action is not discrete series representation, the
modular Hamiltonian is still well-defined.

Here we only consider the case of conformal group $G=SO(n+1,1)$ which is the
group of conformal transformations on $\mathbb{R}^{n}$. Using principal series
representation and Poissonization, we construct nets of type III1 local
Poisson algebras and check that the nets satisfy the desired axioms. Our local
algebras will be defined on the one-point compactification of
$\mathbb{R}^{n}$, namley the sphere $\mathbb{S}^{n}\cong G/P$ (c.f. Example
4.5). Mathematically, the local algebras form a pre-cosheaf over the category
of connected open subsets on $\mathbb{S}^{n}$. 444If the group $G$ is the
conformal group acting on the Minkowski space $\mathbb{R}^{n,1}$, then we must
consider local algerbas on certain distinguished open subsets (i.e. proper
open double cones). One can still define such cones on $\mathbb{R}^{n+1}$. But
since our example is a ”Euclidean” field theory model, there is no causal
constraint that prevents us from considering more general open subsets.  To
state the proposition, we fix a unitary representation of the minimal
parabolic group $\mu_{P}:P\rightarrow U(H_{\mu})$ where the spectrum of the
generator $\log a$ generates the multiplicative group $\mathbb{R}_{+}$.555This
assumption is not mathematically unnecessary. We make this assumption only to
ensure the local algebras are all type III1 factors. It is believed that the
local algebras of QFT should be of type III1. Denote the Poisson algebra
$\mathbb{P}_{\omega_{\mu}\otimes\operatorname{tr}}\mathbb{B}(H_{\mu}\otimes\ell_{2})$
as $\mathbb{P}_{\mu}$ with Poisson state $\phi_{\mu}$. And denote its Haagerup
$L_{2}$-space as $\mathcal{H}_{\mathbb{P}}$. Then we have:

######  Proposition 4.2.

Consider the twisted Poisson algebra bundle $G\times_{P}\mathbb{P}_{\mu}$ over
$G/P=\mathbb{S}^{n}$. The smooth section of this bundle is given by:

(4.14) $C^{\infty}(G/P,\mathbb{P}_{\mu}):=\\{f\in
C^{\infty}(G,\mathbb{P}):f(gp)=\alpha_{p}^{-1}f(g)\\}$

where $\alpha:P\rightarrow Aut(\mathbb{P}_{\mu})$ is the strongly continuous
induced action of $P$ on the Poisson algebra. Analogously consider the twisted
Hilbert bundle $G\times_{P}\mathcal{H}_{\mathbb{P}}$ over
$G/P=\mathbb{S}^{n}$. Then the $L^{2}$-section of the tensor bundle of
$\big{(}G\times_{P}\mathcal{H}_{\mathbb{P}}\big{)}\otimes\Omega^{1/2}_{P}$ is
given by:

(4.15) $\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}}):=\\{f\in
L^{2}(G,\mathcal{H}_{\mathbb{P}}):f(gman)=U_{\mu_{P}}(ma)^{*}e^{-\rho_{G}(\log
a)}f(g),\int_{G/P}|f|^{2}_{\mathcal{H}_{\mathbb{P}}}<\infty\\}$

where $U_{\mu_{P}}:P\rightarrow U(\mathcal{H}_{\mathbb{P}})$ is the unitary
implementation of the automorphism $\alpha$. This is induced by the unitary
representation $\mu_{P}$. $C^{\infty}(G/P,\mathbb{P}_{\mu})$ is a unital
$*$-algebra acting on $\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}})$. Denote its
weak closure as $\mathcal{A}(\mathbb{S}^{n})$.

Then for each connected open set $\mathcal{O}$ there exists a von Neumann
algebra
$\mathcal{A}(\mathcal{O}):=\\{f\in\mathcal{A}(\mathbb{S}^{n}):supp(f)\\!\\!\subset\\!\\!\mathcal{O}\\}$
acting on $\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}})$. The net of local
algebras $\\{\mathcal{A}(\mathcal{O})\\}_{\mathcal{O}\subset\mathbb{S}^{n}}$
satisfies the following properties:

  1. (1)

For any $\mathcal{O}$, $\mathcal{A}(\mathcal{O})$ is type III1 with nontrivial
center $L^{\infty}(\mathcal{O})$;

  2. (2)

(Isotony)
$\mathcal{A}(\mathcal{O}_{1})\\!\\!\subset\\!\\!\mathcal{A}(\mathcal{O}_{2})$
if $\mathcal{O}_{1}\\!\\!\subset\\!\\!\mathcal{O}_{2}$;

  3. (3)

($G$-covariance) There exists a unitary induced representation given by left
regular action:

(4.16) $V:G\rightarrow
U(\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}})):(V_{g}F)(h)=F(g^{-1}h)\hskip
2.84544pt.$

By conjugation, this unitary representation induces a strongly continuous
action $\widehat{\alpha}:G\rightarrow Aut(\mathcal{A}(\mathbb{S}^{n}))$ such
that for any connect open set $\mathcal{O}$:
$\alpha_{g}(\mathcal{A}(\mathcal{O}))=\mathcal{A}(g\mathcal{O})$ where
$g\mathcal{O}=\\{\xi\in\mathbb{S}^{n}:g^{-1}\xi\in\mathcal{O}\\}$;

  4. (4)

(Existence of vacuum state) The normal state
$\varphi:=\int_{G/P}d\xi\otimes\phi_{\mathbb{P}}$ where $d\xi$ is the unique
$K$-invariant measure on $G/P=K/(M\cap K)=\mathbb{S}^{n}$666Note there is no
$G$-invariant measure on $G/P$. is invariant under the group action and it is
represented by the vector
$1\otimes\phi_{\mathbb{P}}^{1/2}\in\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}})$.
In addition, the expectation under this vacuum state does not split into a
product of two-point functions (i.e. the vacuum state is not quasi-free and
the consequently the model cannot be regarded as "generalized free fields");

  5. (5)

(Locality) $[\mathcal{A}(\mathcal{O}_{1}),\mathcal{A}(\mathcal{O}_{2})]=0$ if
$\mathcal{O}_{1}\cap\mathcal{O}_{2}=\emptyset$.

###### Proof.

The statements are all straight-forward based on previous discussions. Using
the strongly continuous automorphism $\alpha:P\rightarrow
Aut(\mathbb{P}_{\mu})$, the Poisson algebra bundle is formed as the quotient:
$G\times_{P}\mathbb{P}_{\mu}:=\\{[(g,T)]:(gp,T)\sim(g,\alpha_{p}(T))\text{ for
}g\in G\text{ , }T\in\mathbb{P}_{\mu}\\}$.

Then the smooth section of this bundle acts on
$\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}})$ by left multiplication. We only
need to check that this action is well-defined:

(4.17)
$\displaystyle\begin{split}(f\xi)(gman)&=f(gman)\xi(gman)=\alpha_{man}^{-1}f(g)\big{(}U_{\mu_{P}}(ma)^{*}e^{-\rho_{G}(\log
a)}\xi(g)\big{)}\\\ &=U_{\mu_{P}}(ma)^{*}e^{-\rho_{G}(\log
a)}f(g)\xi(g)\end{split}$

where $f\in C^{\infty}(G/P,\mathbb{P}_{\mu})$ and
$\xi\in\Gamma^{2}(G/P,\mathcal{H}_{\mathbb{P}})$. Using the same proof that
shows the usual principal series representations are unitary and the fact that
the Poisson state $\phi_{\mu}$ is invariant under $P$-action, it follows that
the left multiplication action is also unitary. All the properties are
straight-forward. The state $\varphi$ is not quasi-free because the Poisson
state $\phi_{\mu}$ is not quasi-free. ∎

###  4.3. Split Inclusion and Hyperfiniteness of Poisson Algebras

In this subsection, we study the structure of Poisson algebra inclusions.
Specifically, we will focus on a pair of von Neumann algebras
$N\\!\\!\subset\\!\\!M$ with a normal semifinite faithful weight $\omega$ on
$M$ such that its restriction to $N$ remains semifinite. Then Poissonization
constructs a pair of Poisson algebras
$(\mathbb{P}_{\omega}N\\!\\!\subset\\!\\!\mathbb{P}_{\omega}M,\phi_{\omega})$,
where the Poisson state on $\mathbb{P}_{\omega}M$ restricts to the Poisson
state on $\mathbb{P}_{\omega}N$. For simplicity, we focus on the case where
both Poisson algebras are factors. There are two situations to consider. If
$\phi_{\omega}$ is tracial, then the Poisson algebras are type II1 and we are
in the classical set-up of the type II subfactor theory [Jon83, PP86, Pop95,
Ocn89]. On the other hand, if $\phi_{\omega}$ is not tracial, then the Poisson
algebras are properly infinite (either containing a type I∞ factor or are of
type III). By a classical theorem of Dixmier and Maréchal [DM71], there exists
a dense subset of vectors in $L_{2}(\mathbb{P}_{\omega}M,\phi_{\omega})$ such
that each vector is simultaneously cyclic and separating for both Poisson
algebras. This is the classical set-up of type III subfactor theory [DL84,
Lon89, Kos, KL92]. Unfortunately, for the Poisson algebra inclusion, the
subfactor index is typically infinite. There is no definitive construction for
the standard invariant of infinite index subfactors, although some possible
answers have been provided [Bur03, Pen13, JP15] by considering the Jones
construction with type II∞ algebras. Following this approach, we calculate
directly the centralizer algebras and the central vectors [Bur03, Pen13, JP15]
of type II1 Poisson subfactors. A planar algebra calculus has also been
developed in the case of infinite index [Pen13]. We shall reserve this and a
general study of bimodules over Poisson algebras for a future paper.

The key observation for Poisson subfactors is the following lemma.

######  Lemma 4.6.

Let $M$ be a von Neumann algebras and let $\omega$ be a n.s.f. weight on $M$.
In addition, let $e\in M_{\omega}$ be a projection in the centralizer. Then
the inclusion of the corner $N:=M_{e}\\!\\!\subset\\!\\!M$ induces Poisson
algebra inclusion
$\mathbb{P}_{\omega}N\\!\\!\subset\\!\\!\mathbb{P}_{\omega}M$ such that
$\phi_{\omega}$ on $\mathbb{P}_{\omega}M$ restricts to the Poisson state on
$\mathbb{P}_{\omega}N$. If $\mathbb{P}_{\omega}N$ is a factor, then the
relative commutant is given by
$\mathbb{P}_{\omega}M\cap(\mathbb{P}_{\omega}N)^{\prime}=\mathbb{P}_{\omega}(M_{1-e})$.
In addition, we have split isomorphism:

(4.18)
$\mu_{e}:\mathbb{P}_{\omega}N\overline{\otimes}\mathbb{P}_{\omega}(M_{1-e})\rightarrow\mathbb{P}_{\omega}N\vee\mathbb{P}_{\omega}(M_{1-e})\cong\mathbb{P}_{\omega}M:\Gamma(e^{ix})\otimes\Gamma(e^{iy})\mapsto\Gamma(e^{i(x+y)})\hskip
2.84544pt.$

In particular, the Poisson state $\phi_{\omega}$ splits:

(4.19)
$\phi_{\omega}(\Gamma(e^{ix})\Gamma(e^{iy}))=\phi_{\omega}(\Gamma(e^{ix}))\phi_{\omega}(\Gamma(e^{iy}))$

where $e^{ix}\in N$ and $e^{iy}\in M_{1-e}$.

###### Proof.

We first show that the Haagerup $L_{2}$ space of $\mathbb{P}_{\omega}M$
factorizes into a tensor product. This is a simple consequence of the well-
known result that the symmetric Fock space construction turns direct sum to
tensor product. Let $n_{\omega}$ be the left ideal of $\omega$-finite elements
in $M$. Then $en_{\omega}$ is the left ideal of $\omega$-finite elements in
$N$:

$\omega(ex^{*}xe)<\infty$

where we used the fact that the definition ideal
$m_{\omega}=n_{\omega}^{*}n_{\omega}$ is a bimodule over the centralizer
$M_{\omega}$ [Tak03]. In addition, we have orthogonality:

$\omega(ex^{*}y(1-e))=\omega(x^{*}y(1-e)e)=0$

where $x,y\in n_{\omega}$ [Tak03]. Then by the Fock space model, we have:

(4.20)
$\displaystyle\begin{split}L_{2}(\mathbb{P}_{\omega}M,\phi_{\omega})&\cong\mathcal{F}_{s}(L_{2}(M,\omega))=\mathcal{F}_{s}(\overline{n_{\omega}\eta_{\omega}})=\mathcal{F}_{s}(\overline{en_{\omega}\eta_{\omega}\oplus(1-e)n_{\omega}\eta_{\omega}})\\\
&\cong\mathcal{F}_{s}(\overline{en_{\omega}\eta_{\omega}})\otimes\mathcal{F}_{s}(\overline{(1-e)n_{\omega}\eta_{\omega}})=\mathcal{F}_{s}(L_{2}(N,\omega))\otimes\mathcal{F}_{s}(L_{2}(M_{1-e},\omega))\\\
&\cong L_{2}(\mathbb{P}_{\omega}N,\phi_{\omega})\otimes
L_{2}(\mathbb{P}_{\omega}M_{1-e},\phi_{\omega})\end{split}$

where $\eta_{\omega}$ is the representing vector in the semi-cyclic
representation of $\omega$. Using the same calculation, we can directly show
that the Poisson state factorizes:

(4.21)
$\displaystyle\begin{split}\phi_{\omega}(\Gamma(e^{ix})\Gamma(e^{iy}))&=\phi_{\omega}(\Gamma(e^{i(x+y)}))=\exp(\omega(e^{i(x+y)}-1))\\\
&=\exp(\omega\big{(}(e^{ix}-1)(e^{iy}-1)\big{)})\exp(\omega(e^{ix}-1))\exp(\omega(e^{iy}-1))\\\
&=\exp(\omega(e^{ix}-1))\exp(\omega(e^{iy}-1))=\phi_{\omega}(\Gamma(e^{ix}))\phi_{\omega}(\Gamma(e^{iy}))\end{split}$

where $e^{ix}-1\in em_{\omega}e=m_{\omega_{e}}\\!\\!\subset\\!\\!N$ and
$e^{iy}-1\in(1-e)m_{\omega}(1-e)=m_{\omega_{1-e}}\\!\\!\subset\\!\\!M_{1-e}$.
Then the factorization follows by density. Combining these two calculations,
we see that the multiplication map is a split isomorphism and it is unitarily
implemented by the Fock space isomorphism. Finally, when
$\mathbb{P}_{\omega}N$ is a factor, using the Fock space isomorphism we have:
$(\mathbb{P}_{\omega}N)^{\prime}\cap\mathbb{P}_{\omega}M=(\mathbb{P}_{\omega}N)^{\prime}\cap(\mathbb{P}_{\omega}N\vee\mathbb{P}_{\omega}M_{1-e})\cong
J_{N}\mathbb{P}_{\omega}NJ_{N}\otimes\mathbb{B}(L_{2}(\mathbb{P}_{\omega}M_{1-e},\phi_{\omega}))\cap\mathbb{P}_{\omega}N\otimes\mathbb{P}_{\omega}M_{1-e}\cong\mathbb{P}_{\omega}M_{1-e}$
where $J_{N}$ is the modular conjugation of the Poisson algebra
$\mathbb{P}_{\omega}N$ in its standard form. ∎

######  Corollary 4.7.

Using the same set-up as Lemma 4.6. If both $\mathbb{P}_{\omega}M$ and
$\mathbb{P}_{\omega}N$ are factors, then the index
$[\mathbb{P}_{\omega}M:\mathbb{P}_{\omega}N]=\infty$.

###### Proof.

If the index is finite, then the relative commutant is finite dimensional
[Lon89]. However in our case, the relative commutant is a Poisson algebra
which is always infinite dimensional. Therefore the index is always infinite.
∎

We pause to comment on the infinite index type II1 subfactors. By Proposition
4.1 and Corollary 4.7, for any projection $e\sim id\in\mathbb{B}(\ell_{2})$
that is Murray-von-Neumann equivalent to the identity, we have an infinite
index type II1 subfactor
$\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2})\\!\\!\subset\\!\\!\mathbb{P}_{\operatorname{tr}}\mathbb{B}(\ell_{2})$.
It is possible to carry out Jones’ construction on infinite index type II1
subfactors. But it is not clear that the construction produces the standard
invariant as in the finite index case. One candidate for the standard
invariant of the infinite index subfactor was proposed in [Bur03, JP15, Pen13]
in terms of the centralizer algebras and the central $L_{2}$-vectors and their
corresponding planar calculi.

######  Definition 4.8.

Given an infinite index type II1 subfactor
$(N\\!\\!\subset\\!\\!M,\operatorname{tr})$ where $\operatorname{tr}$ is the
canonical trace. For $n\geq 0$, the centralizer algebras are defined by:

(4.22)
$\mathcal{Q}_{n}:=N^{\prime}\cap(N^{op})^{\prime}\cap\mathbb{B}(\otimes^{n}_{N}L_{2}(M))$

where $\otimes^{n}_{N}L_{2}(M)$ is the Connes fusion tensor product of
$N$-bimodule $L_{2}(M)$ and $N^{\prime}$ is the commutant of the left
$N$-action on the bimodule $\otimes^{n}_{N}L_{2}(M)$ while $(N^{op})^{\prime}$
is the commutant of the right $N$-action.

The central $L_{2}$-vectors are defined by:

(4.23) $\mathcal{P}_{n}:=N^{\prime}\cap\otimes^{n}_{N}L_{2}(M)$

where $N^{\prime}$ is the commutant of the left $N$-action.

For infinite index subfactors, the centralizer algebras do not coincide with
the central vectors [Bur03]. However, they admit compatible planar calculi
[Jon99, Pen13]. For type II1 Poisson subfactors, we can calculate these
"invariants" explicitly.

######  Corollary 4.9.

Let $e\sim id\in\mathbb{B}(\ell_{2})$ be a projection that is Murray-von-
Neumann equivalent to the identity. Then as an
$\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2})$-bimodule, the Connes
fusion product is given by:

(4.24)
$\otimes^{n}_{\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2})}L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(\ell_{2}),\phi_{\operatorname{tr}})\cong
L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2}),\phi_{\operatorname{tr}})\otimes\bigg{(}\otimes^{n}_{\mathbb{C}}\mathcal{F}_{s}(S_{2}((1-e)\ell_{2}))\bigg{)}$

where $S_{2}((1-e)\ell_{2})$ is the Schatten 2-class on $(1-e)\ell_{2}$. On
the right hand side, the Poisson algebra only acts on the bimodule
$L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2}),\phi_{\operatorname{tr}})$
while the symmetric Fock space is the multiplicity. The tensor product on the
right hand side is the usual tensor product over $\mathbb{C}$.

In addition, the centralizer algebras are the matrix algebra on the symmetric
Fock space:

(4.25)
$\mathcal{Q}_{n}=\mathbb{B}\bigg{(}\otimes^{n}_{\mathbb{C}}\mathcal{F}_{s}(S_{2}((1-e)\ell_{2}))\bigg{)}\hskip
2.84544pt.$

And the central vectors are given by:

(4.26)
$\mathcal{P}_{n}=\widehat{\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2})^{\prime}}\otimes\bigg{(}\otimes^{n}_{\mathbb{C}}\mathcal{F}_{s}(S_{2}((1-e)\ell_{2}))\bigg{)}$

where
$\widehat{\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2})^{\prime}}\\!\\!\subset\\!\\!L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2}),\phi_{\operatorname{tr}})$
are the bounded vectors that commute with the left Poisson algebra action on
its standard representation.

###### Proof.

The claims follow easily once we proved the isomorphism 4.24. However this is
a simple consequence of Lemma 4.6. As
$\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2})$-bimodule,
$L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(\ell_{2}),\phi_{\operatorname{tr}})\cong
L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2}),\phi_{\operatorname{tr}})\otimes\mathcal{F}_{s}(S_{2}((1-e)\ell_{2}))$
where the tensor product on the right hand side is the usual tensor product
over $\mathbb{C}$ and the Poisson algebra only acts on the
$L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2}),\phi_{\operatorname{tr}})$
component. Since
$L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2}),\phi_{\operatorname{tr}})\otimes_{\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2})}L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2}),\phi_{\operatorname{tr}})=L_{2}(\mathbb{P}_{\operatorname{tr}}\mathbb{B}(e\ell_{2}),\phi_{\operatorname{tr}})$,
then the claim follows. ∎

If $rk(1-e):=r<\infty$ has finite rank, then the tensor product can be written
as
$\otimes^{n}\mathcal{F}_{s}(S^{r}_{2})=\mathcal{F}_{s}(\oplus^{n}S^{r}_{2})$.
In the work of [JP15], the authors showed that there exists a faithful
embedding of the gauge-invariant CAR algebras into the centralizer algebras.
In the case of Poisson subfactors, Corollary 4.9 gives faithful
representations of "fermionic" algebras on "bosonic" (symmetric) Fock space.
The relation to the Jordan-Wigner transformation and a more detailed study of
the type II1 Poisson subfactors will be conducted in a seperate paper.

In the final part of this subsection, we turn our attention to the type III
Poisson subfactors. To fix the notation, we consider a pair of von Neumann
algerbas $(N\\!\\!\subset\\!\\!M,\omega)$ where the group generated by the
Arveson spectrum $Sp(\omega_{N})$ in $\mathbb{R}_{+}$ is either
$\mathbb{R}_{+}$ or $\langle\lambda^{n}\rangle$. Then by Corollary 4.1, both
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))$
and
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$
are type III factors. One easy corollary of Lemma 4.6 is the following
observation:

######  Corollary 4.10.

Given a type III Poisson factor of the form
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$,
then for any $k\in\mathbb{N}$, we have normal isomorphism:

(4.27)
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))^{\otimes
k}\cong\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))\hskip
2.84544pt.$

###### Proof.

By induction, it suffices to prove the statement for $k=2$. Fix a projection
$e\in\mathbb{B}(\ell_{2})$ such that $e\sim 1-e\sim 1$. Then since $e$ is in
the centralizer
$(M\overline{\otimes}\mathbb{B}(\ell_{2}))_{\omega\otimes\operatorname{tr}}$,
by Lemma 4.6 we have normal isomorphism:
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(e\ell_{2}))\overline{\otimes}\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}((1-e)\ell_{2}))\cong\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$.
Since
$\mathbb{B}(e\ell_{2})\cong\mathbb{B}((1-e)\ell_{2})\cong\mathbb{B}(\ell_{2})$,
then we have the desired isomorphism. ∎

Recall the definition of a split inclusion [DL84, Lon84]:

######  Definition 4.11.

Let $N\\!\\!\subset\\!\\!M$ be two von Neumann algebras acting on the same
Hilbert space $\mathcal{H}$. Then the inclusion split if there exists a normal
isomorphism: $N\vee M^{\prime}\cong N\overline{\otimes}M^{\prime}$.

The split inclusion is an important concept in algebraic quantum field theory.
Briefly, splitting property can be understood as a mathematical
characterization of physical locality. Here we show that type III Poisson
subfactors of the form
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))\\!\\!\subset\\!\\!\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$.

######  Lemma 4.12.

Let $N\\!\\!\subset\\!\\!M$ be two hyperfinite von Neumann algebras and let
$\omega$ be a n.s.f. weight on $M$ such that the group generated of the
Arveson spectrum $Sp(\omega_{N})$ is either $\mathbb{R}_{+}$ or
$\langle\lambda^{n}\rangle$. Then the inclusion
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))\\!\\!\subset\\!\\!\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$
is a split inclusion. In particular, there exists an intermediate type I
subfactor:

(4.28)
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))\\!\\!\subset\\!\\!\mathbb{B}\bigg{(}L_{2}\big{(}\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2})),\phi_{\omega\otimes\operatorname{tr}}\big{)}\bigg{)}\\!\\!\subset\\!\\!\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))\hskip
2.84544pt.$

###### Proof.

The normal isomorphism in Corollary 4.10
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))^{\otimes
2}\cong\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$
is implemented by the unitary isomorphism:

(4.29)
$U:L_{2}(\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2})),\phi_{\omega\otimes\operatorname{tr}})^{\otimes
2}\rightarrow
L_{2}(\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2})),\phi_{\omega\otimes\operatorname{tr}})\hskip
2.84544pt.$

Notice that the tensor product is over $\mathbb{C}$ not the Connes fusion
product. Then on the left hand side, the faithful left action of the Poisson
subalgebra
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))$
is contained in the type I factor:

(4.30)
$\pi_{l}\bigg{(}\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))\bigg{)}\cong\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))\otimes\mathbb{C}\\!\\!\subset\\!\\!\mathbb{B}\bigg{(}L_{2}\big{(}\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2})),\phi_{\omega\otimes\operatorname{tr}}\big{)}\bigg{)}\otimes\mathbb{C}\hskip
2.84544pt.$

Similarly on the tensor product, the faithful right action of the Poisson
algebra
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$
has a commutant that contains the same type I factor:

(4.31)
$\pi_{r}\bigg{(}\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))\bigg{)}^{\prime}\cong\mathbb{B}\bigg{(}L_{2}\big{(}\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2})),\phi_{\omega\otimes\operatorname{tr}}\big{)}\bigg{)}\otimes\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))\hskip
2.84544pt.$

Since
$U\pi_{r}\bigg{(}\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))\bigg{)}U^{*}=J\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))J$
where $J$ is the modular conjugation of the Poisson state
$\phi_{\omega\otimes\operatorname{tr}}$, then
$U\mathbb{B}\bigg{(}L_{2}\big{(}\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2})),\phi_{\omega\otimes\operatorname{tr}}\big{)}\bigg{)}\otimes\mathbb{C}U^{*}$
is the intermediate type I factor we are looking for.

To finish the proof, the relative commutant
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))^{\prime}\cap\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$
is properly infinite. Again using the unitary isomorphism $U$, both the
commutant
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(N\overline{\otimes}\mathbb{B}(\ell_{2}))^{\prime}$
and the Poisson algebra
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$
absorb the Poisson algebra
$\mathbb{P}_{\omega\otimes\operatorname{tr}}(M\overline{\otimes}\mathbb{B}(\ell_{2}))$
itself. Hence the relative commutant absorbs the type III Poisson factor as
well. Hence as an infinite von Neumann algebra, the relative commutant must be
properly infinite. Hence by a result in [DL84], when the relative commutant is
properly infinite, an inclusion is split if and only if there exists an
intermediate type I factor. Hence we have the desired result. ∎

###  4.4. Relative Entropy in Poisson Algebras

As a last application of Poissonization, we show how to calculate quantum
relative entropy for certain states in Poisson algebras. When the Poisson
algebra is of type III, this calculation gives rigorous formula for the
relative entropy in a type III algebra.

To state the main mathematical result in full generality, we first recall
Araki’s definition of relative entropy and Lindblad’s generalization.

######  Definition 4.13 (Araki [Ara76, Ara77]).

Let $N$ be a von Neumann algebra and $\rho,\psi$ be two states on $N$. Then
the relative entropy is given by:

(4.32)
$D(\rho|\psi):=\langle\eta_{\psi},-\log\Delta_{\rho,\psi}\eta_{\psi}\rangle$

where $\eta_{\psi}\in L_{2}(N,\psi)$ is the vector representation of the state
$\psi$ and $\Delta_{\rho,\psi}$ is the relative modular operator between the
state $\rho$ and $\psi$.

######  Definition 4.14 (Lindblad [Lin74]).

Let $N$ be a finite dimensional von Neumann algebra and $\rho,\psi\in
L_{1}(N)$ be two positive densities (not necessarily normalized). Then the
Lindblad relative entropy is given by:

(4.33) $D_{Lin}(\rho|\psi):=Tr(\rho(\log\rho-\log\psi))+Tr(\psi)-Tr(\rho)$

where $Tr(\cdot)$ is the canonical trace on $N$.

In the case where $N$ is finite dimensional and both $\rho$ and $\psi$ are
normalized, the Lindblad relative entropy reduces to the Araki’s definition.
It is not clear how to define relative entropy when $\rho$ and $\psi$ are
unbounded weights. However in the case where one weight is a "small"
perturbation of the other, the Lindblad relative entropy can be generalized:

######  Definition 4.15.

Let $N$ be a semifinite von Neumann algebra and let $\rho,\psi$ be two n.s.f
weigths such that $\rho\leq\psi$ and $\psi-\rho\in N_{*}$ is a normal bounded
linear functional. Then the generalized Lindblad relative entropy is given by:

(4.34)
$D_{Lin}(\rho|\psi):=\langle\eta_{\psi},-\log\Delta_{\rho,\psi}\eta_{\psi}\rangle+(\psi-\rho)(1)\hskip
2.84544pt.$

When $N$ is finite dimensional and $\rho,\psi$ are bounded functionals, the
generalized Lindblad relative entropy reduces to the familiar Lindblad
relative entropy. Just as compact operators are treated as small perturbations
around bounded operators in noncommutative geometry, the perturbation
$\psi-\rho$ can be understood as a small bounded perturbation around an
unbounded weight.

Using this generalized notion of relative entropy, we will calculate the
relative entropy between two Poisson states $\phi_{\rho}$ and $\phi_{\psi}$.
One difficulty in defining this relative entropy is that on a Poisson algebra
$\mathbb{P}_{\psi}N$, it is not clear that $\phi_{\rho}$ is a normal state
because Poissonization only constructs $\phi_{\rho}$ as a state on another
Poisson algebra, namely $\mathbb{P}_{\rho}N$. However, when $\rho$ is a
bounded perturbation around $\psi$, we can define both states on the Poisson
algebra $\mathbb{P}_{\psi}N$. 777In general, when the two weights $\rho,\psi$
do not satisfy the conditions of Proposition 4.3, there is no guarantee that
the Poisson states $\phi_{\rho},\phi_{\psi}$ can be simultaneously defined on
the same Poisson algebra. In fact, if $\psi=\lambda\rho$ for some constant
$\lambda>0$ and assume both are well-defined states on the Poisson algebra
$\mathbb{P}_{\psi}N$, then using Powers-Størmer inequality and a net of
approximating states, one can show that these two states are necessarily
orthogonal.

######  Proposition 4.3.

Let $N$ be a semifinite von Neumann algebra and let $\rho,\psi$ be two n.s.f.
weights on $N$. In addition let $\mathbb{P}_{\psi}N$ be the Poisson algebra
and $\phi_{\psi}$ be the corresponding Poisson state. Assume $\rho\leq\psi$
and $\psi-\rho$ is a bounded positive functional. Then on
$\mathbb{P}_{\psi}N$, there exists a n.s.f. weight $\widetilde{\phi}_{\rho}$
such that:

(4.35) $(D\widetilde{\phi}_{\rho}:D\phi_{\psi})_{t}=\Gamma((D\rho:D\psi)_{t})$

where $(D\rho:D\psi)_{t}\in N$ is Connes’ cocycle [Tak03] between $\rho$ and
$\psi$ and similarly for $(D\widetilde{\phi}_{\rho}:D\phi_{\psi})_{t}$.
Moreover the Poisson state $\phi_{\rho}$ is a well-defined state on
$\mathbb{P}_{\psi}N$ and the cocycle between $\widetilde{\phi_{\rho}}$ and
$\phi_{\rho}$ is given by:
$(D\phi_{\rho}:D\widetilde{\phi_{\rho}})_{t}=\exp(-it(\psi-\rho)(1))$. Then
the generalized Lindblad entropy is preserved

(4.36) $D(\phi_{\rho}|\phi_{\psi})=D_{Lin}(\rho|\psi)\hskip 2.84544pt.$

###### Proof.

The proof is based on the simple observation that Poissonization naturally
lifts a cocycle in $N$ to a cocycle in the Poisson algebra
$\mathbb{P}_{\psi}N$. But the integrated weight is not the correct Poisson
state. So the assumption on bounded perturbation is necessary to ensure that
there exists another cocycle on the Poisson algebra to produce the correct
Poisson state.

Let $u_{t}:=(D\rho:D\psi)_{t}\in N$ be the cocycle between $\rho,\psi$ in N.
Poissonization produces a unitary $\Gamma(u_{t})\in\mathbb{P}_{\psi}N$ such
that the cocycle condition holds:

(4.37)
$\Gamma(u_{t})\sigma_{t}^{\phi_{\psi}}(\Gamma(u_{s}))=\Gamma(u_{t})\Gamma(\sigma^{\psi}_{t}(u_{s}))=\Gamma(u_{t}\sigma^{\psi}_{t}(u_{s}))=\Gamma(u_{t+s})$

where $\sigma^{\phi_{\psi}}_{t}$ is the modular automorphism of $\phi_{\psi}$
and similarly for $\sigma^{\psi}_{t}$. Hence by Connes-Matsuda theorem, there
exists a n.s.f. weight $\widetilde{\phi_{\rho}}$ on the Poisson algebra
$\mathbb{P}_{\psi}N$ such that:
$(D\widetilde{\phi_{\rho}}:D\phi_{\psi})_{t}=\Gamma(u_{t})$.

To calculate the $\widetilde{\phi_{\rho}}$, we first observe that
$\Gamma(u_{-i/2})$ is a well-defined contraction on
$L_{2}(\mathbb{P}_{\psi}N,\phi_{\psi})$. Since $\rho\leq\psi$,
$||u_{-i/2}||\leq 1$ [Tak03] and using the isomorphism
$\mathcal{F}_{s}(L_{2}(N,\psi))\cong L_{2}(\mathbb{P}_{\psi}N,\phi_{\psi})$,
the contraction $u_{-i/2}$ can be lifted to a contraction $\Gamma(u_{-i/2})$
such that:

(4.38)
$\displaystyle\begin{split}\big{(}\Gamma(u_{-i/2})\cdot\phi_{\psi}\cdot\Gamma(u_{-i/2}),\Gamma(x)\big{)}&=\phi_{\psi}(\Gamma(u_{-i/2})^{*}\Gamma(x)\Gamma(u_{-i/2}))=\exp(\psi(u_{-i/2}^{*}xu_{-i/2}-1))\\\
&=\exp(\rho(x-1))\exp((\rho-\psi)(1))\end{split}$

where $x\in S^{\psi}_{ana}=\\{y\in\mathcal{U}^{\psi}_{ana}:y-1\in
m_{\psi},||y||\leq 1\\}$ and $\big{(}\cdot,\cdot\big{)}$ is the canonical
pairing between the Poisson algebra and its predual. Since
$\widetilde{\phi_{\rho}}=\Gamma(u_{-i/2})\cdot\phi_{\psi}\cdot\Gamma(u_{-i/2})$,
then by density we see that the twisted version of $\widetilde{\phi_{\rho}}$
coincides with the Poisson state $\phi_{\rho}$:

(4.39)
$\exp((\rho-\psi)(1))\widetilde{\phi_{\rho}}(\Gamma(x))=\exp(\rho(x-1))=\phi_{\rho}(\Gamma(x))\hskip
2.84544pt.$

Hence $\phi_{\rho}$ is well-defined state on the Poisson algebra
$\mathbb{P}_{\psi}N$ and we have cocycle:
$(D\phi_{\rho}:D\widetilde{\phi_{\rho}})_{t}=\exp(it(\rho-\psi)(1))$. In
particular,
$(D\phi_{\rho}:D\phi_{\psi})_{t}=\exp(it(\rho-\psi)(1))\Gamma(u_{t})$.
Therefore by the definition of Araki’s relative entropy, we have:

(4.40)
$\displaystyle\begin{split}D(\phi_{\rho}|\phi_{\psi})&=-\langle\phi_{\psi}^{1/2},\log\Delta_{\phi_{\rho},\phi_{\psi}}\phi_{\psi}^{1/2}\rangle=-\langle\phi_{\psi}^{1/2},\lambda(\log\Delta_{\rho,\psi})\phi_{\psi}^{1/2}\rangle+(\psi-\rho)(1)\\\
&=-\langle\psi^{1/2},\log\Delta_{\rho,\psi}\psi^{1/2}\rangle+(\psi-\rho)(1)=D_{Lin}(\rho|\psi)\hskip
2.84544pt.\end{split}$

∎

A similar calculation also works for the Renyi entropies. In a separate paper,
we will use this observation to study several other quantum information
quantities on the Poisson algebras. It turns out that the Poisson algebra is a
good toy model to study the so-call chaotic quantum systems.

## References

  * [AAS22] andA natoly Dymarsky Alexander Avdoshkin and Michael Smolkin.  Krylov complexity in quantum field theory, and beyond.  arXiv prepring arXiv:2212.14429, 2022.
  * [AD20] Alexander Avdoshkin and Anatoly Dymarsky.  Euclidean operator growth and quantum chaos.  Phys. Rev. Res., 2:043234, Nov 2020.
  * [AH14] Hiroshi Ando and Uffe Haagerup.  Ultraproducts of von neumann algebras.  Journal of Functional Analysis, 266(12):6842–6913, 2014.
  * [Akh20] N. I. Akhiezer.  The Classical Moment Problem and Some Related Questions in Analysis.  Society for Industrial and Applied Mathematics, Philadelphia, PA, 2020\.
  * [Ara76] H. Araki.  Relative entropy of states of von neumann algebras.  Publ. Res. Inst. Math. Sci. Kyoto, 1976:809–833, 1976.
  * [Ara77] H. Araki.  Relative entropy for states of von neumann algebras ii.  Publ. Res. Inst. Math. Sci. Kyoto, 1:173–192, 1977.
  * [BCK66] Jean Bretagnolle, Didier Dacunha Castelle, and Jean-Louis Krivine.  Lois stables et espaces lp.  Annales de l’institut Henri Poincare, 2(3):231–259, 1965-1966.
  * [BDH15] Arthur Bartels, Christopher L. Douglas, and Andre Henriques.  Conformal nets i: coordinate-free nets.  International Mathematics Research Notices, 2015:4975–5952, 2015\.
  * [BDL90a] Detlev Buchholz, Claudio D’Antoni, and Roberto Longo.  Nuclear maps and modular structures. i. general properties.  Journal of Functional Analysis, 88(2):233–250, 1990.
  * [BDL90b] Detlev Buchholz, Claudio D’Antoni, and Roberto Longo.  Nuclear maps and modular structures. II. Applications to quantum field theory.  Communications in Mathematical Physics, 129(1):115 – 138, 1990\.
  * [Ber92] Joseph Bernstein.  Representations of p-adic groups.  1992\.  Lecture note draft.
  * [Bor92] H.-J. Borchers.  The cpt-theorem in two-dimensional theories of local observables.  Communications in Mathematical Physics, 143(2):315–332, 1992.
  * [Bou15] Gabriel Bouch.  Complex-time singularity and locality estimates for quantum lattice systems.  Journal of Mathematical Physics, 56(12):123303, 2015.
  * [BR81] Ola Bratteli and Derek W. Robinson.  Operator Algebras and Quantum Statistical Mechanics II.  Theoretical and Mathematical Physics. Springer Berlin, Heidelberg, 1 edition, 1981.
  * [BR87] Ola Bratteli and Derek W. Robinson.  Operator Algebras and Quantum Statistical Mechanics I.  Theoretical and Mathematical Physics. Springer Berlin, Heidelberg, 2 edition, 1987.
  * [Bur03] Michael Burns.  Subfactors, planar algebras and rotations.  arXiv preprint arXiv:1111.1362, 2003.  PhD thesis at the University of California, Berkeley.
  * [Cao21] Xiangyu Cao.  A statistical mechanism for operator growth.  Journal of Physics A: Mathematical and Theoretical, 54(14), 2021\.
  * [CH15] Tyrone Crisp and Nigel Higson.  Parabolic induction, categories of representations and operator spaces.  arXiv preprint arXiv:1506.09111, 2015.
  * [Chr95] Lance E. Christopher.  Hilbert C*-Modules: A Toolkit for Operator Algebraists.  London Mathematical Society Lecture Note Series. Cambridge University Press, 1995.
  * [DG20] Anatoly Dymarsky and Alexander Gorsky.  Quantum chaos as delocalization in krylov space.  Phys. Rev. B, 102:085137, Aug 2020.
  * [DL] S. Doplicher and R. Longo.  Inventiones mathematicae.
  * [DL84] S. Doplicher and R. Longo.  Inventiones Mathematicae, (75):493–536, 1984.
  * [DM71] J. Dixmier and O. Maréchal.  Vecteurs totalisateurs d’une algèbre de von Neumann.  Communications in Mathematical Physics, 22(1):44 – 50, 1971.
  * [DS21] Anatoly Dymarsky and Michael Smolkin.  Krylov complexity in conformal field theory.  Phys. Rev. D, 104:L081702, Oct 2021.
  * [EHF14] Tarek A. Elsayed, Benjamin Hess, and Boris V. Fine.  Signatures of chaos in time series generated by many-spin systems at high temperatures.  Phys. Rev. E, 90:022910, Aug 2014.
  * [Flo98] Martin Florig.  On borchers’ theorem.  Letters in Mathematical Physics, 46:289–293, 1998.
  * [GKZ22] Y. Gu, A. Kitaev, and P. Zhang.  A two-way approach to out-of-time-order correlations.  Journal of High Energy Physics, (133), 2022.
  * [HA05] Laszlo Zsido Huzihiro Araki.  Extension of the structure theorem of borchers and its application to half-sided modular inclusions.  Reviews in Mathematical Physics, 17(05):491–543, 2005.
  * [Haa96] Rudolf Haag.  Local Quantum Physics.  Theoretical and Mathematical Physics. Springer Berlin, Heidelberg, 1996\.
  * [HQRY16] P. Hosur, X. L. Qi, D. A. Roberts, and Beni Yoshida.  Chaos in quantum channels.  Journal of High Energy Physics, (4), 2016.
  * [Hus94] Dale. Husemöller.  Fibre Bundles.  Graduate Texts in Mathematics, 20. Springer New York, New York, NY, 3rd ed. 1994. edition, 1994.
  * [JJB75] Eyvind H. Wichmann Joseph J. Bisognano.  On the duality condition for a hermitian scalar field.  Journal of Mathematical Physics, 16(4):985–1007, 1975.
  * [Jon83] V.F.R. Jones.  Index for subfactors.  Inventiones mathematicae, 72:1–26, 1983.
  * [Jon99] Vaughan F. R. Jones.  Planar algebras i.  arXiv preprint arXiv:math/9909027, 1999.
  * [JP09] Marius Junge and Javier Parcet.  Mixed-Norm Inequalities and Operator Space Lp Embedding Theory, volume 203 of Memoirs of the American Mathematical Society.  2009\.
  * [JP15] V.F.R. Jones and D. Penneys.  Infinite index subfactors and the gicar categories.  Communications in Mathematical Physics, (339):729–768, 2015.
  * [Jun] Marius Junge.  Noncommutative poisson measure.  unpublished.
  * [Jun06] Marius Junge.  Operator spaces and Araki-Woods factors: A quantum probabilistic approach.  International Mathematics Research Papers, 2006, 01 2006.
  * [KL92] Hideki Kosaki and Roberto Longo.  A remark on the minimal index of subfactors.  Journal of Functional Analysis, 107(2):458–470, 1992.
  * [KLM01] Y. Kawahigashi, R. Longo, and M. Muger.  Multi-interval subfactors and modularity of representations in conformal field theory.  Communications in Mathematical Physics, (219):631–669, 2001.
  * [Kna86] Anthony W. Knapp.  Representation Theory of Semisimple Groups: An Overview Based on Examples (PMS-36).  Princeton University Press, rev - revised edition, 1986.
  * [Kna96] Anthony W. Knapp.  Lie Groups Beyond an Introduction.  Progress in Mathematics. Birkhauser Boston, MA, 1 edition, 1996.
  * [Kos] Hideki Kosaki.  Relative entropy of states: A variational expression.  Journal of Operator Theory, (2):335–348.
  * [Lie83] Elliott H. Lieb.  Sharp constants in the hardy-littlewood-sobolev and related inequalities.  Annals of Mathematics, 118(2):349–374, 1983.
  * [Lin74] G. Lindblad.  Expectations and entropy inequalities for finite quantum systems.  Communications in Mathematical Physics, (39):111–119, 1974.
  * [LMVR19] Tyler LeBlond, Krishnanand Mallayya, Lev Vidmar, and Marcos Rigol.  Entanglement and matrix elements of observables in interacting integrable systems.  Phys. Rev. E, 100:062134, Dec 2019.
  * [Lon] Roberto Longo.  Lectures on conformal nets.  unpublished.
  * [Lon84] R. Longo.  Solution of the factorial stone-weierstrass conjecture. an application of the theory of standard split w*-inclusions.  Inventiones mathematicae, 76:145–156, 1984.
  * [Lon89] Roberto Longo.  Index of subfactors and statistics of quantum fields. I.  Communications in Mathematical Physics, 126(2):217 – 247, 1989\.
  * [LTZ22] Chang Liu, Haifeng Tang, and Hui Zhai.  Krylov complexity in open quantum systems.  arXiv preprint arXiv:2207.13603, 2022.
  * [Mey95] Paul-Andre Meyer.  Quantum Probability for Probabilists.  Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 1995.
  * [MSS16] J. Maldacena, S. H. Shenker, and D. Stanford.  A bound on chaos.  Journal of High Energy Physics, (106), 2016.
  * [NN20] T. Nosaka and T. Numasawa.  Quantum chaos, thermodynamics and black hole microstates in the mass deformed syk model.  Journal of High Energy Physics, (81), 2020.
  * [Ocn89] A. Ocneanu.  Quantized groups, string algebras, and Galois theory for algebras, volume 2 of London Mathematical Society Lecture Note Series, page 119–172.  Cambridge University Press, 1989.
  * [Par92] K. R. Parthasarathy.  An Introduction to Quantum Stochastic Calculus.  Modern Birkhauser Classics. Birkhauser Basel, 1 edition, 1992.
  * [Par12] K. R. Parthasarathy.  An Introduction to Quantum Stochastic Calculus.  Modern Birhauser Classics. Birkhauser Basel, 2012.
  * [Pas73] William L. Paschke.  Inner product modules over b*-algebras.  Transactions of the American Mathematical Society, 182:443–468, 1973.
  * [PCA+19] Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, and Ehud Altman.  A universal operator growth hypothesis.  Phys. Rev. X, 9:041017, Oct 2019.
  * [Pen13] D. Penneys.  A planar calculus for infinite index subfactors.  Communications in Mathematical Physics, (319):595–648, 2013.
  * [Pop95] Sorin Popa.  volume 86 of CBMS Regional Conference Series in Mathematics.  1995\.
  * [PP86] Mihai Pimsner and Sorin Popa.  Entropy and index for subfactors.  Annales scientifiques de l’Ecole Normale Superieure, 19(1):57–106, 1986.
  * [Rep00] Representation theory of Lie groups.  IAS/Park City mathematics series ; v. 8. American Mathematical Society, Providence, RI, 2000.
  * [RSGSS21] E. Rabinovici, A. Sanchez-Garrido, R. Shir, and J. Sonner.  Operator complexity: a journey to the edge of krylov space.  Journal of High Energy Physics, (62), 2021.
  * [RSGSS22] E. Rabinovici, A. Sanchez-Garrido, R. Shir, and J. Sonner.  Krylov complexity from integrability to chaos.  Journal of High Energy Physics, (151), 2022.
  * [RSS18] D. A. Roberts, D. Stanford, and A. Streicher.  Operator growth in the syk model.  (122), 2018.
  * [RvD77] Marc A. Rieffel and Alfons van Daele.  A bounded operator approach to tomita-takesaki theory.  Pacific Journal of Mathematics, 69(1):187 – 221, 1977.
  * [Sat99] Kenichi Sato.  Levy Processes and Infinitely Divisible Distributions.  Cambridge studies in advanced mathematics. Cambridge University Press, Cambridge, U.K., 1999.
  * [SML18] Akshay Seshadri, Vaibhav Madhok, and Arul Lakshminarayan.  Tripartite mutual information, entanglement, and scrambling in permutation symmetric systems with an application to quantum chaos.  Phys. Rev. E, 98:052205, Nov 2018.
  * [SPQ+19] C. Sunderhauf, L. Piroli, X. L. Qi, N. Schuch, and J. I. Cirac.  Quantum chaos in the brownian syk model with large finite n: Otocs and tripartite information.  Journal of High Energy Physics, (38), 2019.
  * [ST] Gennady Samoradnitsky and M. S. Taqqu.  Stable Non-Gaussian Random Processes Stochastic Models with Infinite Variance.  Chapman and Hall, 1 edition.
  * [Tak79] Masamichi Takesaki.  Theory of operator algebras. I.  Springer-Verlag, New York, 1979.
  * [Tak03] Masamichi Takesaki.  Theory of operator algebras. II.  Encyclopaedia of Mathematical Sciences, vol. 127. Springer-Verlag, Berlin, 2003.
  * [Ter82] Marianne Terp.  Interpolation spaces between a von neumann algebra and its predual.  Journal of Operator Theory, 8(2):327–360, 1982.
  * [VND92] Dan Voiculescu, Alexandru Nica, and Kenneth J. Dykema.  Free Random Variables, volume 1 of CRM Monograph Series.  1992\.
  * [Was98] Antony Wassermann.  Operator algebras and conformal field theory iii. fusion rules of positive energy representations of lsu(n) using bounded operators.  Inventiones Mathematicase, (133):467–538, 1998.
  * [Wid46] David Vernon Widder.  The Laplace Transform.  Princeton University Press, 1946.
  * [Zel05] S. Zelditch.  Quantum ergodicity and mixing.  arXiv preprint arXiv:math-ph/0503026, 2005.
  * [Zel09] S. Zelditch.  Recent developments in mathematical quantum chaos.  arXiv preprint arXiv:0911.4312, 2009.