$\displaystyle=\varphi(L_{\varphi}(\zeta_{i})^{\ast}L_{\varphi}(x\zeta_{j})\sigma^{\varphi}_{-i/2}(y))$
$\displaystyle=\varphi(L_{\varphi}(\zeta_{i})^{\ast}L_{\varphi}(x\zeta_{j}y))$
$\displaystyle=\langle\zeta_{i},x\zeta_{j}y\rangle_{\mathcal{H}}$

for $x\in M$, $y\in\operatorname{dom}(\sigma^{\varphi}_{-i/2})$ and $i,j\in
I$, there exists a unique unitary bimodule map $U\colon
F_{\varphi}\to\mathcal{H}$ such that $U\xi_{i}=\zeta_{i}$. Then it is not hard
to see that
$\mathcal{F}(U)X^{\varphi}_{i}\mathcal{F}(U)^{\ast}=s_{0}(\zeta_{i})$.

As $\mathcal{F}(U)$ is a unitary bimodule map that leaves $L^{2}(M)$
invariant, we conclude that $\mathcal{F}(U)\,\cdot\,\mathcal{F}(U)^{\ast}$ is
a $\ast$-isomorphism between $\Phi(M,\eta)$ and
$\Gamma_{0}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ that leaves the copies
of $M$ invariant and intertwines the canonical conditional expectations. ∎

###  6.2. Operator-Valued Twisted Araki–Woods Algebras over Type I Factors

In this section we give a complete description of operator-valued twisted
Araki–Woods algebras in the case when the base algebra is a type I factor (and
the Tomita correspondence is separable). More precisely, we show that they
decompose as a tensor product of the base algebra and a (scalar-valued)
twisted Araki–Woods factor. In particular, in the light of the previous
subsection, this yields a complete description of the von Neumann algebras
generated by operator-valued semicircular variables over type I factors for
which the canonical conditional expectation is faithful.

######  Theorem 6.3.

If $M$ is a type I factor, $\varphi$ a normal semifinite faithful weight on
$M$, $(H,\mathcal{J},(\mathcal{U}_{t}))$ a separable Tomita correspondence
over $(M,\varphi)$ and $\mathcal{T}$ a local braided compatible twist on
$\mathcal{H}\otimes_{\varphi}\mathcal{H}$, then there exists a Hilbert space
$H$, an anti-unitary involution $J$ on $H$, a strongly continuous unitary
group $(U_{t})$ on $H$ satisfying $[J,U_{t}]=0$ for all $t\in\mathbb{R}$, and
a local braided compatible twist $T$ on $H\otimes H$ such that

$(M\subset\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t})),E_{0})\cong(M\subset
M\overline{\otimes}\Gamma_{T}(H,J,(U_{t})),\mathrm{id}\otimes\psi),$

where $\psi$ is the vacuum state on the scalar-valued twisted Araki–Woods
algebra $\Gamma_{T}(H,J,(U_{t}))$.

###### Proof.

By Theorem 3.7 there exists a Hilbert space $H$, an anti-unitary involution
$J\colon H\to H$, a strongly continuous unitary group $(U_{t})$ on $H$
satisfying $[J,U_{t}]=0$ for all $t\in\mathbb{R}$ and a unitary bimodule map
$V\colon\mathcal{H}\to L^{2}(M)\otimes H$ such that
$V\mathcal{J}=(J_{\varphi}\otimes J)V$ and
$V\mathcal{U}_{t}=(\Delta_{\varphi}^{it}\otimes U_{t})V$.

As discussed in Example 4.5, there is a canonical identification
$(L^{2}(M)\otimes H)^{\otimes_{\varphi}2}\cong L^{2}(M)\otimes H\otimes H$,
which we will tacitly make use in the following. The map
$(V\otimes_{\varphi}V)\mathcal{T}(V^{\ast}\otimes_{\varphi}V^{\ast})$ is a
bounded bimodule map on $L^{2}(M)\otimes H\otimes H$, that is, it belongs to
$(M\otimes\mathbb{C}1_{H\otimes
H})^{\prime}\cap(M^{\prime}\otimes\mathbb{C}1_{H\otimes H})^{\prime}$. Since
$M$ is a factor, this implies that there exists a bounded linear map $T$ on
$H\otimes H$ such that
$(V\otimes_{\varphi}V)\mathcal{T}(V^{\ast}\otimes_{\varphi}V^{\ast})=1_{L^{2}(M)}\otimes
T$.

A direct computation shows that
$V^{\otimes_{k}\varphi}\mathcal{T}_{k}(V^{\ast})^{\otimes_{\varphi}k}=1_{L^{2}(M)}\otimes
T_{k}$ and
$V^{\otimes_{\varphi}n}P_{\mathcal{T},n}(V^{\ast})^{\otimes_{\varphi}n}=1_{L^{2}(M)}\otimes
P_{T,n}$. Thus $T$ is a twist, and a strict twist if $\mathcal{T}$ is a strict
twist. Moreover, $V^{\otimes_{\varphi}n}$ extends to a unitary bimodule map
$V^{(n)}$ between $H_{\mathcal{T},n}$ and $L^{2}(M)\otimes H_{T,n}$ for all
$n\in\mathbb{N}$. We write $\mathcal{F}_{\mathcal{T}}(V)$ for the unitary
bimodule map between $\mathcal{F}_{\mathcal{T}}(\mathcal{H})$ and
$L^{2}(M)\otimes\mathcal{F}_{T}(H)$ that restricts to the identity on
$L^{2}(M)$ and to $V^{(n)}$ on $\mathcal{H}_{\mathcal{T},n}$.

Furthemore,

$1_{L^{2}(M)}\otimes
T_{1}T_{2}T_{1}=\mathcal{F}_{\mathcal{T}}(V)\mathcal{T}_{1}\mathcal{T}_{2}\mathcal{T}_{1}\mathcal{F}_{\mathcal{T}}(V)^{\ast}=\mathcal{F}_{\mathcal{T}}(V)\mathcal{T}_{2}\mathcal{T}_{1}\mathcal{T}_{2}\mathcal{F}_{\mathcal{T}}(V)^{\ast}=1_{L^{2}(M)}\otimes
T_{2}T_{1}T_{2}.$

Hence $T$ is braided. The compatibility of $T$ follows directly from the
tensor product decompositions of $\mathcal{T}$, $\mathcal{J}$ and
$(\mathcal{U}_{t})$.

To show that $T$ is local, let $\xi,\eta\in H$ and
$x\in\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast}$ and non-zero. A
direct computation shows
$a^{\ast}_{\mathcal{T},n}(V^{\ast}(\Lambda_{\varphi}(x)\otimes\xi))=\mathcal{F}_{\mathcal{T}}(V)(x\otimes
a^{\ast}_{T,n}(\xi))\mathcal{F}_{\mathcal{T}}(V)^{\ast}$ and likewise
$b^{\ast}_{\mathcal{T},n}(V^{\ast}(\Lambda_{\varphi}^{\prime}(x)\otimes\eta))=\mathcal{F}_{\mathcal{T}}(V)(J_{\varphi}x^{\ast}J_{\varphi}\otimes
b^{\ast}_{T,n}(\eta))\mathcal{F}_{\mathcal{T}}(V)^{\ast}$. Hence the locality
of $T$ follows from the locality of $\mathcal{T}$, and the spatial isomorphism
induced by $\mathcal{F}_{\mathcal{T}}(V)$ maps
$\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ onto
$M\overline{\otimes}\Gamma_{T}(H,J,(U_{t}))$ and interwines the conditional
expectations. ∎

######  Corollary 6.4.

If $M$ is a type I factor, $I$ a countable set and $\eta\colon M\to
M\overline{\otimes}B(\ell^{2}(I))$ a covariance matrix such that the canonical
conditional expectation from $\Phi(M,\eta)$ onto $M$ is faithful, then
$\Phi(M,\eta)$ is isomorphic to a tensor product of $M$ with a free
Araki–Woods algebra.

######  Remark 6.5.

By [Shl97, Corollary 6.11], if $(U_{t})$ is non-trivial, then
$\Gamma_{0}(H,J,(U_{t}))$ is a type III factor and in particular properly
infinite. Thus
$M\overline{\otimes}\Gamma_{0}(H,J,(U_{t}))\cong\Gamma_{0}(H,J,(U_{t}))$ for
every type I factor $M$. If $(U_{t})$ is trivial, then
$\Gamma_{0}(H,J,(U_{t}))\cong L(\mathbb{F}_{\dim H})$ and
$M\overline{\otimes}\Gamma_{0}(H,J,(U_{t}))$ is an interpolated free group
factor if $M$ is finite.

###  6.3. Gaussian Actions on Twisted Araki–Woods Algebras

Let $H$ be a Hilbert space, $J$ an anti-unitary involution on $H$, $(U_{t})$ a
strongly continuous unitary group on $H$ such that $[U_{t},J]=0$ for all
$t\in\mathbb{R}$ and $T$ a local compatible braided twist on $H\otimes H$. We
consider group actions on the twisted Araki–Woods algebra
$\Gamma_{T}(H,J,(U_{t}))$ that are induced by unitary actions on $H$
compatible with $J$ and $(U_{t})$.

More precisely, let $G$ be a discrete group and $\pi$ a unitary representation
of $G$ on $H$ such that $[\pi(g),J]=0$ for all $g\in G$. In other words, $\pi$
is the complexification of an orthogoncal action of $G$ on $H^{J}$. If
$[\pi(g)\otimes\pi(g),T]=0$ for all $g\in G$ and $[\pi(g),U_{t}]=0$ for all
$g\in G$, $t\in\mathbb{R}$, then $\pi(g)^{\otimes n}$ is a unitary operator on
$H^{\otimes n}$ that commutes with $P_{T,n}$. Hence it gives rise to a unitary
operator $\mathcal{F}_{T}(\pi(g))$ on $\mathcal{F}_{T}(H)$. The Gaussian
action $\Gamma_{T}(\pi)$ of $G$ on the twisted Araki–Woods algebra
$\Gamma_{T}(H,J,(U_{t}))$ induced by $\pi$ is given by

$\Gamma_{T}(\pi(g))\colon B(\mathcal{F}_{T}(H))\to
B(\mathcal{F}_{T}(H)),\,\Gamma_{T}(g)x=\mathcal{F}_{T}(\pi(g))x\mathcal{F}_{T}(\pi(g))^{\ast}.$

######  Lemma 6.6.

If $g\in G$ and $x\in\Gamma_{T}(H,J,(U_{t}))$, then
$\Gamma_{T}(\pi(g))x\in\Gamma_{T}(H,J,(U_{t}))$.

###### Proof.

Since $T$ is local, it suffices to show that
$\Gamma_{T}(\pi(g))(b_{T}^{\ast}(\eta)+b_{T}(\eta))\in\Gamma_{T}(H,J,(U_{t}))^{\prime}$
for all $\eta\in\operatorname{dom}(U_{i/2})$ with $JU_{i/2}\eta=\eta$. For
$\xi_{1},\dots,\xi_{n}\in H$ we have

$\displaystyle\Gamma_{T}(\pi(g))(b_{T}^{\ast}(\eta))(\xi_{1}\otimes\dots\xi_{n})$
$\displaystyle=\mathcal{F}_{T}(\pi(g))(\pi(g)^{\ast}\xi_{1}\otimes\pi(g)^{\ast}\xi_{n}\otimes\eta)$
$\displaystyle=\xi_{1}\otimes\dots\otimes\xi_{n}\otimes\pi(g)\eta$
$\displaystyle=b_{T}^{\ast}(\pi(g)\eta)(\xi_{1}\otimes\dots\otimes\xi_{n}).$

Hence
$\Gamma_{T}(\pi(g))(b_{T}^{\ast}(\eta)+b_{T}(\eta))=b_{T}^{\ast}(\pi(g)\eta)+b_{T}(\pi(g)\eta)$.
Since $\pi(g)$ commutes with $J$ and $(U_{t})$ by assumption, we have
$\pi(g)\eta\in\operatorname{dom}(U_{i/2})$ and
$JU_{i/2}\pi(g)\eta=\pi(g)\eta$. Thus
$b_{T}^{\ast}(\pi(g)\eta)+b_{T}(\pi(g)\eta)\in\Gamma_{T}(H,J,(U_{t}))^{\prime}$.
∎

In the following we consider the correspondence $\mathcal{H}_{\pi}$ from
$L(G)$ to itself that was defined in Example 4.6. As discussed there, one can
identify $\mathcal{H}_{\pi}^{\otimes_{\tau}n}$ with $\ell^{2}(G)\otimes
H^{\otimes n}$, and we will tacitly do so in the following.

To fix notations, let us also briefly recall the definition of crossed
products. If $N$ is a von Neumann algebra, $\Gamma$ a discrete group and
$\rho\colon\Gamma\to\mathrm{Aut}(N)$ a group homomorphism, let

$\alpha\colon N\to B(\ell^{2}(\Gamma)\otimes
L^{2}(N)),\,\alpha(x)(\delta_{\gamma}\otimes\eta)=\delta_{\gamma}\otimes\rho(\gamma^{-1})x\eta.$

The crossed product $N\rtimes_{\rho}\Gamma$ is the von Neumann algebra
generated by $L(\Gamma)\otimes\mathbb{C}1$ and $\alpha(N)$ inside
$B(\ell^{2}(\Gamma)\otimes L^{2}(N))$.

If $\psi$ is a faithful normal state on $N$ that is invariant under $\rho$,
$\Omega_{\psi}\in L^{2}_{+}(N)$ the corresponding cyclic vector and

$\iota_{\psi}\colon\ell^{2}(\Gamma)\to\ell^{2}(\Gamma)\otimes
L^{2}(N),\,\delta_{\gamma}\mapsto\delta_{\gamma}\otimes\Omega_{\psi},$

then $E_{\psi}=\iota_{\psi}^{\ast}\,\cdot\,\iota_{\psi}$ is a faithful normal
conditional expectation from $N\rtimes_{\rho}\Gamma$ onto $L(\Gamma)$.

######  Proposition 6.7.

Let $T$ be an equivariant local compatible braided twist on $H\otimes H$. If
$\mathcal{T}=1_{\ell^{2}(G))}\otimes T$,
$\mathcal{U}_{t}=1_{\ell^{2}(G)}\otimes U_{t}$ and $\psi$ is the vacuum state
on $\Gamma_{T}(H,J,(U_{t}))$, then the operator-valued $W^{\ast}$-probability
spaces
$(L(G)\subset\Gamma_{\mathcal{T}}(\mathcal{H}_{\pi},\mathcal{J}_{\pi},(\mathcal{U}_{t})),E)$
and $(L(G)\subset\Gamma_{T}(H,J,(U_{t}))\rtimes_{\Gamma_{T}(\pi)}G,E_{\psi})$
are isomorphic.

###### Proof.

Let

$V_{n}\colon\ell^{2}(G)\otimes H_{T,n}\to\ell^{2}(G)\otimes
H_{T,n},\,\delta_{g}\otimes\xi\mapsto\delta_{g}\otimes\pi(g^{-1})^{\otimes
n}\xi.$

If $g,h\in G$ and $\xi,\eta\in H^{\otimes n}$, then

$\displaystyle\langle
V_{n}(\delta_{g}\otimes\xi),V_{n}(\delta_{h}\otimes\eta)\rangle_{\ell^{2}(G)\otimes
H_{T,n}}$ $\displaystyle=\delta_{g,h}\langle\pi(g^{-1})^{\otimes
n}\xi,\pi(g^{-1})^{\otimes n}\eta\rangle_{H_{T,n}}$
$\displaystyle=\delta_{g,h}\langle\pi(g^{-1})^{\otimes
n}\xi,\pi(g^{-1})^{\otimes n}P_{T,n}\eta\rangle_{H^{\otimes n}}$
$\displaystyle=\delta_{g,h}\langle\xi,P_{T,n}\eta\rangle_{H^{\otimes n}}$
$\displaystyle=\langle\delta_{g}\otimes\xi,\delta_{h}\otimes\eta\rangle_{\ell^{2}(G)\otimes
H_{T,n}},$

where we used that $\pi(g^{-1})^{\otimes n}$ commutes with $P_{T,n}$ since $T$
is equivariant. As the range of $V_{n}$ is dense for all $n\in\mathbb{N}$, we
conclude that $V_{n}$ is unitary for all $n\in\mathbb{N}$.

Moreover,

$\displaystyle V_{n}(\delta_{gh}\otimes\pi(g)^{\otimes
n}\xi)=\delta_{gh}\otimes\pi(h^{-1})\xi=(\lambda_{g}\otimes
1)V_{n}(\delta_{h}\otimes\xi)$

and

$\displaystyle V_{n+1}((1\otimes a^{\ast}_{T}(\xi))(\delta_{g}\otimes\eta))$
$\displaystyle=V_{n+1}(\delta_{g}\otimes\xi\otimes\eta)$
$\displaystyle=\delta_{g}\otimes\pi(g^{-1})\xi\otimes\pi(g^{-1})^{\otimes
n}\eta$
$\displaystyle=(1\otimes\Gamma_{T}(\pi(g^{-1}))a^{\ast}_{T}(\xi))(\delta_{g}\otimes\pi(g^{-1})^{\otimes}n\eta)$
$\displaystyle=(1\otimes\Gamma_{T}(\pi(g^{-1}))a^{\ast}_{T}(\xi))V_{n}(\delta_{g}\otimes\eta).$

Moreover, if $\xi\in\operatorname{dom}(U_{-i/2})$, then

$\mathcal{J}_{\pi}\mathcal{U}_{-i/2}(\delta_{e}\otimes\xi)=\mathcal{J}_{\pi}(\delta_{e}\otimes
U_{-i/2}\xi)=\delta_{e}\otimes JU_{-i/2}\xi.$

Hence
$\mathcal{J}_{\pi}\mathcal{U}_{-i/2}(\delta_{e}\otimes\xi)=(\delta_{e}\otimes\xi)$
if and only if $JU_{-i/2}\xi=\xi$.

Let

$V\colon\ell^{2}(G)\otimes\mathcal{F}_{T}(H)\to\ell^{2}(G)\otimes\mathcal{F}_{T}(H),\,V=1_{\ell^{2}(G)}\oplus\bigoplus_{n=1}^{\infty}V_{n}.$

Then $V$ is unitary and the previous computations show

$\displaystyle V(\lambda_{g}\cdot\xi)$ $\displaystyle=(\lambda_{g}\otimes
1)V\xi$ $\displaystyle Vs_{\mathcal{T}}(\delta_{e}\otimes\xi)$
$\displaystyle=1\otimes\alpha(s_{T}(\xi))V,$

where $\alpha\colon\Gamma_{T}(H,J,(U_{t}))\to
B(\ell^{2}(G)\otimes\mathcal{F}_{T}(H))$ is the representation in the
definition of the crossed product. Thus
$V\Gamma_{\mathcal{T}}(\mathcal{H}_{\pi},\mathcal{J}_{\pi},(\mathcal{U}_{t}))V^{\ast}=\Gamma_{T}(H,J,(U_{t}))\rtimes_{\Gamma_{T}(\pi)}G$
and $VL(G)V^{\ast}=L(G)\otimes\mathbb{C}1$. Furthermore,
$V\circ\iota_{\mathcal{T}}=\iota_{\psi}$, which implies
$E(\cdot)=E_{\psi}(V\,\cdot\,V^{\ast})$. ∎

######  Remark 6.8.

In the free case $T=0$, the structure of the crossed product
$\Gamma_{0}(H,J,(U_{t}))\rtimes_{\Gamma(\pi)}G$ has been studied in [HT21]. In
particular, the authors characterize factoriality, determine the type
classification and give sufficient criteria for fullness and strong solidity
for these crossed product algebras.

##  7\. Factoriality

In this section we study when the operator-valued Araki–Woods algebras are
factors. In contrast to the scalar-valued case,
$\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ can fail to
be a factor even for strict twists – for example, if
$\mathcal{H}=L^{2}(M)\otimes\mathbb{C}^{d}$, $\mathcal{J}=J_{\varphi}\otimes
J$ and $\mathcal{U}_{t}=\Delta_{\varphi}^{it}\otimes 1$, then it is not hard
to see that $\Gamma_{0}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))\cong
M\overline{\otimes}L(\mathbb{F}_{d})$ (compare with the proof of Theorem 6.3),
which is only a factor if $M$ is a factor. However, there are also interesting
cases when $\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$
is a factor even though $M$ is not (see [Shl99, Section 3] or Example 7.4
below). This already suggests that the factoriality question is more subtle
than in the scalar-valued case.

Here we give two sufficient criteria for factoriality. The first is based on
mixing properties of the Tomita correspondence, while the second one relies on
a suitable notion of the Tomita correspondence having sufficiently large
dimension as bimodule.

We start with the mixing property. Let $M$ be a von Neumann algebra. Following
the terminology used for example in [BBM19], a correspondence $\mathcal{H}$
over $M$ is called _jointly mixing_ if
$\pi_{l}^{\mathcal{H}}(x_{n})\pi_{r}^{\mathcal{H}}(y_{n})\to 0$ in the weak∗
topology whenever $(x_{n})$, $(y_{n})$ are bounded sequences in $M$ such that
$x_{n}\to 0$ or $y_{n}\to 0$ in the weak∗ topology.

######  Example 7.1.

If $\pi$ is a normal representation of $M$ on a Hilbert space $H$, then
$\mathrm{HS}(H)$ with the left and right action given by $x\cdot R\cdot
y=\pi(x)R\pi(y)$ is a jointly mixing correspondence from $M$ to itself. In
particular, the coarse bimodule $L^{2}(M)\otimes L^{2}(M)$ is jointly mixing.

Note that if $(\pi,H,(U_{t}))$ is a covariant representation of
$(M,\sigma^{\varphi}_{t})$, that is,
$U_{t}\pi(x)U_{-t}=\pi(\sigma^{\varphi}_{t}(x))$, then one can turn
$\mathrm{HS}(H)$ into a Tomita correspondence by defining
$\mathcal{J}(R)=R^{\ast}$ and $\mathcal{U}_{t}(R)=U_{t}RU_{-t}$.

######  Lemma 7.2.

If $\mathcal{H}$ is a jointly mixing Tomita correspondence, then
$\mathcal{H}_{\mathcal{T},k}$ is jointly mixing for all $k\geq 1$ and all
strict twists $\mathcal{T}$.

###### Proof.

We first prove that the relative tensor product
$\mathcal{H}\otimes_{\varphi}\mathcal{K}$ is jointly mixing if $\mathcal{K}$
is jointly mixing. Let $(x_{n})$, $(y_{n})$ be bounded sequences in $M$ such
that one of them converges to $0$ in the weak∗ topology.

Since the sequences are bounded, it suffices to show that

$\langle\xi_{1}\otimes_{\varphi}\eta_{1},x_{n}\xi_{2}\otimes_{\varphi}\eta_{2}y_{n}\rangle_{\mathcal{H}\otimes_{\varphi}\mathcal{K}}\to
0$

for all $\xi_{1},\xi_{2}\in\mathcal{H}$ left $\varphi$-bounded and
$\eta_{1},\eta_{2}\in\mathcal{K}$.

We have

$\langle\xi_{1}\otimes_{\varphi}\eta_{1},x_{n}\xi_{2}\otimes_{\varphi}\eta_{2}y_{n}\rangle_{\mathcal{H}\otimes_{\varphi}\mathcal{K}}=\langle\eta_{1},\pi_{l}^{\mathcal{K}}(L_{\varphi}(\xi_{1})^{\ast}x_{n}L_{\varphi}(\xi_{2}))\pi_{r}^{\mathcal{K}}(y_{n})\eta_{2}\rangle_{\mathcal{K}}.$

By assumption, $L_{\varphi}(\xi_{1})^{\ast}x_{n}L_{\varphi}(\xi_{2})\to 0$ or
$y_{n}\to 0$ in the weak∗ topology. As $\mathcal{K}$ is jointly mixing, it
follows that
$\langle\xi_{1}\otimes_{\varphi}\eta_{1},x_{n}\xi_{2}\otimes_{\varphi}\eta_{2}y_{n}\rangle_{\mathcal{H}\otimes_{\varphi}\mathcal{K}}\to
0$.

It follows by induction that $\mathcal{H}^{\otimes_{\varphi}k}$ is jointly
mixing for all $k\geq 1$. To prove that $\mathcal{H}_{\mathcal{T},k}$ is
jointly mixing, let again $(x_{n})$, $(y_{n})$ be bounded sequences in $M$
such that one of them converges to $0$ in the weak∗ topology. Again, by a
density argument it suffices to show that

$\langle\xi,x_{n}\eta y_{n}\rangle_{\mathcal{T},k}\to 0$

for all $\xi,\eta\in\mathcal{H}^{\otimes_{\varphi}k}$. Since
$\mathcal{P}_{\mathcal{T},k}$ is a bimodule map, we have $\langle\xi,x_{n}\eta
y_{n}\rangle_{\mathcal{T},k}=\langle\xi,x_{n}(\mathcal{P}_{\mathcal{T},k}\eta)y_{n}\rangle_{\mathcal{H}^{\otimes_{\varphi}k}}$,
and the claimed convergence follows immediately from the fact that
$\mathcal{H}^{\otimes_{\varphi}k}$ is jointly mixing. ∎

The following result is an adaptation of [KV19, Theorem 5.1], where the result
was proved under the assumption that $\varphi$ is a trace, $\mathcal{T}=0$ and
$\mathcal{U}_{t}=\mathrm{id}_{\mathcal{H}}$ (with a slightly different mixing
condition). We call an element of $M$ _diffuse_ if it is self-adjoint and its
spectral measure has no atoms.

######  Proposition 7.3.

Let $(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ be a jointly mixing Tomita
correspondence over $(M,\varphi)$ with $\varphi$ finite and let $\mathcal{T}$
be a local compatible braided strict twist on
$\mathcal{H}\otimes_{\varphi}\mathcal{H}$. If the centralizer $M^{\varphi}$ of
$\varphi$ contains a diffuse element, then

$\mathcal{Z}(\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t})))=\\{z\in\mathcal{Z}(M)\mid
z\xi=\xi z\text{ for all }\xi\in\mathcal{H}\\}.$

In particular, if $M$ is a factor, so is
$\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$.

###### Proof.

If $x\in M^{\varphi}$ is diffuse, then there exists a sequence $(u_{n})$ of
unitaries in $\\{x\\}^{\prime\prime}$ that converges to $0$ in the weak∗
topology. Let
$z\in\mathcal{Z}(\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t})))$
and let $\Omega\in L^{2}_{+}(M)$ denote the cyclic and separating vector
representing $\varphi$. We have

$(z-E_{\mathcal{T}}(z))\Omega=(u_{n}zu_{n}^{\ast}-E_{\mathcal{T}}(u_{n}zu_{n}^{\ast}))\Omega=u_{n}(z-E_{\mathcal{T}}(z))u_{n}^{\ast}\Omega=u_{n}(z-E_{\mathcal{T}}(z))\Omega
u_{n}^{\ast}.$

Note that
$(z-E_{\mathcal{T}}(z))\Omega\in\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus
L^{2}(M)$. It follows from Lemma 7.2 that $u_{n}(z-E(z))\Omega u_{n}^{\ast}\to
0$ weakly. Thus $z=E_{\mathcal{T}}(z)\in M\cap M^{\prime}$.

Moreover,
$z\in\mathcal{Z}(\Gamma_{M}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$
implies

$z\xi=zs_{\mathcal{T}}(\xi)\Omega=s_{\mathcal{T}}(\xi)z\Omega=\xi z$

for all bounded vectors $\xi\in\mathcal{H}$. As bounded vectors are dense in
$\mathcal{H}$, it follows that $z\xi=\xi z$ for all $\xi\in\mathcal{H}$.

Conversely, if $x\in\mathcal{Z}(M)$ and $x\xi=\xi x$ for all
$\xi\in\mathcal{H}$, then

$xs_{\mathcal{T}}(\xi)\Omega=x\xi=\xi
x=s_{\mathcal{T}}(\xi)J_{\varphi}x^{\ast}\Omega=s_{\mathcal{T}}(\xi)x\Omega$

for all left-bounded vectors $\xi\in\mathcal{H}$ with
$\mathcal{J}\mathcal{U}_{-i/2}\xi=\xi$. Therefore,
$x\in\mathcal{Z}(\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t})))$.
∎

######  Example 7.4.

Let $(\pi,H,(U_{t}))$ be a normal faithful covariant representation of
$(M,\sigma^{\varphi})$ and let
$(\mathrm{HS}(H),\mathcal{J},(\mathcal{U}_{t}))$ be the Tomita correspondence
described in Example 7.1. This correspondence is jointly mixing. Moreover, if
$\pi(x)R=R\pi(x)$ for all $R\in\mathrm{HS}(H)$, then $\pi(x)\in\mathbb{C}1$,
hence $x\in\mathbb{C}1$ since $\pi$ was assumed to be faithful. Thus
$\Gamma_{\mathcal{T}}(\mathrm{HS}(H),\mathcal{J},(\mathcal{U}_{t}))$ is a
factor if $M^{\varphi}$ contains a diffuse element and $\mathcal{T}$ is a
local compatible braided strict twist on
$\mathcal{H}\otimes_{\varphi}\mathcal{H}$.

For the second factoriality criterion we need the operator inequalities from
Lemma 5.1. For the ease of notation, we write

$c(q)=\prod_{k=1}^{\infty}\frac{1-q^{k}}{1+q^{k}},\quad d(q)=(1-q)^{-1}$

for $0\leq q<1$. Note that these are the limits as $n\to\infty$ of the
constants appearing in Lemma 5.1.

The following result is a direct consequence of Lemma 5.1.

######  Lemma 7.5.

Let $M$ be a von Neumann algebra, $\varphi$ a normal semi-finite faithful
weight on $M$ and $\mathcal{H}$ a correspondence from $M$ to itself. If
$\mathcal{T}$ is a braided twist on $\mathcal{H}\otimes_{\varphi}\mathcal{H}$
with $\lVert\mathcal{T}\rVert<1$, then the identity map on
$L^{\infty}(\mathcal{H}_{M},\varphi)\odot\mathcal{F}_{\mathcal{T}}(\mathcal{H})$
(resp. $\mathcal{F}_{\mathcal{T}}(\mathcal{H})\odot
L^{\infty}(_{M}\mathcal{H},\varphi)$) extends to a bounded invertible operator
$j_{l}$ (resp. $j_{r}$) from
$\mathcal{H}\otimes_{\varphi}\mathcal{F}_{\mathcal{T}}(\mathcal{H})$ (resp.
$\mathcal{F}_{\mathcal{T}}(\mathcal{H})\otimes_{\varphi}\mathcal{H}$) to
$\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus L^{2}(M)$ with $\lVert
j_{l/r}\rVert\leq d(\lVert\mathcal{T}\rVert)^{1/2}$, $\lVert
j_{l/r}^{-1}\rVert\leq c(\lVert\mathcal{T}\rVert)^{-1/2}$.

######  Proposition 7.6 (Spectral Gap).

Let $(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ be a Tomita correspondence
over $(M,\varphi)$ and assume there exists a finite family $(\xi_{i})_{i\in
I}$ in $\mathcal{H}_{0}$ such that

  * •

$\mathcal{J}\xi_{i}=\xi_{i}$ for all $i\in I$,

  * •

$L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\xi_{j})=\delta_{ij}1$ for all $i,j\in
I$,

  * •

there exists a family $(p_{i})_{i\in I}$ of projections in $M$ such that
$L_{\varphi}(\mathcal{U}_{i/2}\xi_{i})^{\ast}L_{\varphi}(\mathcal{U}_{i/2}\xi_{j})=\delta_{ij}p_{i}$
for all $i,j\in I$.

There exists a function $f\colon[0,1)\to[0,\infty)$ such that whenever
$\mathcal{T}$ is a local braided compatible twist on
$\mathcal{H}\otimes_{\varphi}\mathcal{H}$ with $\lVert\mathcal{T}\rVert<1$ and
$\lvert I\rvert>f(\lVert\mathcal{T}\rVert)$, then there exists a constant
$\kappa>0$ such that

$\sum_{i\in
I}\lVert\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\mathcal{F}_{\mathcal{T}}(\mathcal{J})\xi_{i}-x\xi_{i}\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}\geq\kappa^{2}\hat{\varphi}_{\mathcal{T}}(\lvert
x-E_{\mathcal{T}}(x)\rvert^{2})$

for all $x$ in the centralizer of $\hat{\varphi}_{\mathcal{T}}$.

###### Proof.

The idea of the proof follows [Śni04]. First note that the assumptions imply
$\mathcal{J}\mathcal{U}_{-i/2}\xi_{i}=\mathcal{U}_{i/2}\xi_{i}$,
$\mathcal{J}\mathcal{U}_{i/2}\xi_{i}=\mathcal{U}_{-i/2}\xi_{i}$ and
$R_{\varphi}(\xi_{i})^{\ast}R_{\varphi}(\xi_{j})=\delta_{ij}1$,
$R_{\varphi}(\mathcal{U}_{-i/2}\xi_{i})^{\ast}R_{\varphi}(\mathcal{U}_{-i/2}\xi_{j})=\delta_{ij}J_{\varphi}p_{i}J_{\varphi}$.

Define

$\displaystyle S$
$\displaystyle\colon\mathcal{F}_{\mathcal{T}}(\mathcal{H})\to\ell^{2}(I)\otimes\mathcal{F}_{\mathcal{T}}(\mathcal{H}),\,S\eta=\sum_{i\in
I}e_{i}\otimes(a_{\mathcal{T}}(\mathcal{U}_{i/2}\xi_{i})-b_{\mathcal{T}}(\mathcal{U}_{-i/2}\xi_{i}))\eta,$
$\displaystyle T$
$\displaystyle\colon\mathcal{F}_{\mathcal{T}}(\mathcal{H})\to\ell^{2}(I)\otimes\mathcal{F}_{\mathcal{T}}(\mathcal{H}),\,T\eta=\sum_{i\in
I}e_{i}\otimes(a_{\mathcal{T}}^{\ast}(\xi_{i})-b_{\mathcal{T}}^{\ast}(\xi_{i}))\eta,$

where $(e_{i})$ is the canonical orthonormal basis for $\ell^{2}(I)$.

_Step 1:_ We first show that $\lVert S\rVert\leq
2d(\lVert\mathcal{T}\rVert)^{1/2}$.

For that purpose let $S_{l}\eta=\sum_{i\in I}e_{i}\otimes
a_{\mathcal{T}}(\mathcal{U}_{i/2}\xi_{i})\eta$ and $S_{r}\eta=\sum_{i\in
I}e_{i}\otimes b_{\mathcal{T}}(\mathcal{U}_{-i/2}\xi_{i})\eta$. With

$\Pi_{i}\colon\ell^{2}(I)\otimes\mathcal{F}_{\mathcal{T}}(\mathcal{H})\to\mathcal{F}_{\mathcal{T}}(\mathcal{H}),\,e_{j}\otimes\eta\mapsto\delta_{ij}\eta$

we have $S_{l}^{\ast}=\sum_{i\in
I}a_{\mathcal{T}}^{\ast}(\mathcal{U}_{i/2}\xi_{i})\Pi_{i}$ and
$S_{r}^{\ast}=\sum_{i\in
I}b_{\mathcal{T}}^{\ast}(\mathcal{U}_{-i/2}\xi_{i})\Pi_{i}$.

From Lemma 5.1 we deduce

$\displaystyle\left\lVert S_{l}^{\ast}\left(\sum_{i\in
I}e_{i}\otimes\eta_{i}\right)\right\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}$
$\displaystyle=\left\lVert\sum_{i\in
I}a_{\mathcal{T}}^{\ast}(\mathcal{U}_{i/2}\xi_{i})\eta_{i}\right\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}$
$\displaystyle=\left\lVert\sum_{i\in
I}\mathcal{U}_{i/2}\xi_{i}\otimes_{\varphi}\eta_{i}\right\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}$
$\displaystyle\leq d(\lVert\mathcal{T}\rVert)\left\lVert\sum_{i\in
I}\mathcal{U}_{i/2}\xi_{i}\otimes_{\varphi}\eta_{i}\right\rVert_{\mathcal{H}\otimes_{\varphi}\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}$
$\displaystyle=d(\lVert\mathcal{T}\rVert)\sum_{i,k\in
I}\langle\eta_{i},L_{\varphi}(\mathcal{U}_{i/2}\xi_{i})^{\ast}L_{\varphi}(\mathcal{U}_{i/2}\xi_{k})\eta_{k}\rangle_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}$
$\displaystyle=d(\lVert\mathcal{T}\rVert)\sum_{i\in I}\lVert
p_{i}\eta_{i}\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}$
$\displaystyle\leq d(\lVert\mathcal{T}\rVert)\left\lVert\sum_{i\in
I}e_{i}\otimes\eta_{i}\right\rVert_{\ell^{2}(I)\otimes\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}.$

It follows that $\lVert S_{l}\rVert\leq d(\lVert\mathcal{T}\rVert)^{1/2}$. An
analogous calculation shows $\lVert S_{r}\rVert\leq
d(\lVert\mathcal{T}\rVert)^{1/2}$. Thus $\lVert S\rVert\leq
2d(\lVert\mathcal{T}\rVert)^{1/2}$.

_Step 2:_ We prove that $\lvert T\rvert\geq\lvert
I\rvert^{1/2}c(\lVert\mathcal{T}\rVert)^{1/2}-\lvert
I\rvert^{-1/2}d(\lVert\mathcal{T}\rVert)^{1/2}$ on
$\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus L^{2}(M)$.

For $\eta_{1},\dots,\eta_{m}\in\mathcal{H}$ left-bounded and
$\zeta_{1},\dots,\zeta_{m}\in\mathcal{F}_{\mathcal{T}}(\mathcal{H})$ let

$\displaystyle X\sum_{k=1}^{m}\eta_{k}\otimes_{\varphi}\zeta_{k}=\sum_{i\in
I}\sum_{k=1}^{m}L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\eta_{k})\zeta_{k}\otimes_{\varphi}\xi_{i}.$

We have

$\displaystyle\left\lVert\sum_{i\in
I}\sum_{k=1}^{m}L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\eta_{k})\zeta_{k}\otimes_{\varphi}\xi_{i}\right\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})\otimes_{\varphi}\mathcal{H}}^{2}$
$\displaystyle=\sum_{i\in
I}\left\lVert\sum_{k=1}^{m}L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\eta_{k})\zeta_{k}\right\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}$
$\displaystyle=\sum_{k,l=1}^{m}\left\langle\zeta_{k},L_{\varphi}(\eta_{k})^{\ast}\sum_{i\in
I}L_{\varphi}(\xi_{i})L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\eta_{l})\zeta_{l}\right\rangle_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}$

Since

$\left(\sum_{i\in
I}L_{\varphi}(\xi_{i})L_{\varphi}(\xi_{i})^{\ast}\right)^{2}=\sum_{i,j\in
I}L_{\varphi}(\xi_{i})L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\xi_{j})L_{\varphi}(\xi_{j})^{\ast}=\sum_{i\in
I}L_{\varphi}(\xi_{i})L_{\varphi}(\xi_{i})^{\ast},$

we have $\sum_{i\in I}L_{\varphi}(\xi_{i})L_{\varphi}(\xi_{i})^{\ast}\leq 1$.

Therefore

$\displaystyle\left\lVert\sum_{i\in
I}\sum_{k=1}^{m}L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\eta_{k})\zeta_{k}\otimes_{\varphi}\xi_{i}\right\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})\otimes_{\varphi}\mathcal{H}}^{2}$
$\displaystyle\leq\sum_{k,l=1}^{m}\left\langle\zeta_{k},L_{\varphi}(\eta_{k})^{\ast}L_{\varphi}(\eta_{l})\zeta_{l}\right\rangle_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}$
$\displaystyle=\left\langle\sum_{k=1}^{m}\eta_{k}\otimes_{\varphi}\zeta_{k},\sum_{l=1}^{m}\eta_{l}\otimes_{\varphi}\zeta_{l}\right\rangle_{\mathcal{H}\otimes_{\varphi}\mathcal{F}_{\mathcal{T}}(\mathcal{H})}.$

Thus $X$ extends to a contraction from
$\mathcal{H}\otimes_{\varphi}\mathcal{F}_{\mathcal{T}}(\mathcal{H})$ to
$\mathcal{F}_{\mathcal{T}}(\mathcal{H})\otimes_{\varphi}\mathcal{H}$. It
follows from Lemma 7.5 that $\tilde{X}=j_{r}Xj_{l}^{-1}$ is a bounded linear
operator on $\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus L^{2}(M)$ with
$\lVert\tilde{X}\rVert\leq
c(\lVert\mathcal{T}\rVert)^{-1/2}d(\lVert\mathcal{T}\rVert)^{1/2}$.

For $\eta\in\mathcal{F}_{\mathcal{T}}(\mathcal{H})$ let

$Y\eta=\sum_{i\in I}e_{i}\otimes\xi_{i}\otimes_{\varphi}\eta.$

We have

$\displaystyle\lVert
Y\eta\rVert_{\ell^{2}(I)\otimes\mathcal{H}\otimes_{\varphi}\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}=\sum_{i\in
I}\lVert\xi_{i}\otimes_{\varphi}\eta\rVert_{\mathcal{H}\otimes_{\varphi}\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}=\sum_{i\in
I}\lVert\eta\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}=\lvert
I\rvert\lVert\eta\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}.$

Thus $Y$ is a bounded operator from $\mathcal{F}_{\mathcal{T}}(\mathcal{H})$
to
$\ell^{2}(I)\otimes\mathcal{H}\otimes_{\varphi}\mathcal{F}_{\mathcal{T}}(\mathcal{H})$
with norm $\lvert I\rvert^{1/2}$. Therefore, another application of Lemma 7.5
shows that $\tilde{Z}=Y^{\ast}(1\otimes j_{l}^{-1})$ defines a bounded linear
operator from
$\ell^{2}(I)\otimes(\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus L^{2}(M))$
to $\mathcal{F}_{\mathcal{T}}(\mathcal{H})$ with norm
$\lVert\tilde{Z}\rVert\leq\lvert
I\rvert^{1/2}c(\lVert\mathcal{T}\rVert)^{-1/2}$.

A direct computation shows that $\tilde{Z}T=\lvert I\rvert-\tilde{X}$ on
$\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus L^{2}(M)$. Hence, whenever
$\eta\in\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus L^{2}(M)$, then

$\displaystyle\lvert I\rvert^{1/2}c(\lVert\mathcal{T}\rVert)^{-1/2}\lVert
T\eta\rVert\geq\lVert\tilde{Z}T\eta\rVert=\lVert(\lvert
I\rvert-\tilde{X})\eta\rVert\geq(\lvert
I\rvert-c(\lVert\mathcal{T}\rVert)^{-1/2}d(\lVert\mathcal{T}\rVert)^{1/2})\lVert\eta\rVert.$

Therefore,

$\lvert T\rvert\geq c(\lVert\mathcal{T}\rVert)^{1/2}\lvert
I\rvert^{1/2}-d(\lVert\mathcal{T}\rVert)^{1/2}\lvert I\rvert^{-1/2}.$

_Step 3:_ We show that $\lvert S+T\rvert\geq
c(\lVert\mathcal{T}\rVert)^{1/2}\lvert
I\rvert^{1/2}-d(\lVert\mathcal{T}\rVert)^{1/2}\lvert
I\rvert^{-1/2}-2d(\lVert\mathcal{T}\rVert)^{1/2}$ on
$\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus L^{2}(M)$.

If $\eta\in\mathcal{F}_{\mathcal{T}}(\mathcal{H})\ominus L^{2}(M)$, then the
inequalities from Step 1 and Step 2 combined imply

$\lVert(S+T)\eta\rVert\geq\lVert T\eta\rVert-\lVert
S\eta\rVert\geq(c(\lVert\mathcal{T}\rVert)^{1/2}\lvert
I\rvert^{1/2}-d(\lVert\mathcal{T}\rVert)^{1/2}\lvert
I\rvert^{-1/2})\lVert\eta\rVert-2d(\lVert\mathcal{T}\rVert)^{1/2}\lVert\eta\rVert.$

_Step 4:_ Let $\kappa=c(\lVert\mathcal{T}\rVert)^{1/2}\lvert
I\rvert^{1/2}-d(\lVert\mathcal{T}\rVert)^{1/2}\lvert
I\rvert^{-1/2}-2d(\lVert\mathcal{T}\rVert)^{1/2}$. We show that if $x$ is in
the centralizer of $\hat{\varphi}_{\mathcal{T}}$ and $\kappa\geq 0$, then

$\sum_{i\in
I}\lVert\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\mathcal{F}_{\mathcal{T}}(\mathcal{J})\xi_{i}-x\xi_{i}\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}\geq\kappa^{2}\hat{\varphi}_{\mathcal{T}}(\lvert
x-E_{\mathcal{T}}(x)\rvert^{2}).$

Since $x$ is in the centralizer of $\hat{\varphi}_{\mathcal{T}}$ and
$E_{\mathcal{T}}$ is a weight-preserving conditional expectation,
$E_{\mathcal{T}}(x)$ is also in the centralizer of
$\hat{\varphi}_{\mathcal{T}}$. Hence we can assume without loss of generality
$E_{\mathcal{T}}(x)=0$. Moreover, the assumption
$L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\xi_{i})=1$ implies

$\varphi(1)=\varphi(L_{\varphi}(\xi_{i})^{\ast}L_{\varphi}(\xi_{i}))=\lVert\xi_{i}\rVert_{\mathcal{H}}^{2}<\infty$

by Lemma 5.14. Let $\Omega\in L^{2}_{+}(M)$ denote the cyclic and separating
vector representing $\varphi$. Since $x$ is in the centralizer of
$\hat{\varphi}_{\mathcal{T}}$, it follows from Theorem 5.18 that
$\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\Omega=x\Omega$.

Now we use
$\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\mathcal{F}_{\mathcal{T}}(\mathcal{J})$
commutes with
$a_{\mathcal{T}}^{\ast}(\xi_{i})+a_{\mathcal{T}}(\mathcal{U}_{i/2}\xi_{i})$
and $x$ commutes with
$b_{\mathcal{T}}^{\ast}(\xi_{i})+b_{\mathcal{T}}(\mathcal{U}_{-i/2}\xi_{i})$
for all $i\in I$ to deduce

$\displaystyle\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\mathcal{F}_{\mathcal{T}}(\mathcal{J})\xi_{i}$
$\displaystyle=\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\mathcal{F}_{\mathcal{T}}(\mathcal{J})(a_{\mathcal{T}}^{\ast}(\xi_{i})+a_{\mathcal{T}}(\mathcal{U}_{i/2}\xi_{i})\Omega$
$\displaystyle=(a_{\mathcal{T}}^{\ast}(\xi_{i})+a_{\mathcal{T}}(\mathcal{U}_{i/2}\xi_{i}))\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\Omega$
$\displaystyle=(a_{\mathcal{T}}^{\ast}(\xi_{i})+a_{\mathcal{T}}(\mathcal{U}_{i/2}\xi_{i}))x\Omega$

and

$x\xi_{i}=x(b_{\mathcal{T}}^{\ast}(\xi_{i})+b_{\mathcal{T}}(\mathcal{U}_{-i/2}\xi_{i}))(\xi_{i})\Omega=(b_{\mathcal{T}}^{\ast}(\xi_{i})+b_{\mathcal{T}}(\mathcal{U}_{-i/2}\xi_{i}))x\Omega.$

Therefore,

$\sum_{i\in
I}\lVert\mathcal{F}_{\mathcal{T}}(\mathcal{J})x\mathcal{F}_{\mathcal{T}}(\mathcal{J})\xi_{i}-x\xi_{i}\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2}=\lVert(S+T)x\Omega\rVert_{\mathcal{F}_{\mathcal{T}}(\mathcal{H})}^{2},$

and $x\Omega\in L^{2}(M)^{\perp}$ because $E_{\mathcal{T}}(x)=0$. Now the
claim follows from Step 3. ∎

######  Corollary 7.7.

Under the assumptions of Proposition 7.6, the center of the operator-valued
twisted Araki–Woods algebra is given by

$\mathcal{Z}(\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t})))=\\{x\in\mathcal{Z}(M)\mid\xi
x=x\xi\text{ for all }\xi\in\mathcal{H}\\}.$

In particular, if $M$ is a factor, then
$\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ is a factor.

###### Proof.

If $x$ is in the center of
$\Gamma_{\mathcal{T}}(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$, then $x$ is
in the centralizer of $\hat{\varphi}_{\mathcal{T}}$. Moreover,

$\displaystyle x\xi$
$\displaystyle=x(a_{\mathcal{T}}^{\ast}(\xi)+a_{\mathcal{T}}(\mathcal{J}\mathcal{U}_{-i/2}\xi))\Omega$
$\displaystyle=(a_{\mathcal{T}}^{\ast}(\xi)+a_{\mathcal{T}}(\mathcal{J}\mathcal{U}_{-i/2}\xi))x\Omega$
$\displaystyle=(a_{\mathcal{T}}^{\ast}(\xi)+a_{\mathcal{T}}(\mathcal{J}\mathcal{U}_{-i/2}\xi))\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\mathcal{F}_{\mathcal{T}}(\mathcal{J})\Omega$
$\displaystyle=\mathcal{F}_{\mathcal{T}}(\mathcal{J})x^{\ast}\mathcal{F}_{\mathcal{T}}(\mathcal{J})\xi.$

for all $\xi\in\mathcal{H}_{0}$. Thus $x=E_{\mathcal{T}}(x)\in M$ by
Proposition 7.6. Hence $x\in\mathcal{Z}(M)$ and $x\xi=\xi x$ for all
$\xi\in\mathcal{H}$ follows from the computation above by the density of
$\mathcal{H}_{0}$ in $\mathcal{H}$.

The converse inclusion can be shown exactly as in the proof of Proposition
7.3. ∎

##  8\. Application to Quantum Markov Semigroups

Tomita correspondences occur naturally in the study of generators of GNS-
symmetric quantum Markov semigroups and their relation to derivations [Wir22,
Wir24a]. Operator-valued twisted Araki–Woods algebras enter the picture if one
wants to obtain derivations with values in a von Neumann superalgebra instead
of a bimodule.

In [JRS] in the tracial case and in [Wir22, Section 7] in the general case,
operator-valued free Araki–Woods algebras were employed for this purpose. One
disadvantage is that they are always infinite-dimensional (unless the Tomita
correspondence is trivial), even if the base algebra is finite-dimensional. In
this section we used a twisted algebra to show that one can always choose the
von Neumann superalgebra to be finite-dimensional if the base algebra is a
matrix algebra.

Moreover, we use the disintegration theory from Section 3 to introduce the
notion of Bohr spectrum of a quantum Markov semigroup on a semi-finite von
Neumann algebra in a way that it coincides with the set of Bohr frequencies
occurring in Alicki’s theoremm for quantum Markov semigroups on matrix
algebras.

Let us briefly recall the relevant terminology regarding quantum Markov
semigroups. A _quantum Markov semigroup_ on a von Neumann algebra $M$ is a
family $(P_{t})_{t\geq 0}$ of normal unital completely positive maps on $M$
such that $P_{0}=\mathrm{id}_{M}$, $P_{s}P_{t}=P_{s+t}$ for all $s,t\geq 0$
and $P_{t}\to 0$ as $t\to 0$ in the point-weak∗ topology.

If $\varphi$ is a normal semi-finite faithful weight on $M$, then the quantum
Markov semigroup $(P_{t})$ is called _GNS-symmetric with respect to $\varphi$_
if $\varphi\circ P_{t}\leq\varphi$ and

$\varphi(P_{t}(x)^{\ast}y)=\varphi(x^{\ast}P_{t}(y))$

for all $x,y\in\mathfrak{n}_{\varphi}$, $t\geq 0$.

A GNS-symmetric quantum Markov semigroup $(P_{t})$ gives rise to a strongly
continuous symmetric contraction semigroup $(T_{t})$ on $L^{2}(M)$
characterized by $T_{t}(\Lambda_{\varphi}(x))=\Lambda_{\varphi}(P_{t}(x))$ for
$x\in\mathfrak{n}_{\varphi}$, $t\geq 0$. By semigroup theory, $(T_{t})$ is of
the form $T_{t}=e^{-t\mathcal{L}}$ for some positive self-adjoint operator
$\mathcal{L}$ on $L^{2}(M)$. By [Wir22, Theorem 6.3], the set

$\mathfrak{a}_{\mathcal{L}}=\left\\{x\in\bigcap_{z\in\mathbb{C}}\operatorname{dom}(\sigma^{\varphi}_{z}):\sigma^{\varphi}_{z}(x)\in\mathfrak{n}_{\varphi}\cap\mathfrak{n}_{\varphi}^{\ast},\,\Lambda_{\varphi}(\sigma^{\varphi}_{z}(x))\in\operatorname{dom}(\mathcal{L}^{1/2})\text{
for all }z\in\mathbb{C}\right\\}$

is a weak∗ dense $\ast$-subalgebra of $M$ such that
$\Lambda_{\varphi}(\mathfrak{a}_{\mathcal{L}})$ is a core for
$\mathcal{L}^{1/2}$.

Following the terminology introduced in the tracial case by Junge, Li and
LaRacuente in [LJL20] (see also [BGJ22]), a _derivation triple_ for $(P_{t})$
is a a triple $(M\subset\hat{M},E,\delta)$ consisting of

  * •

a unital inclusion $M\subset\hat{M}$ of von Neumann algebras,

  * •

a faithful normal conditional expectation $E\colon\hat{M}\to M$, and

  * •

a closed operator $\delta\colon\operatorname{dom}(\mathcal{L}^{1/2})\to
L^{2}(\hat{M})$ satisfying $\delta\circ J_{\varphi}=J_{\varphi\circ
E}\circ\delta$,
$\delta\circ\Delta_{\varphi}^{it}=\Delta_{\hat{\varphi}}^{it}\circ\delta$ for
all $t\in\mathbb{R}$, and

$\delta(\Lambda_{\varphi}(x)y)=\delta(x\Lambda_{\varphi}^{\prime}(y))=x\delta(\Lambda_{\varphi}^{\prime}(y))+\delta(\Lambda_{\varphi}(x))y$

for $x,y\in\mathfrak{a}_{\mathcal{L}}$

such that $\mathcal{L}=\delta^{\ast}\delta$.

It was shown in [Wir22, Corollary 7.6] that a GNS-symmetric quantum Markov
semigroup admits a derivation triple if and only if it satisfies a property
called $\Gamma$-regularity. Every GNS-symmetric quantum Markov semigroup on a
finite-dimensional von Neumann algebra or more generally every GNS-symmetric
quantum Markov semigroup with bounded generator is $\Gamma$-regular.

######  Proposition 8.1.

If $M$ is a type I factor, $\varphi$ a normal semifinite faithful weight on
$M$ and $(P_{t})$ a quantum Markov semigroup that is GNS-symmetric with
respect to $\varphi$ and $\Gamma$-regular, then $(P_{t})$ admits a derivation
triple $(M\subset\hat{M},E,\delta)$ with
$\hat{M}=M\overline{\otimes}\Gamma_{T}(H,J,(U_{t}))$ and
$E=\mathrm{id}\otimes\psi$, where $\psi$ is the vacuum state on
$\Gamma_{T}(H,J,(U_{t}))$. Moreover, if $M$ is finite, then $H$ can be chosen
finite-dimensional.

###### Proof.

Let $(M\subset N,F,\partial)$ be any derivation triple for $(P_{t})$ and let
$\mathcal{H}$ be the closed linear hull of $\\{x\partial(a)y\mid x,y\in
M,\,a\in\operatorname{dom}(\mathcal{L}^{1/2})\\}$. By definition of a
derivation triple, $J_{\varphi\circ F}$ and $\Delta^{it}_{\varphi\circ F}$
leave $\mathcal{H}$ invariant, making it into a Tomita correspondence
$(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ over $(M,\varphi)$. Moreover, if
$M$ is finite-dimensional, so is $\mathcal{H}$.

By Theorem 3.7 we can write $\mathcal{H}=L^{2}(M)\otimes H$,
$\mathcal{J}=J_{\varphi}\otimes J$,
$\mathcal{U}_{t}=\Delta_{\varphi}^{it}\otimes U_{t}$. Taking $\delta=\partial$
viewed as map into
$L^{2}(M\overline{\otimes}\Gamma_{\mathcal{T}}(H,J,(U_{t}))$ and $\hat{M}$,
$E$ as in the statement of the theorem does the job. ∎

######  Corollary 8.2.

Every GNS-symmetric quantum Markov semigroup $(P_{t})$ on $M_{n}(\mathbb{C})$
admits a derivation triple $(M\subset\hat{M},E,\delta)$ with $\hat{M}$ finite-
dimensional.

###### Proof.

If one takes $T$ to be minus the flip map in the previous theorem, then
$\Gamma_{T}(H,J,(U_{t}))$ is contained in the bounded linear operators on the
fermionic Fock space over $H$. In particular, if $H$ is finite-dimensional, so
is $\Gamma_{T}(H,J,(U_{t}))$ and then
$M_{n}(\mathbb{C})\otimes\Gamma_{T}(H,J,(U_{t}))$ is also finite dimensional.
∎

As the previous results show, derivation triples for a given quantum Markov
semigroup are not unique, even if one makes the natural minimality assumption
that the range of $\delta$ generate $L^{2}(\hat{M})$ as a correspondence. What
is unique (up to isomorphism) however is the Tomita correspondence
$(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ generated by the range of
$\delta$ with the restriction of the modular data from $\hat{M}$ [Wir22,
Theorem 6.9].

In particular, if $M$ is semi-finite with normal faithful trace $\tau$ and
$\varphi=\tau(h^{1/2}\,\cdot\,h^{1/2})$, then the spectrum of
$\mathcal{U}_{t}(h^{-it}\,\cdot\,h^{it})$ is an invariant of $(P_{t})$. This
leads us to the following definition.

######  Definition 8.3.

If $(P_{t})$ is a GNS-symmetric quantum Markov semigroup and
$(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ the associated Tomita
correspondence, then the _Bohr spectrum_ of $(P_{t})$ is the set of all
$\omega\in\mathbb{R}$ for which the operator
$\mathcal{U}_{t}(h^{-it}\,\cdot\,h^{it})-e^{i\omega t}$ on $\mathcal{H}$ is
not invertible for some (or all) $t\in\mathbb{R}$.

To give an interpretation of the Bohr spectrum, we show how it is related to
the Bohr frequencies occurring in Alicki’s theorem on generators of GNS-
symmetric quantum Markov semigroups on $M_{n}(\mathbb{C})$.

######  Proposition 8.4.

If $h\in M_{n}(\mathbb{C})$ is positive definite and $P_{t}=e^{-t\mathcal{L}}$
with $\mathcal{L}$ given by

$\mathcal{L}\colon M_{n}(\mathbb{C})\to
M_{n}(\mathbb{C}),\,\mathcal{L}(x)=\sum_{j=1}^{d}e^{-\omega_{j}/2}(v_{j}^{\ast}[v_{j},x]-[v_{j}^{\ast},x]v_{j})$

with $v_{j}\in M_{n}(\mathbb{C})$ non-zero and $\omega_{j}\in\mathbb{R}$ such
that $\\{v_{j}\mid 1\leq j\leq d\\}=\\{v_{j}^{\ast}\mid 1\leq j\leq d\\}$ and
$hv_{j}h^{-1}=e^{-\omega_{j}}v_{j}$, then $(P_{t})$ is a quantum Markov
semigroup, which is GNS-symmetric with respect to $\mathrm{tr}(\,\cdot\,h)$,
and the Bohr spectrum of $(P_{t})$ is $\\{\omega_{j}\mid 1\leq j\leq d\\}$.

###### Proof.

For $j\in\\{1,\dots,d\\}$ let $j^{\ast}$ denote the unique index in
$\\{1,\dots,d\\}$ such that $v_{j^{\ast}}=v_{j}^{\ast}$. We write $\varphi$
for $\mathrm{tr}(h\,\cdot\,)$. As shown in [Wir22, Section 8.2], the Tomita
correspondence $(\mathcal{H},\mathcal{J},(\mathcal{U}_{t}))$ associated with
$(P_{t})$ is given by
$\mathcal{H}=\mathrm{HS}(\mathbb{C}^{n})\otimes\mathbb{C}^{d}$,
$\mathcal{J}=J_{\varphi}\otimes J$ with
$(\mathcal{J}\xi)_{j}=\overline{\xi_{j^{\ast}}}$, and
$\mathcal{U}_{t}=\Delta_{\varphi}^{it}\otimes U_{t}$ with
$(U_{t}\xi)_{j}=e^{i\omega_{j}t}\xi_{j}$. Thus $\mathcal{V}_{t}=1\otimes
U_{t}$ and the Bohr spectrum of $(P_{t})$ is $\\{\omega_{j}\mid 1\leq j\leq
d\\}$. ∎

######  Remark 8.5.

By Alicki’s theorem (see [Ali76, Theorem 3], [CM17, Theorem 3.1]), every GNS-
symmetric quantum Markov semigroup on $M_{n}(\mathbb{C})$ is of the form given
in the previous theorem. In this case, the Bohr spectrum is always contained
in the spectrum of $\log\Delta_{\varphi}$. It is an interesting question
whether this is true for the Bohr spectrum of GNS-symmetric quantum Markov
semigroups on an arbitrary semi-finite von Neumann algebras.

## References

  * [Ali76] Robert Alicki  “On the detailed balance condition for non-Hamiltonian systems”  In _Rep. Math. Phys._ 10.2, 1976, pp. 249–258  DOI: 10.1016/0034-4877(76)90046-X
  * [Ana90] Claire Anantharaman-Delaroche  “On relative amenability for von Neumann algebras”  In _Compositio Math._ 74.3, 1990, pp. 333–352  URL: http://www.numdam.org/item?id=CM_1990__74_3_333_0
  * [BBM19] Jon Bannon, Panchugopal Bikram and Kunal Mukherjee  “A note on coarse bimodules”  In _Journal of Mathematical Analysis and Applications_ 475.2, 2019, pp. 1404–1415  DOI: https://doi.org/10.1016/j.jmaa.2019.03.022
  * [BGJ22] Michael Brannan, Li Gao and Marius Junge  “Complete logarithmic Sobolev inequalities via Ricci curvature bounded below” Id/No 108129  In _Adv. Math._ 394, 2022, pp. 60  DOI: 10.1016/j.aim.2021.108129
  * [BHV18] Rémi Boutonnet, Cyril Houdayer and Stefaan Vaes  “Strong solidity of free Araki-Woods factors”  In _Amer. J. Math._ 140.5, 2018, pp. 1231–1252  DOI: 10.1353/ajm.2018.0029
  * [Bik+21] Panchugopal Bikram, Rahul Kumar, Rajeeb Mohanta, Kunal Mukherjee and Diptesh Saha  “On the von Neumann algebras associated to Yang-Baxter operators”  In _Proc. Roy. Soc. Edinburgh Sect. A_ 151.4, 2021, pp. 1331–1354  DOI: 10.1017/prm.2020.62
  * [Bik+23] Panchugopal Bikram, Kunal Mukherjee, Éric Ricard and Simeng Wang  “On the factoriality of $q$-deformed Araki-Woods von Neumann algebras”  In _Comm. Math. Phys._ 398.2, 2023, pp. 797–821  DOI: 10.1007/s00220-022-04535-2
  * [BKM23] Panchugopal Bikram, Rahul Kumar R and Kunal Mukherjee  “Mixed $q$-deformed Araki-Woods von Neumann algebras”  In _J. Noncommut. Geom._ 17.4, 2023, pp. 1231–1297  DOI: 10.4171/jncg/513
  * [BKM24] Panchugopal Bikram, Rahul Kumar and Kunal Mukherjee  “On noninjectivity of mixed $q$-deformed Araki-Woods von Neumann algebras”  In _Kyoto J. Math._ 64.2, 2024, pp. 501–518  DOI: 10.1215/21562261-2023-0021
  * [BKS97] Marek Bożejko, Burkhard Kümmerer and Roland Speicher  “$q$-Gaussian processes: non-commutative and classical aspects”  In _Comm. Math. Phys._ 185.1, 1997, pp. 129–154  DOI: 10.1007/s002200050084
  * [BM17] Panchugopal Bikram and Kunal Mukherjee  “Generator masas in $q$-deformed Araki-Woods von Neumann algebras and factoriality”  In _J. Funct. Anal._ 273.4, 2017, pp. 1443–1478  DOI: 10.1016/j.jfa.2017.03.005
  * [Boż98] Marek Bożejko  “Completely positive maps on Coxeter groups and the ultracontractivity of the $q$-Ornstein-Uhlenbeck semigroup”  In _Quantum probability (Gdańsk, 1997)_ 43, Banach Center Publ.  Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 87–93
  * [BS94] Marek Bożejko and Roland Speicher  “Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces”  In _Math. Ann._ 300.1, 1994, pp. 97–120  DOI: 10.1007/BF01450478
  * [Cas23] Martijn Caspers  “On the isomorphism class of $q$-Gaussian $\rm W^{*}$-algebras for infinite variables”  In _C. R. Math. Acad. Sci. Paris_ 361, 2023, pp. 1711–1716  DOI: 10.5802/crmath.489
  * [CL23] Ricardo Correa da Silva and Gandalf Lechner  “Modular structure and inclusions of twisted Araki-Woods algebras”  In _Comm. Math. Phys._ 402.3, 2023, pp. 2339–2386  DOI: 10.1007/s00220-023-04773-y
  * [CM17] Eric A. Carlen and Jan Maas  “Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance”  In _J. Funct. Anal._ 273.5, 2017, pp. 1810–1869  DOI: 10.1016/j.jfa.2017.05.003
  * [GS14] A. Guionnet and D. Shlyakhtenko  “Free monotone transport”  In _Invent. Math._ 197.3, 2014, pp. 613–661  DOI: 10.1007/s00222-013-0493-9
  * [Hal13] Brian C. Hall  “Quantum theory for mathematicians” 267, Graduate Texts in Mathematics  Springer, New York, 2013, pp. xvi+554  DOI: 10.1007/978-1-4614-7116-5
  * [Hia03] Fumio Hiai  “$q$-deformed Araki-Woods algebras”  In _Operator algebras and mathematical physics (Constanţa, 2001)_ Theta, Bucharest, 2003, pp. 169–202
  * [HR11] Cyril Houdayer and Éric Ricard  “Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors”  In _Adv. Math._ 228.2, 2011, pp. 764–802  DOI: 10.1016/j.aim.2011.06.010
  * [HT21] Cyril Houdayer and Benjamin Trom  “Structure of extensions of free Araki-Woods factors”  In _Ann. Inst. Fourier (Grenoble)_ 71.2, 2021, pp. 473–514  DOI: 10.5802/aif.3391
  * [JRS] Marius Junge, Eric Ricard and Dimitri Shlyakhtenko  “Noncommutative diffusion semigroups and free probability” Unpublished preprint
  * [Kró06] Ilona Królak  “Factoriality of von Neumann algebras connected with general commutation relations—finite dimensional case”  In _Quantum probability_ 73, Banach Center Publ.  Polish Acad. Sci. Inst. Math., Warsaw, 2006, pp. 277–284  DOI: 10.4064/bc73-0-20
  * [KSW23] Manish Kumar, Adam Skalski and Mateusz Wasilewski  “Full solution of the factoriality question for $q$-Araki-Woods von Neumann algebras via conjugate variables”  In _Comm. Math. Phys._ 402.1, 2023, pp. 157–167  DOI: 10.1007/s00220-023-04734-5
  * [Kum23] Manish Kumar  “Conjugate variables approach to mixed $q$-Araki-Woods algebras: factoriality and non-injectivity”  In _J. Math. Phys._ 64.9, 2023, pp. Paper No. 09350619  DOI: 10.1063/5.0158660
  * [KV19] Anna Sofie Krogager and Stefaan Vaes  “Thin ${\rm II}_{1}$ factors with no Cartan subalgebras”  In _Kyoto J. Math._ 59.4, 2019, pp. 815–867  DOI: 10.1215/21562261-2019-0028
  * [Lec23] Gandalf Lechner  “Twisted Araki-Woods Algebras, the Yang-Baxter Equation, and quantum field theory”  In _XL Workshop on Geometric Methods in Physics 2023_ , 2023
  * [LJL20] Haojian Li, Marius Junge and Nicholas LaRacuente  “Graph Hörmander Systems”, 2020  arXiv:2006.14578
  * [MS23] Akihiro Miyagawa and Roland Speicher  “A dual and conjugate system for $q$-Gaussians for all $q$”  In _Adv. Math._ 413, 2023, pp. Paper No. 10883436  DOI: 10.1016/j.aim.2022.108834
  * [Nel15] Brent Nelson  “Free monotone transport without a trace”  In _Comm. Math. Phys._ 334.3, 2015, pp. 1245–1298  DOI: 10.1007/s00220-014-2148-0
  * [Nel59] Edward Nelson  “Analytic vectors”  In _Ann. of Math. (2)_ 70, 1959, pp. 572–615  DOI: 10.2307/1970331
  * [Nou04] Alexandre Nou  “Non injectivity of the $q$-deformed von Neumann algebra”  In _Math. Ann._ 330.1, 2004, pp. 17–38  DOI: 10.1007/s00208-004-0523-4
  * [Pim97] Michael V. Pimsner  “A class of $C^{*}$-algebras generalizing both Cuntz-Krieger algebras and crossed products by ${\bf Z}$”  In _Free probability theory (Waterloo, ON, 1995)_ 12, Fields Inst. Commun.  Amer. Math. Soc., Providence, RI, 1997, pp. 189–212
  * [Ric05] Éric Ricard  “Factoriality of $q$-Gaussian von Neumann algebras”  In _Comm. Math. Phys._ 257.3, 2005, pp. 659–665  DOI: 10.1007/s00220-004-1266-5
  * [Sch02] Jürgen Schweizer  “Hilbert $C^{\ast}$-modules with a predual”  In _J. Operator Theory_ 48.3, 2002, pp. 621–632
  * [Shl00] Dimitri Shlyakhtenko  “Free entropy with respect to a completely positive map”  In _Amer. J. Math._ 122.1, 2000, pp. 45–81  URL: http://muse.jhu.edu/journals/american_journal_of_mathematics/v122/122.1shlyakhtenko.pdf
  * [Shl97] Dimitri Shlyakhtenko  “Free quasi-free states”  In _Pacific J. Math._ 177.2, 1997, pp. 329–368  DOI: 10.2140/pjm.1997.177.329
  * [Shl99] Dimitri Shlyakhtenko  “$A$-valued semicircular systems”  In _J. Funct. Anal._ 166.1, 1999, pp. 1–47  DOI: 10.1006/jfan.1999.3424
  * [Ske01] Michael Skeide  “Hilbert Modules and Applications in Quantum Probability”, 2001  URL: http://web.unimol.it/skeide/_MS/downloads/habil.pdf
  * [Ske98] Michael Skeide  “Hilbert modules in quantum electrodynamics and quantum probability”  In _Comm. Math. Phys._ 192.3, 1998, pp. 569–604  DOI: 10.1007/s002200050310
  * [Śni04] Piotr Śniady  “Factoriality of Bożejko-Speicher von Neumann algebras”  In _Comm. Math. Phys._ 246.3, 2004, pp. 561–567  DOI: 10.1007/s00220-003-1031-1
  * [SW18] Adam Skalski and Simeng Wang  “Remarks on factoriality and $q$-deformations”  In _Proc. Amer. Math. Soc._ 146.9, 2018, pp. 3813–3823  DOI: 10.1090/proc/13715
  * [Tak03] Masamichi Takesaki  “Theory of operator algebras. II” Operator Algebras and Non-commutative Geometry, 6 125, Encyclopaedia of Mathematical Sciences  Springer-Verlag, Berlin, 2003, pp. xxii+518  DOI: 10.1007/978-3-662-10451-4
  * [Tak79] Masamichi Takesaki  “Theory of operator algebras. I”  Springer-Verlag, New York–Heidelberg, 1979, pp. vii+415
  * [Vas24] Ezio Vasselli  “Twisted tensor products of field algebras”, 2024  arXiv:2401.17753
  * [VDN92] D.. Voiculescu, K.. Dykema and A. Nica  “Free random variables” A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups 1, CRM Monograph Series  American Mathematical Society, Providence, RI, 1992, pp. vi+70  DOI: 10.1090/crmm/001
  * [Wir22] Melchior Wirth  “The Differential Structure of Generators of GNS-symmetric Quantum Markov Semigroups”, 2022  arXiv:2207.09247
  * [Wir24] Melchior Wirth  “Christensen–Evans theorem and extensions of GNS-symmetric quantum Markov semigroups”  In _Journal of Functional Analysis_ 287.3, 2024, pp. 110475  DOI: 10.1016/j.jfa.2024.110475
  * [Wir24a] Melchior Wirth  “Modular Completely Dirichlet forms as Squares of Derivations”  In _International Mathematics Research Notices. IMRN_ , 2024, pp. rnae092  DOI: 10.1093/imrn/rnae092
  * [Yan23] Zhiyuan Yang  “A Conjugate System for Twisted Araki-Woods von Neumann Algebras of finite dimensional spaces”, 2023  arXiv:2304.13856


# Measurable Imbeddings, Free Products, and Graph Products

Özkan Demir

###### Abstract.

We study Measurable Imbeddability between groups, which is an order-like
generalization of Measure Equivalence that allows the imbedded group to have
an infinite measure fundamental domain. We prove if $\Lambda_{1}$ measurably
imbeds into $\Gamma_{1}$, and $\Lambda_{2}$ measurably imbeds into
$\Gamma_{2}$ under an additional assumption that lets the corresponding
fundamental domains to be arranged in a special way, then
$\Lambda_{1}*\Lambda_{2}$ measurably imbeds into $\Gamma_{1}*\Gamma_{2}$.
Building upon the techniques we used, we show that the analogous result holds
for graph products of groups.

##  1\. Introduction

Measure equivalence is the measured analogue of quasi-isometry in geometric
group theory. We study a slight generalization of this notion, called
Measurable Imbedding, and its behavior under free products and graph products
of groups. Throughout the paper all groups will be discrete countable, all
measure spaces will be standard, and all set equalities are almost everywhere
(a.e.) equalities (i.e. the symmetric difference between the two sets is null
with respect to the ambient measure), unless otherwise stated. We first recall
the definition of Measure Equivalence (introduced by Gromov in [Gro]):

######  Definition 1.1.

Two groups $\Lambda$ and $\Gamma$ are called Measure Equivalent (abbreviated
as ME, and denoted $\Lambda\overset{\mathrm{ME}}{\sim}\Gamma$) if there exists
an infinite measure space $(\Sigma,\mu)$ with a measurable, measure preserving
action of $\Lambda\times\Gamma$, so that both actions
$\Lambda\curvearrowright\Sigma$ and $\Gamma\curvearrowright\Sigma$ admit
finite-measure Borel fundamental domains
$Y\cong\Sigma/\Lambda,X\cong\Sigma/\Gamma$, i.e.:

$\displaystyle\displaystyle{\Sigma=\bigsqcup_{\gamma\in\Gamma}\gamma
X=\bigsqcup_{\lambda\in\Lambda}\lambda Y}$

The space $(\Sigma,\mu)$ is called a $(\Lambda,\Gamma)$-coupling or ME-
coupling. The index of this coupling is the ratio of the fundamental domains:

$\displaystyle[\Gamma:\Lambda]_{\Sigma}=\frac{\mu(Y)}{\mu(X)}\in(0,\infty)$

######  Remarks.

(See [Furman] for details)

  1. (1)

For a given coupling, the index is well-defined, i.e., it does not depend on
the choices of the fundamental domains.

  2. (2)

Let $\Lambda\overset{\mathrm{ME}}{\sim}\Delta$ and
$\Delta\overset{\mathrm{ME}}{\sim}\Gamma$ with couplings $\Sigma_{1}$ and
$\Sigma_{2}$. Then $\Lambda\overset{\mathrm{ME}}{\sim}\Gamma$ with the
coupling
$\Sigma=\Sigma_{1}\times_{\Delta}\Sigma_{2}\mathrel{\mathop{:}}=(\Sigma_{1}\times\Sigma_{2})/\Delta$
where $\Delta$ acts diagonally on the product. Moreover, the index of this
coupling can be computed as:

$[\Gamma:\Lambda]_{\Sigma}=[\Gamma:\Delta]_{\Sigma_{1}}[\Delta:\Lambda]_{\Sigma_{2}}$

In particular, $\overset{\mathrm{ME}}{\sim}$ is an equivalence relation on
groups.

  3. (3)

If $[\Gamma:\Lambda]_{\Sigma}=c\geq 1$, and the coupling is ergodic, then we
can choose fundamental domains $X$ and $Y$ (possibly after rescaling the
measure $\mu$) so that $X\subset Y$, $\mu(X)=1$, and $\mu(Y)=c$.

  4. (4)

Given a $(\Lambda,\Gamma)$-coupling $\Sigma$ and choice of fundamental domains
$Y\cong\Sigma/\Lambda,X\cong\Sigma/\Gamma$, we obtain a measurable measure-
preserving action $\Lambda\curvearrowright X$ where for $\lambda\in\Lambda$
and $x\in X$, $\lambda\cdot x$ is the (essentially) unique element in
$\Gamma\lambda x\cap X$, and a cocycle $\alpha:\Lambda\times
X\rightarrow\Gamma$ where $\alpha(\lambda,x)$ is the unique element in
$\Gamma$ satisfying $\gamma\cdot x=\alpha(\lambda,x)\lambda x$. We similarly
have $\Gamma\curvearrowright Y$ and corresponding cocycle $\beta:\Gamma\times
Y\rightarrow\Lambda$. (We use $\cdot$ to denote those two actions, to
distinguish them from the original action
$\Lambda\times\Gamma\curvearrowright\Sigma$)

We now give the definition of Measurable Imbedding between groups, which can
be thought as ME but with index in $(0,\infty]$.

######  Definition 1.2.

$\Lambda$ Measurably Imbeds into $\Gamma$ (abbreviated as MI, and denoted
$\Lambda\overset{\mathrm{MI}}{\rightarrowtail}\Gamma$) if there exists an
infinite measure space $(\Sigma,\mu)$ with a measurable, measure preserving
action of $\Lambda\times\Gamma$, so that both actions
$\Lambda\curvearrowright\Sigma$ and $\Gamma\curvearrowright\Sigma$ admit Borel
fundamental domains $Y,X\subset\Sigma$ such that $X$ has finite measure.

The space $(\Sigma,\mu)$ is called a $(\Lambda\rightarrowtail\Gamma)$-coupling
or MI-coupling. The index of this coupling is the ratio of the fundamental
domains:

$\displaystyle[\Gamma:\Lambda]_{\Sigma}=\frac{\mu(Y)}{\mu(X)}\in(0,\infty]$

Here are analogous remarks for MI (the proofs are similar to the ME case):

######  Remarks.

  1. (1)

For a given coupling, the index is well-defined, i.e., it does not depend on
the choices of the fundamental domains.

  2. (2)

Let $\Lambda\overset{\mathrm{MI}}{\rightarrowtail}\Delta$ and
$\Delta\overset{\mathrm{MI}}{\rightarrowtail}\Gamma$ with couplings
$\Sigma_{1}$ and $\Sigma_{2}$. Then
$\Lambda\overset{\mathrm{MI}}{\rightarrowtail}\Gamma$ with the coupling
$\Sigma=\Sigma_{1}\times_{\Delta}\Sigma_{2}$. Same as the ME case, the index
of this coupling can be computed as:

$[\Gamma:\Lambda]_{\Sigma}=[\Gamma:\Delta]_{\Sigma_{1}}[\Delta:\Lambda]_{\Sigma_{2}}$

In particular, $\overset{\mathrm{MI}}{\rightarrowtail}$ is an order relation
on groups.

  3. (3)

We have actions $\Lambda\curvearrowright X$ , $\Gamma\curvearrowright Y$ and
corresponding cocycles $\alpha:\Lambda\times X\rightarrow\Gamma$ ,
$\beta:\Gamma\times Y\rightarrow\Lambda$ as in the ME case.

Arranging fundamental domains so that the large group’s fundamental domain is
contained in the smaller one’s plays an important role in our constructions,
which is not possible for this general definition if index is less than 1.
Even if the index is greater than 1, it is not always possible to arrange
fundamental domains like that, as the following example demonstrates:

######  Example 1.

Let $(\Sigma_{i},\mu_{i})$ be a $(\Lambda\rightarrowtail\Gamma)$-couplings
with indices $c_{i}$ so that $c_{1}<1$, $c_{2}>1$. Possibly after rescaling
the measures $\mu_{i}$, we can choose fundamental domains
$Y_{i}\cong\Sigma_{i}/\Lambda,X_{i}\cong\Sigma_{i}/\Gamma$ for $i=1,2$, so
that $\mu_{1}(Y_{1})=c_{1}$, $\mu_{1}(X_{1})=1$, $\mu_{2}(Y_{2})=ac_{2}$,
$\mu_{2}(X_{2})=a$, where $a$ is chosen so that the ratio
$\displaystyle{c=\frac{ac_{2}+c_{1}}{a+1}>1}$. Then the coupling
$\Sigma=\Sigma_{1}\bigsqcup\Sigma_{2}$ has index $c>1$, but it is not possible
to arrange fundamental domains so that the one with the bigger measure
contains the other.

######  Definition 1.3.

$\Lambda$ Strictly Measurably Imbeds into $\Gamma$ (abbreviated as SMI, and
denoted $\Lambda\overset{\mathrm{SMI}}{\rightarrowtail}\Gamma$) if there
exists an infinite measure space $(\Sigma,\mu)$ with a measurable, measure
preserving action of $\Lambda\times\Gamma$, so that both actions
$\Lambda\curvearrowright\Sigma$ and $\Gamma\curvearrowright\Sigma$ admit Borel
fundamental domains $Y,X\subset\Sigma$ such that $X$ has finite measure and
$X\subset Y$.

The space $(\Sigma,\mu)$ is called a $(\Lambda\rightarrowtail\Gamma)$-coupling
or SMI-coupling. The index of this coupling is the ratio of the fundamental
domains:

$\displaystyle[\Gamma:\Lambda]_{\Sigma}=\frac{\mu(Y)}{\mu(X)}\in[1,\infty]$

We similarly have well-definedness of the coupling index, transitivity, and
cocycles for SMI. Note that in Example 1, what prevented the MI-coupling from
being a SMI-coupling was the fact that $\Sigma_{1},\Sigma_{2}\subset\Sigma$
were invariant disjoint subsets, so the coupling was not ergodic. Indeed in
general, if $\Sigma$ is an ergodic MI-coupling with index $c\geq 1$, then
$\Sigma$ is an SMI-coupling with the same index.

##  2\. Free Products

A special case of ME is Orbit Equivalence (OE), which corresponds to having a
coupling with index 1. If additionally the actions are ergodic, one can choose
the same set as the fundamental domain for the actions of both of the groups.
(See [Furman] for the definition and proof of this correspondence.) In [Gab],
Gaboriau proves the following result:

######  Theorem 2.1.

($\mathbf{P_{ME}6}$ in [Gab]) If
$\Lambda_{1}\overset{\mathrm{OE}}{\sim}\Gamma_{1}$ and
$\Lambda_{2}\overset{\mathrm{OE}}{\sim}\Gamma_{2}$, then
$\Lambda_{1}*\Lambda_{2}\overset{\mathrm{OE}}{\sim}\Gamma_{1}*\Gamma_{2}$

There, he actually proves the same result for infinite free products, from
which the above result follows. He also gives counterexamples to show that the
above result is not true if we replace OE with ME. We show that the result is
true if we replace OE with SMI.

######  Theorem 2.2.

If $\Lambda_{1}\overset{\mathrm{SMI}}{\rightarrowtail}\Gamma_{1}$ and
$\Lambda_{2}\overset{\mathrm{SMI}}{\rightarrowtail}\Gamma_{2}$, then
$\Lambda_{1}*\Lambda_{2}\overset{\mathrm{SMI}}{\rightarrowtail}\Gamma_{1}*\Gamma_{2}$
Moreover, for given $(\Lambda_{i},\Gamma_{i})$-couplings $\Sigma_{i}$ with
indices $[\Gamma_{i}:\Lambda_{i}]_{\Sigma_{i}}=c_{i}$ for $i=1,2$, one can
find a $(\Lambda_{1}*\Lambda_{2},\Gamma_{1}*\Gamma_{2})$-coupling $\Sigma$
where:

$[\Gamma_{1}*\Gamma_{2}:\Lambda_{1}*\Lambda_{2}]_{\Sigma}=\begin{cases}1,&\text{if}\
c_{1}=c_{2}=1\\\ \infty,&\text{otherwise}\end{cases}$

###### Proof.

Using transitivity of SMI, it is enough to prove the following statement:

For groups $\Lambda$, $\Gamma$, $G$; if
$\Lambda\overset{\mathrm{SMI}}{\rightarrowtail}\Gamma$, then
$G*\Lambda\overset{\mathrm{SMI}}{\rightarrowtail}G*\Gamma$ with coupling index
$\infty$, unless $[\Gamma:\Lambda]=1$, in which case the new coupling index is
also 1.

Fix a $(\Lambda,\Gamma)$-coupling $(\Sigma,\mu)$, fundamental domains
$Y\cong\Sigma/\Lambda,X\cong\Sigma/\Gamma$ such that $X\subset Y$ and
$\mu(X)=1$. Then we have the action $\Lambda\curvearrowright X$ and the
corresponding cocycle $\alpha:\Lambda\times X\rightarrow\Gamma$ so that
$\lambda\cdot x=\alpha(\lambda,x)\lambda x$.

Let $G$ act on $X$ trivially to get an action of $G*\Lambda$ on $X$, still
denoted by $\cdot$. Extend the cocycle $\alpha$ to $\alpha:(G*\Lambda)\times
X\rightarrow G*\Gamma$ by setting $\alpha(g,x)=g$, for $g\in G$ and using the
cocycle identity $\alpha(gh,x)=\alpha(g,h\cdot x)\alpha(h,x)$

Set $\widetilde{\Sigma}=(G*\Gamma)\times X$. With an abuse of notation, we
name the induced product measure on $\widetilde{\Sigma}$ as $\mu$ again. We
define the following actions:

$\displaystyle
G*\Gamma\curvearrowright\widetilde{\Sigma}\qquad\;w\cdot(\bar{w},x)$
$\displaystyle=(w\,\bar{w},x)$ $\displaystyle
G*\Lambda\curvearrowright\widetilde{\Sigma}\qquad\;w\cdot(\bar{w},x)$
$\displaystyle=(\bar{w}\,\alpha(w,x)^{-1},w\cdot x)$

We claim $\widetilde{\Sigma}$ is a $(G*\Lambda,G*\Gamma)$-coupling. Clearly,
those two actions commute, and $\\{e\\}\times X$ is a $G*\Gamma$ fundamental
domain. We will describe a $G*\Lambda$ fundamental domain, and calculate its
measure, which will give the result.

First, we will look at $\widetilde{\Sigma}$ from another perspective which
will make the description of this fundamental domain easier. The idea is to
see $\widetilde{\Sigma}$ as a collection of disjoint copies of the original
coupling $\Sigma$, using the natural identification $\Sigma\cong\Gamma\times
X$. Letting $\Gamma\curvearrowright G*\Gamma$ by right multiplication, we have
$G*\Gamma\cong G*\Gamma/\Gamma\times\Gamma$. By seeing elements of $G*\Gamma$
as reduced words, and choosing the representatives of minimal word length from
$G*\Gamma/\Gamma$ gives the identification $G*\Gamma/\Gamma\cong W$, where:

$W=\\{w\in G*\Gamma|\,w\text{ is a reduced word ending with an element of
}G\\}$

This gives us the identification $\widetilde{\Sigma}\cong W\times\Gamma\times
X$. The action of $G*\Lambda$ making this identification
$G*\Lambda$-equivariant is:

$\displaystyle g\cdot(w,\gamma,x)$ $\displaystyle=(w\gamma g^{-1},e,x)\quad$
$\displaystyle\text{for}\,g\in G$ $\displaystyle\lambda\cdot(w,\gamma,x)$
$\displaystyle=(w,\gamma\alpha(\lambda,x)^{-1},\lambda\cdot x)\quad$
$\displaystyle\text{for}\,\lambda\in\Lambda$

Identifying $\Gamma\times X\cong\Sigma$, we claim
$\widetilde{Y}=(\\{e\\}\times X)\bigsqcup(W\times(Y\setminus X))$ is a
fundamental domain for $G*\Lambda$. It is enough to show for any nontrivial
element $w\in G*\Lambda$ and $y\in\widetilde{Y}$, $w\cdot
y\notin\widetilde{Y}$ An arbitrary element of $G*\Lambda$ can be written of
the form $\lambda z$ where $\lambda\in\Lambda$ and $z\in G*\Lambda$ is a
reduced word starting with an element of G. Also note that
$(w,\gamma,x)\in\widetilde{Y}$ if and only if $w=\gamma=e$ corresponding to
$\\{e\\}\times X$ or $\gamma x\in Y$ as an element in $\Sigma$. We also note
the following simple observation coming from the choice of the fundamental
domains so that $X\subset Y$:

(1) $\alpha(\lambda,x)\neq e\quad\text{for}\quad\lambda\neq e$

Letting $p:G*\Gamma\rightarrow\Gamma$ be the retraction homomorphism onto
$\Gamma$, note that for $w\in G*\Gamma$ and $x\in X$, $w\cdot x=p(w)\cdot x$,
as $G$ acts trivially on $X$. So, we have the following equation, which we
will use a lot:

(2) $w\cdot x=\alpha(p(w),x)p(w)x$

We have three cases with two subcases for each case:

######  Case 1:

Both $\lambda$ and $z$ are non-trivial.

$\lambda z\cdot(e,e,x)=(\alpha(z,x)^{-1},\alpha(\lambda,z\cdot x)^{-1},\lambda
z\cdot x)\notin\widetilde{Y}$

as $\alpha(\lambda,z\cdot x)\neq e$ by (1), and

$\displaystyle\alpha(\lambda,z\cdot x)^{-1}(\lambda z\cdot x)$
$\displaystyle=\alpha(\lambda,p(z)\cdot x)^{-1}\alpha(\lambda p(z),x)\lambda
p(z)x)\quad$ $\displaystyle\text{by }(\ref{eq:2})$
$\displaystyle=\alpha(p(z),x)\lambda p(z)x$ by cocycle identity
$\displaystyle=\lambda(p(z)\cdot x)$ $\displaystyle\text{by }(\ref{eq:2})$

$\lambda(p(z)\cdot x)\notin\widetilde{Y}$ as $p(z)\cdot x\in X\subset Y$ and
$\lambda\neq e$.

For the second subcase, let $(w,\gamma,x)\in W\times(Y\setminus X)$, meaning
$\gamma\neq e$ and $\gamma x\in Y$. Then,

$\lambda z\cdot(w,\gamma,x)=(w\gamma\alpha(z,x)^{-1},\alpha(\lambda,z\cdot
x)^{-1},\lambda z\cdot x)\notin\widetilde{Y}$

as $\alpha(\lambda,z\cdot x)\neq e$ by (1) and $\alpha(\lambda,z\cdot
x)^{-1}(\lambda z\cdot x)\notin\widetilde{Y}$ as before.

######  Case 2:

$\lambda=e$ and $z$ is non-trivial.

$z\cdot(e,e,x)=(\alpha(z,x)^{-1},e,z\cdot x)\notin\widetilde{Y}$

as $\alpha(z,x)\neq e$ by (1).

For $\gamma\neq e$ and $\gamma x\in Y$:

$z\cdot(w,\gamma,x)=(w\gamma\alpha(z,x)^{-1},e,z\cdot x)\notin\widetilde{Y}$

as $w\gamma\alpha(z,x)^{-1}\neq e$. Indeed, $\gamma x\in Y$ implies
$\alpha(\lambda,x)\neq\gamma$ for all $\lambda\in\Lambda$ and for a.e. $x\in
X$, $\gamma$ inside the word will not be cancelled.

######  Case 3:

$z=e$ and $\lambda$ is non-trivial.

$\lambda\cdot(e,e,x)=(e,\alpha(\lambda,x)^{-1},\lambda\cdot
x)\notin\widetilde{Y}$

as $\alpha(\lambda,x)\neq e$ by (1), and

$\alpha(\lambda,x)^{-1}\lambda\cdot x=\lambda x\notin Y$

For $\gamma\neq e$ and $\gamma x\in Y$:

$\lambda\cdot(w,\gamma,x)=(w,\gamma\alpha(\lambda,x)^{-1},\lambda\cdot
x)\notin\widetilde{Y}$

as $\gamma\alpha(\lambda,x)^{-1}\neq e$ as in Case 2, and

$\gamma\alpha(\lambda,x)^{-1}\lambda\cdot x=\gamma\lambda x=\lambda(\gamma
x)\notin Y$

as $\gamma x\in Y$ and $\lambda\neq e$.

This shows $\widetilde{Y}$ is a $G*\Lambda$ fundamental domain. Note if
$[\Gamma:\Lambda]=1$, then $Y=X$, so $\mu(\widetilde{Y})=\mu(\\{e\\}\times
X)=1$. Otherwise, $[\Gamma:\Lambda]>1$, so $\mu(Y\setminus X)>0$, so
$\mu(\widetilde{Y})=\mu(\\{e\\}\times X)+\mu(W\times(Y\setminus
X))=1+|W|\mu(Y\setminus X)=\infty$, as $W$ is infinite as long as $G$ is a
non-trivial group. This proves the final statement of the theorem about the
coupling indices. ∎

##  3\. Graph Products

We first state some preliminary definitions for graph products, and a normal
form theorem for the elements in a graph product. For proofs and further
details, see [Green].

######  Definition 3.1.

(Definition 3.1 in [Green]) Let $\Theta$ be a graph, with vertex set
$V=\\{v_{i}|i=1,2,...,n\\}$. Let $A_{k}$ be groups, k = 1,2, …, n. Let
$G\Theta=\langle A_{k}\,|\;[\,A_{i},A_{j}]\,\>\forall(\,v_{i},v_{j})\,\in
E\rangle$, where $E$ is the edge set of $\Theta$. We call $G\Theta$ the graph
product of the $A_{k}$ given by $\Theta$. We call $\Theta$ the underlying
graph, and the $A_{k}$ are called the generating groups.

######  Definition 3.2.

(Definition 3.5 in [Green]) Let $G\Theta$ be a graph product of the groups
$A_{1},A_{2},...,A_{n}$. A sequence $g_{1},g_{2},...g_{m}$ of elements from
$G\Theta$ is called reduced if:

  1. (i)

$g_{i}\in A_{i}$, for $i=1,2,...,n$ .

  2. (ii)

$g_{i}\neq e$, for $i=1,2,...,n$ .

  3. (iii)

$\forall g_{i},g_{j}$ with $i<j$ for which $\exists k\;i\leq k<j$ with

$\displaystyle[\,g_{i},g_{i+1}]\,$
$\displaystyle=[\,g_{i},g_{i+2}]\,=...=[\,g_{i},g_{k}]\,=e$
$\displaystyle[\,g_{k+1},g_{j}]\,$
$\displaystyle=[\,g_{k+2},g_{j}]\,=...=[\,g_{j-1},g_{j}]\,=e$

$g_{i}$ and $g_{j}$ are in different generating groups.

Part (iii) means that repeatedly swapping elements from generating groups
whose corresponding vertices are adjacent in $\Theta$ cannot bring together
two terms from the same generating group. We will refer to this swapping of
commuting terms as syllable shuffling.

######  Definition 3.3.

(Definition 3.7 in [Green]) Let $G\Theta$ be a graph product of the groups
$A_{1},A_{2},...,A_{n}$. Let $w$ be a word in the generators of the $A_{i}$.
If $w=w_{1}w_{2}...w_{r}$ where each $w_{j}$ is a word in the generators of
only one of the generating groups, no $w_{j}$ is the empty word, and $w_{j}$,
$w_{j+1}$ are not in the same generating group for $j=1,2,...,r-1$, then the
syllable length, $\lambda(w)$ of $w$ is $r$, and $w_{1},w_{2},...,w_{r}$ are
called the syllables of $w$. The syllable length $\lambda(g)$ of an element
$g\in G\Theta$ is the minimum syllable length of a word defining $g$.

######  Theorem 3.4.

(Theorem 3.9 in [Green]) Let $G\Theta$ be a graph product of the groups
$A_{1},A_{2},...,A_{n}$. Each element $g\neq e$ of $G\Theta$ can be uniquely
(up to syllable shuffling) expressed as a product

$g=g_{1}g_{2}...g_{r}\quad\text{where}\;g_{1},g_{2},...,g_{r}\;\text{is a
reduced sequence.}$

######  Proposition 3.5.

Let $G\Theta$ be a graph product of the groups $A_{1},A_{2},...,A_{n}$. Fix a
vertex generator $A_{k}$, and let $L$ be the standard subgroup of $G\Theta$
generated by $lk(v_{k})$, where $v_{k}$ is the vertex corresponding to
$A_{k}$. Then, each element $g\in G\Theta$ can be expressed as a product
$g=alh$ where:

  1. (i)

$a\in A_{k}$

  2. (ii)

$l$ is a reduced word in $L$.

  3. (iii)

$h$ is a reduced word starting with an element of $A_{m}$, where the
corresponding vertex is not $v_{k}$ and not in $lk(v_{k})$, or $h=e$

###### Proof.

For $g=e$, we just have $a=l=g=e$.

For $g\neq e$, write $g=g_{1}g_{2}...g_{r}$ as in Theorem 3.4. Note that,
either it is possible to have $g_{1}\in A_{k}$ after syllable shuffling, in
which case $a=g_{1}$; or not, in which case $a=e$. Then, let $g_{k}$ be the
first element not in $A_{k}$ or $H$. In case $a=g_{1}$, taking
$l=g_{2}g_{3}...g_{k-1}$ and $h=g_{k}g_{k+1}...g_{r}$ gives the result. (If
$k=2$, $l=e$) In case $a=e$, taking $l=g_{1}g_{2}g_{3}...g_{k-1}$ and
$h=g_{k}g_{k+1}...g_{r}$ gives the result. (If $k=1$, $l=e$) If such a $g_{k}$
does not exist, just take $l=a^{-1}g$ and $h=e$. These gives the result in all
possible cases. ∎

The following theorem is a generalization of Horbez and Huang’s result [HH]
concerning graph products of orbit equivalent groups.

######  Theorem 3.6.

Let $\Theta$ be a finite simple irreducible graph with vertex set $V$. Let $H$
and $G$ be two graph products over $\Theta$, with vertex groups
$\\{H_{v}\\}_{v\in V}$ and $\\{G_{v}\\}_{v\in V}$, respectively. Suppose for
each $v\in V$, $H_{v}\overset{\mathrm{SMI}}{\rightarrowtail}G_{v}$, with
coupling index $c_{v}$. Then $H\overset{\mathrm{SMI}}{\rightarrowtail}G$ with
coupling index $\infty$, unless when all $c_{v}=1$, in which case the coupling
index is also 1.

###### Proof.

We first reduce to the case where $H_{v}\cong G_{v}$ except for a
distinguished vertex $w\in V$, and set $\Lambda=H_{w}$, $\Gamma=G_{w}$.

Fix a $(\Lambda,\Gamma)$-coupling $(\Sigma,\mu)$, fundamental domains
$Y\cong\Sigma/\Lambda,X\cong\Sigma/\Gamma$ such that $X\subset Y$ and
$\mu(X)=1$. Then we have the action $\Lambda\curvearrowright X$ and the
corresponding cocycle $\alpha:\Lambda\times X\rightarrow\Gamma$ so that
$\lambda\cdot x=\alpha(\lambda,x)\lambda x$.

For each $v\neq w$, let $H_{v}$ act on $X$ trivially. That gives an action of
the graph product $H$ on $X$, still denoted by $\cdot$. Extend the cocycle
$\alpha$ to $\alpha:H\times X\rightarrow G$ by setting $\alpha(h,x)=g$, for
$h\in H_{v}$, $v\neq w$ and using the cocycle identity
$\alpha(gh,x)=\alpha(g,h\cdot x)\alpha(h,x)$

Set $\widetilde{\Sigma}=G\times X$. With an abuse of notation, we name the
induced product measure on $\widetilde{\Sigma}$ as $\mu$ again. We define the
following actions:

$\displaystyle G\curvearrowright\widetilde{\Sigma}\quad w\cdot(\bar{w},x)$
$\displaystyle=(w\,\bar{w},x)$ $\displaystyle
H\curvearrowright\widetilde{\Sigma}\quad w\cdot(\bar{w},x)$
$\displaystyle=(\bar{w}\,\alpha(w,x)^{-1},w\cdot x)$

Same as before, those two actions commute, and $\\{e\\}\times X$ is a
$G*\Gamma$ fundamental domain. So, we need to describe an $H$ fundamental
domain, and calculate its measure, which will give the result.

Letting $\Gamma\curvearrowright G*\Gamma$ by right multiplication, we have
$G*\Gamma\cong G*\Gamma/\Gamma\times\Gamma$. By seeing elements of $G$ as
reduced words (as in …) up to equality , and choosing a representatives of
minimal word length from $G/\Gamma$ gives the identification $G/\Gamma\cong
W$, where

$W=\\{w\in G|\,w\text{ is not equal to a reduced word ending with an element
of }\Gamma\\}$

This gives us the identification $\widetilde{\Sigma}\cong W\times\Gamma\times
X$. Let $H_{lk(v)}<H$ be the standard subgroup of $H$ generated by $lk(v)$.
The action of $H$ making this identification $H$-equivariant is:

$\displaystyle\lambda\cdot(w,\gamma,x)$
$\displaystyle=(w,\gamma\alpha(\lambda,x)^{-1},\lambda\cdot x)\quad$ for
$\displaystyle\,\lambda\in\Lambda$ $\displaystyle h\cdot(w,\gamma,x)$
$\displaystyle=(w\gamma h^{-1},e,x)\quad$ for $\displaystyle\,h\in
H\setminus(H_{lk(v)}\cup\Lambda)$ $\displaystyle h\cdot(w,\gamma,x)$
$\displaystyle=(wh^{-1},\gamma,x)\quad$ for $\displaystyle\,h\in H_{lk(v)}$

Note $H_{lk(v)}\curvearrowright W$ by right multiplication. Let
$\widetilde{W}\subset W$ be a fundamental domain for this action, for example,
choose the minimal length word from each $H_{lk(v)}$-orbit. Identifying
$\Gamma\times X\cong\Sigma$, we claim $\widetilde{Y}=(\\{e\\}\times
X)\bigsqcup(\widetilde{W}\times(Y\setminus X))$ is a fundamental domain for
$H$. It is enough to show for any nontrivial element $w\in H$ and
$y\in\widetilde{Y}$, $w\cdot y\notin\widetilde{Y}$.

Using Proposition 3.5, such an element $w$ can be written of the form
$w=\lambda gh$ where $\lambda\in\Lambda$, $g$ is a reduced word in
$H_{lk(v)}$, and $h\in H$ is a reduced word starting with an element not in
$H_{lk(v)}$ or $\Lambda$. Also note that $(w,\gamma,x)\in\widetilde{Y}$ if and
only if $w=\gamma=e$ corresponding to $\\{e\\}\times X$ or $w\in\widetilde{W}$
and $\gamma x\in Y$ as an element in $\Sigma$. As in the proof of previous
theorem, we have (1), and letting $p:H\rightarrow\Gamma$ be the retraction, we
have (2).

This time, we have seven cases with two subcases for each case:

######  Case 1:

$\lambda$, $g$, $h$ are non-trivial.

$\lambda gh\cdot(e,e,x)=(\alpha(h,x)^{-1}g^{-1},\alpha(\lambda,h\cdot
x)^{-1},\lambda h\cdot x)\notin\widetilde{Y}$

as $\alpha(\lambda,h\cdot x)\neq e$ by (1), and

$\displaystyle\alpha(\lambda,h\cdot x)^{-1}(\lambda h\cdot x)$
$\displaystyle=\alpha(\lambda,p(h)\cdot x)^{-1}\alpha(\lambda p(h),x)\lambda
p(h)x)\quad$ $\displaystyle\text{by }(\ref{eq:2})$
$\displaystyle=\alpha(p(h),x)\lambda p(h)x$ by cocycle identity
$\displaystyle=\lambda(p(h)\cdot x)$ $\displaystyle\text{by }(\ref{eq:2})$

$\lambda(p(h)\cdot x)\notin\widetilde{Y}$ as $p(h)\cdot x\in X\subset Y$ and
$\lambda\neq e$.

For the second subcase, let $(w,\gamma,x)\in\widetilde{W}\times(Y\setminus
X)$, so $\gamma\neq e$ and $\gamma x\in Y$. Then,

$\lambda
gh\cdot(w,\gamma,x)=(w\gamma\alpha(h,x)^{-1}g^{-1},\alpha(\lambda,h\cdot
x)^{-1},\lambda h\cdot x)\notin\widetilde{Y}$

as $\alpha(\lambda,h\cdot x)\neq e$ by (1) and $\alpha(\lambda,h\cdot
x)^{-1}(\lambda h\cdot x)\notin\widetilde{Y}$ as before.

######  Case 2:

$\lambda=e$ and $g$, $h$ are non-trivial.

$gh\cdot(e,e,x)=(\alpha(h,x)^{-1}g^{-1},e,h\cdot x)\notin\widetilde{Y}$

as $\alpha(h,x)^{-1}g^{-1}\neq e$

For $w\in\widetilde{W}$, $\gamma\neq e$ and $\gamma x\in Y$:

$gh\cdot(w,\gamma,x)=(w\gamma\alpha(h,x)^{-1}g^{-1},e,h\cdot
x)\notin\widetilde{Y}$

as $w\gamma\alpha(h,x)^{-1}g^{-1}\neq e$. Indeed, $\gamma x\in Y$ implies
$\alpha(\lambda,x)\neq\gamma$ for all $\lambda\in\Lambda$ and for a.e. $x\in
X$, so $\gamma$ inside the word will not be cancelled.

######  Case 3:

$g=e$ and $\lambda$, $h$ are non-trivial.

Same as Case 1 from the proof of Theorem 2.2

######  Case 4:

$h=e$ and $\lambda$, $g$ are non-trivial.

$\lambda g\cdot(e,e,x)=(g^{-1},\alpha(\lambda,x)^{-1},\lambda\cdot
x)\notin\widetilde{Y}$

as $g\neq e$, and $\alpha(\lambda,x)^{-1}\lambda\cdot x=\lambda x\notin Y$.

For $w\in\widetilde{W}$, $\gamma\neq e$ and $\gamma x\in Y$:

$\lambda g\cdot(w,\gamma,x)=(wg^{-1},\gamma\alpha(\lambda,x)^{-1},\lambda\cdot
x)\notin\widetilde{Y}$

as $\gamma\alpha(\lambda,x)^{-1}\neq e$ and
$\gamma\alpha(\lambda,x)^{-1}\lambda\cdot x=\lambda\gamma x\notin Y$.

######  Case 5:

$\lambda=g=e$ and $h$ is non-trivial.

Same as Case 2 from the proof of Theorem 2.2.

######  Case 6:

$h=g=e$ and $\lambda$ is non-trivial.

Same as Case 3 from the proof of Theorem 2.2.

######  Case 7:

$\lambda=h=e$ and $g$ is non-trivial.

$g\cdot(e,e,x)=(g^{-1},e,x)\notin\widetilde{Y}$

as $g\neq e$.

For $w\in\widetilde{W}$, $\gamma\neq e$ and $\gamma x\in Y$:

$g\cdot(w,\gamma,x)=(wg^{-1},\gamma,x)\notin\widetilde{Y}$

as $wg^{-1}\notin\widetilde{W}$ by the definition of $\widetilde{W}$.

∎

## References