# Biexact von Neumann algebras

Changying Ding  and  Jesse Peterson  Department of Mathematics, University of
California, Los Angeles, Los Angeles, CA 90095, USA<EMAIL_ADDRESS>Department of Mathematics, Vanderbilt University, 1326 Stevenson Center,
Nashville, TN 37240, USA<EMAIL_ADDRESS>
###### Abstract.

We introduce the notion of biexactness for general von Neumann algebras,
naturally extending the notion from group theory. We show that biexactness
implies solidity for von Neumann algebras, and that many of the examples of
solid von Neumann algebras contained in the literature are, in fact, biexact.
We also give examples of certain crossed products arising from Gaussian
actions that are solid but not biexact, and we give examples of certain
$q$-Gaussian von Neumann algebras that are strongly solid but not biexact. The
techniques developed involve studying a certain weak form of nuclear
embeddings, and we use this setting to give a new description of weak
exactness for von Neumann algebras, which allows us to answer several open
problems in the literature about weakly exact von Neumann algebras.

##  1\. Introduction

Recall that a finite von Neumann algebra $M$ is solid if, for any diffuse von
Neumann subalgebra $A\subset M$, the relative commutant $A^{\prime}\cap M$ is
amenable. A seminal result of Ozawa [Oza04] is that free group factors are
solid and, as a consequence, every nonamenable subfactor of a free group
factor is prime. Ozawa’s proof, which is based on ${\rm C}^{*}$-algebra
theory, holds more generally for the class of biexact groups [BO08, Definition
15.1.2], which are groups that admit a topologically amenable action on a
certain small-at-infinity boundary. Biexact groups have since been shown to
have many applications to the theory of von Neumann algebras [Oza04, Oza06b,
Oza06a]. Combining the techniques of biexact groups with Popa’s
deformation/rigidity theory has produced a number of striking results where
certain structural properties of a von Neumann algebra can be recovered as in
[OP04, CS13, PV14a, CdSS16, CIW21, HIK22, Mar17].

In this article we introduce the notion of biexactness in the setting of von
Neumann algebras, thereby allowing for the previous results to be put in a
systematic framework. This allows for a more integrated use of ${\rm C}^{*}$
and von Neumann algebraic techniques, and allows us to find a common von
Neumann algebraic setting for many solidity results such as those obtained in
[Oza04, Pop07, Pet09, CI18, Bou13, HR15, Iso15b, Shl04]. This then leads to
natural extensions of these results to a larger class of von Neumann algebras.

To describe biexactness for von Neumann algebras, we first recall in greater
detail the corresponding property for groups (see [BO08, Chapter 15] for full
details). Given a countable group $\Gamma$, we have commuting actions of
$\Gamma$ on $\ell^{\infty}\Gamma$ given by left and right multiplication,
respectively, i.e., $L_{t}(f)(x)=f(t^{-1}x)$ and $R_{t}(f)(x)=f(xt)$ for
$f\in\ell^{\infty}\Gamma$ and $x,t\in\Gamma$. We denote

$S(\Gamma)=\\{f\in\ell^{\infty}\Gamma\mid f-R_{t}(f)\in c_{0}\Gamma,\ \forall\
t\in\Gamma\\},$

which is a left-invariant ${\rm C}^{*}$-subalgebra of $\ell^{\infty}\Gamma$.
By Gelfand duality, we may identify $S(\Gamma)$ with continuous functions on
its spectrum $\overline{\Gamma}$, and we then have an action of $\Gamma$ on
$\overline{\Gamma}$ by homeomorphisms. The space $\overline{\Gamma}$ is called
the small-at-infinity compactification of $\Gamma$. The group $\Gamma$ is
biexact if the action $\Gamma{\,\curvearrowright\,}\overline{\Gamma}$ is
topologically amenable in the sense of [AD87], which is equivalent to the
inclusion $C^{*}_{\lambda}\Gamma\subset C(\overline{\Gamma})\rtimes_{r}\Gamma$
being nuclear. Topological amenability is inherited through factor maps, and
so in practice one usually checks biexactness of a group via natural boundary
type actions, e.g., hyperbolic groups acting on their Gromov boundary.

Let now $M$ be a von Neumann algebra and $X\subset{\mathbb{B}}(\mathcal{H})$
an operator $M$-bimodule. Magajna introduced in [Mag98, Mag00] the
$M$-topology on $X$, which is a locally convex topology intermediate between
the uniform and ultraweak topologies (see Section 3 for the precise
definition). As ${\mathbb{B}}(L^{2}M)$ is both an operator $M$-bimodule and an
$M^{\prime}$-bimodule, we may let
${\mathbb{K}}^{\infty,1}(M)\subset{\mathbb{B}}(L^{2}M)$ denote the closure of
${\mathbb{K}}(L^{2}M)$ under both the $M$-topology and the
$M^{\prime}$-topology. In the case when $M$ has a normal faithful trace
$\tau$, we may view $M\subset L^{2}(M,\tau)\subset L^{1}(M,\tau)$, and then by
[DKEP22, Proposition 3.1] we have that ${\mathbb{K}}^{\infty,1}(M)$ is the
closure of ${\mathbb{K}}(L^{2}M)$ in the topology given by the norm
$\|T\|_{\infty,1}=\sup_{x\in M,\|x\|\leq 1}\|T\hat{x}\|_{1}$.

In [DKEP22] we introduced a von Neumann analog of the small-at-infinity
compactification as

${\mathbb{S}}(M)=\\{T\in{\mathbb{B}}(L^{2}M)\mid[T,a]\in{\mathbb{K}}^{\infty,1}(M),\forall\
a\in M^{\prime}\\}.$

The advantage of working with the generalized compact operators from
${\mathbb{K}}^{\infty,1}(M)$ is that to check that an operator $T$ is
contained in ${\mathbb{S}}(M)$, it suffices to check that
$[T,a]\in{\mathbb{K}}^{\infty,1}(M)$ for $a$ in some generating set of $M$
(generated as a von Neumann algebra). In particular, if $[T,a]$ is compact for
each element $a$ in some generating set, then $T\in{\mathbb{S}}(M)$. This
shows that the space ${\mathbb{S}}(M)$ is robust enough so that the inclusion
$M\subset{\mathbb{S}}(M)$ may have interesting properties. For example, if
$M=L\Gamma$ for a group $\Gamma$, then we have a natural inclusion
$S(\Gamma)\rtimes_{r}\Gamma\subset{\mathbb{S}}(L\Gamma)$. Also, the space
${\mathbb{S}}(M)$ is large enough so that it admits non-trivial 1-cohomology
[HPP23, Theorem 7.3]. A consequence of this robustness, though, is that the
the space ${\mathbb{S}}(M)$ is not a ${\rm C}^{*}$-algebra, but is rather an
$M$-system, i.e., an operator system that is also an operator $M$-bimodule.

Since ${\mathbb{S}}(M)$ is an operator $M$-bimodule, we may again consider the
$M$-topology on ${\mathbb{S}}(M)$. We say that the von Neumann algebra $M$ is
biexact if the inclusion $M\subset{\mathbb{S}}(M)$ is $M$-nuclear in the sense
that there exists nets of u.c.p. maps
$\phi_{i}:M\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to{\mathbb{S}}(M)$ so that for each
$x\in M$ we have $\psi_{i}\circ\phi_{i}(x)\to x$ in the $M$-topology.

This notion is flexible enough to show that many von Neumann algebras already
studied in the literature are biexact. In particular, we show in Theorem 7.17
that all von Neumann algebras satisfying condition $(AO)^{+}$ or strong
property $(AO)$ are biexact. We therefore have that each of the following
classes of von Neumann algebras is contained in the class of biexact von
Neumann algebras:

  1. (1)

Amenable von Neumann algebras.

  2. (2)

Group von Neumann algebras associated to locally compact second countable
biexact groups (Proposition 6.2 in [Dep20]).

  3. (3)

Von Neumann algebras of universal orthogonal and unitary discrete quantum
groups (Proposition 3.1.2 in [Iso15a]).

  4. (4)

Free Araki-Woods von Neumann algebras (The proof of Theorem 4.4 in [Hou07]).

  5. (5)

$q$-Gaussian von Neumann algebras associated to finite-dimensional real
Hilbert space (Section 4 in [Shl04] for some $q$, and [Kuz23] in general).

  6. (6)

Free products of biexact von Neumann algebras with respect to normal faithful
states (Proposition 6.16).

  7. (7)

Von Neumann subalgebras with expectation of biexact von Neumann algebras
(Proposition 6.10).

  8. (8)

Amplifications and commutants of biexact von Neumann algebras (Proposition
6.5).

In the case of a group von Neumann algebra associated to a discrete group
$M=L\Gamma$, we have a canonical diagonal embedding of $\ell^{\infty}\Gamma$
into ${\mathbb{B}}(\ell^{2}\Gamma)$, as well as a normal conditional
expectation
$E_{\ell^{\infty}\Gamma}:{\mathbb{B}}(\ell^{2}\Gamma)\to\ell^{\infty}\Gamma$.
We show that $E_{\ell^{\infty}\Gamma}({\mathbb{S}}(L\Gamma))\subset
S(\Gamma)$, and so if $L\Gamma$ is biexact, then by considering the positive
type functions $\Gamma\ni t\mapsto
E_{\ell^{\infty}}(\psi_{i}\circ\phi_{i}(\lambda_{t})\lambda_{t}^{*})\in
S(\Gamma)$ we deduce that the action
$\Gamma{\,\curvearrowright\,}\overline{\Gamma}$ is amenable. The converse
implication also holds by using local reflexivity of $C^{*}_{\lambda}\Gamma$
to show that nuclearity of the inclusion $C^{*}_{\lambda}\Gamma\subset
C(\overline{\Gamma})\rtimes_{r}\Gamma$ can be upgraded to give
$L\Gamma$-nuclearity for the inclusion $L\Gamma\subset{\mathbb{S}}(L\Gamma)$.
A consequence, therefore, of this new framework is that we deduce that
biexactness for groups is stable under $W^{*}$-equivalence (see Corollary 6.3
for the complete details), i.e. if $\Gamma$ and $\Lambda$ are countable groups
such that $L\Gamma\cong L\Lambda$, then $\Gamma$ is biexact if and only if
$\Lambda$ is biexact. This should be compared with Sako’s result [Sak09] where
he shows that biexactness for groups is stable under measure equivalence.

One of the technical achievements of our approach (carried out in Section 4)
is that, in fact, we do not need to use local reflexivity of
$C^{*}_{\lambda}\Gamma$ directly. If we have a von Neumann algebra $M$ and an
ultraweakly dense ${\rm C}^{*}$-subalgebra $M_{0}$, we consider a version of
the $M$-topology called the $(M_{0}\subset M)$-topology that takes the dense
${\rm C}^{*}$-algebra $M_{0}$ into account. If the inclusion
$M_{0}\subset{\mathbb{S}}(M)$ is $(M_{0}\subset M)$-nuclear, we then have a
general upgrading result showing that $M$ is biexact. While this general
result is not needed to show that biexactness is stable under
$W^{*}$-equivalence, it is crucial in showing how biexactness interacts with
general constructions. For instance, this allows us to show that the free
product of biexact von Neumann algebras (with respect to arbitrary normal
faithful states) is again biexact. This also allows us to show that tensor
products of biexact von Neumann algebras satisfy a relative biexactness
property from which one deduces the unique prime decomposition results of
Ozawa and Popa [OP04]. This holds, in fact, for general biexact von Neumann
algebras, and puts this into the framework of Houdayer and Isono [HI17] so
that unique prime decomposition results can be deduced also in the type III
setting as well (see Corollary 6.15).

If we consider the larger $M$-bimodule ${\mathbb{B}}(L^{2}M)$ instead of
$\mathbb{S}(M)$, then $(M_{0}\subset M)$-nuclearity for the inclusion
$M\subset{\mathbb{B}}(L^{2}M)$ is related to Isono’s notion of $M_{0}$ being
weakly exact in $M$ [Iso13]. In Section 5 we make this connection explicit,
and as an application we show in Corollary 5.11 that the free product (with
respect to arbitrary normal faithful states) of weakly exact von Neumann
algebras is again weakly exact, which was previously shown by Isono to hold in
many cases. We also use this perspective to provide a new characterization of
weak exactness for von Neumann algebras, answering Problem 10.4.3 from [BO08]
(see Theorem 5.8).

Biexact von Neumann algebras in general also have many of the
indecomposability type properties that are known to hold for the group von
Neumann algebras associated to biexact groups. We show a biexact von Neumann
algebra $M$ is always full, solid, and properly proximal. Moreover, if $M$ is
finite and weakly amenable, then $M$ is also strongly solid in the sense of
Ozawa and Popa [OP10b].

It was shown in [DKEP22] that if $\Gamma$ is a nonamenable group and we have
an embedding $L\Gamma\subset L\mathbb{F}_{2}$, then it follows that $\Gamma$
is properly proximal and consequently is not inner amenable (an alternate
proof of this fact was also found recently in [Dri23]). Since biexactness is
inherited to subalgebras it follows that $\Gamma$ is even biexact. In fact,
what we establish is a general principle that higher-rank lattices cannot
“$W^{*}$-embed” into rank one lattices. Specifically, if $G$ is a connected
real semisimple Lie group with finite center and $\mathbb{R}{\rm-rank}(G)\geq
2$, and if $\Gamma<G$ is a lattice, then by [Mor15, Exercise 8.2.7], together
with [Sak09, Dep20], it follows that $\Gamma$ is not biexact and hence
$L\Gamma$ cannot embed into $L\Lambda$ for any lattice $\Lambda$ in a rank one
connected real simple Lie group with finite center.

It was shown recently by Caspers [Cas22] that if $\mathcal{H}$ is an infinite
dimensional real Hilbert space and $-1<q<1$ with $q\not=0$, then the
$q$-Gaussian von Neumann algebra $M_{q}(\mathcal{H})$ is not isomorphic to a
free group factor. Casper’s result shows, in fact, that $M_{q}(\mathcal{H})$
is not biexact, and hence it then follows that $M_{q}(\mathcal{H})$ cannot
even embed into a free group factor. This last fact can also be deduced
directly from [Oza10], together with Theorem 7.20 below, which is independent
of the results in the earlier sections. We remark that this can also be
deduced by combining Caspers work with Corollary 3.5 in [Mar23].

Despite the fact that $M_{q}(\mathcal{H})$ is not biexact when $-1<q<1$,
$q\not=0$, and $\mathcal{H}$ is infinite dimensional, we show in Section 8,
that $M_{q}(\mathcal{H})$ shares many of the same indecomposability results
that biexact von Neumann algebras have. In particular, it is strongly solid
and every von Neumann subalgebra not having an amenable summand is properly
proximal. Strong solidity for $M_{q}(\mathcal{H})$ was shown previously by
Avsec [Avs11] in the case when $\mathcal{H}$ is finite-dimensional. Our proof
in the infinite dimensional case uses deformation/rigidity techniques,
together with techniques developed in [DKEP22], to reduce the problem to the
finite-dimensional case where Avsec’s results may then be applied.

In Section 8 we also give another source of solid von Neumann algebras that
are not biexact through the use of Gaussian actions. Specifically, we show
that if $\Gamma$ is a biexact group and
$\pi:\Gamma\to\mathcal{O}(\mathcal{H})$ is an orthogonal representation
satisfying $\pi\not\prec\lambda$, but $\pi^{\otimes k}\prec\lambda$ for some
$k>1$, then the crossed product of the Gaussian action
$A_{\mathcal{H}\,\overline{\otimes}\,\ell^{2}\mathbb{N}}\rtimes^{\sigma_{\pi\otimes
1}}\Gamma$ is solid, but is not biexact. Solidity of this von Neumann algebra
is a consequence of Boutonnet’s solid ergodicity result from [Bou12], and in
Section 8 we also give a relatively direct proof of this result.

### Acknowledgements

We thank Cyril Houdayer and Amine Marrakchi for useful comments.

##  2\. Preliminaries and notation

For ${\rm C}^{*}$-algebras or operator systems $A$ and $B$, we denote the
algebraic tensor product by $A\odot B$ and the minimal tensor product by
$A\otimes B$. We will use the abbreviation c.c. for completely contractive,
c.p. for completely positive, c.c.p. for contractive completely positive, and
u.c.p. for unital completely positive.

For a von Neumann algebra $M$, we denote by $(M,L^{2}M,J,\mathfrak{B})$ its
standard form [Haa75]. If $M$ is $\sigma$-finite and we have a normal faithful
state $\mu$ on $M$, then the standard form may be realized as
$(M,L^{2}(M,\mu),J_{\mu},\mathfrak{B}_{\mu})$, where the representation of $M$
on $L^{2}(M,\mu)$ is given by the GNS-construction, $J_{\mu}$ is modular
conjugation on $L^{2}(M,\mu)$, and $\mathfrak{B}_{\mu}$ is the positive part
of $L^{2}(M,\mu)$. We denote by $\mu^{1/2}\in L^{2}(M,\mu)$ the canonical
cyclic vector so that $\mu(x)=\langle x\mu^{1/2},\mu^{1/2}\rangle$ for $x\in
M$. The centralizer algebra with respect to $\mu$ is denoted by $M^{\mu}$.

For von Neumann algebras $M$ and $N$, a normal Hilbert $M$-$N$ bimodule is a
Hilbert space $\mathcal{H}$, together with a unital $*$-representation
$\pi_{\mathcal{H}}$ of the algebraic tensor product $M\odot N^{\rm op}$ such
that $\pi_{\mathcal{H}}$ is normal and faithful when restricted to $M$ and
$N^{\rm op}$ separately. We will write simply $x\xi
y=\pi_{\mathcal{H}}(x\otimes y^{\rm op})\xi$, for $x\in M$, $y\in N$, and
$\xi\in\mathcal{H}$. A normal Hilbert $M$-$N$ bimodule $\mathcal{H}$ is weakly
contained in another normal Hilbert $M$-$N$ bimodule $\mathcal{K}$ (written
$\mathcal{H}\prec\mathcal{K}$) if the identity map on $M\odot N^{\rm op}$
extends to a $*$-homomorphism from $C^{*}(\pi_{\mathcal{K}}(M\odot N^{\rm
op}))$ to $C^{*}(\pi_{\mathcal{H}}(M\odot N^{\rm op}))$. The trivial bimodule
for $M$ is given by $L^{2}M$ with the bimodule structure $x\xi y=xJy^{*}J\xi$,
and the coarse $M$-$N$ bimodule is given by
$L^{2}M\,\overline{\otimes}\,L^{2}N$ with the bimodule structure given by the
canonical representation of $M\otimes N^{\rm op}$ in
${\mathbb{B}}(L^{2}M\,\overline{\otimes}\,L^{2}N)$.

We will say that a von Neumann subalgebra $N\subset M$ is with expectation if
there exists a normal faithful conditional expectation $E:M\to N$. Given a
normal faithful conditional expectation $E:M\to N$, and a normal faithful
semifinite weight $\psi_{N}$ on $N$, we obtain a normal faithful semifinite
weight on $M$ by $\psi=\psi_{N}\circ E$, and this gives rise to an inclusion
$L^{2}(N,\psi_{N})\subset L^{2}(M,\psi)$. The Jones projection
$e_{N}:L^{2}(M,\psi)\to L^{2}(N,\psi_{N})$ is the corresponding orthogonal
projection. The Jones projection does not depend on the normal faithful
semifinite weight $\psi_{N}$ [HI17, Appendix A] and so we may consider it
abstractly as a coisometry $e_{N}:L^{2}M\to L^{2}N$. The Jones projection
$e_{N}$ satisfies

  1. (1)

$e_{N}^{*}xe_{N}=E(x)$, for $x\in M$.

  2. (2)

$e_{N}J_{M}=J_{N}e_{N}$.

If we have a group action $\Gamma{\,\curvearrowright\,}^{\sigma}M$ by
automorphisms, then by [Haa75, Theorem 3.2] there exists a unique unitary
representation $\sigma^{0}:\Gamma\to\mathcal{U}(L^{2}M)$ so that for each
$t\in\Gamma$ we have

  1. (1)

$\sigma_{t}(x)=\sigma_{t}^{0}x(\sigma_{t}^{0})^{*}$, for $x\in M$.

  2. (2)

$J\sigma_{t}^{0}=\sigma_{t}^{0}J$.

  3. (3)

$\sigma_{t}^{0}(\mathfrak{B})=\mathfrak{B}$.

We call the representation $\sigma^{0}:\Gamma\to\mathcal{U}(L^{2}M)$ the
Koopman representation associated to the action $\sigma$.

If we have an action $\Gamma{\,\curvearrowright\,}A$ of a group on a ${\rm
C}^{*}$-algebra, then there exists a faithful nondegenerate representation
$A\subset{\mathbb{B}}(\mathcal{H})$ and a covariant representation of
$\pi:\Gamma\to\mathcal{U}(\mathcal{H})$. The reduced crossed-product
$A\rtimes_{r}\Gamma$ is the ${\rm C}^{*}$-subalgebra generated by
$\\{(a\otimes 1)\pi_{t}\otimes\lambda_{t}\mid a\in A,t\in\Gamma\\}$, where
$\lambda:\Gamma\to\mathcal{U}(\ell^{2}\Gamma)$ denotes the left-regular
representation. This ${\rm C}^{*}$-algebra is independent of the covariant
representation, and we identify $A$ with $A\otimes\mathbb{C}$ so that
$A\subset A\rtimes_{r}\Gamma$.

If we have an $\Gamma{\,\curvearrowright\,}M$ of a group on a von Neumann
algebra, then we denote by
$M\rtimes\Gamma\subset{\mathbb{B}}(L^{2}M\,\overline{\otimes}\,\ell^{2}\Gamma)$
the von Neumann crossed-product, which is the von Neumann completion of
$M\rtimes_{r}\Gamma\subset{\mathbb{B}}(L^{2}M\,\overline{\otimes}\,\ell^{2}\Gamma)$.
We denote by $u_{t}$ the unitary $\sigma^{0}_{t}\otimes\lambda_{t}\in
M\rtimes_{r}\Gamma$, although if $M=\mathbb{C}$, then in this case we have
$u_{t}=\lambda_{t}$ and we will use both notations interchangeably. The von
Neumann algebra $M\rtimes\Gamma$ is standardly represented on
$L^{2}M\,\overline{\otimes}\,\ell^{2}\Gamma$, and we may explicitly compute
the modular conjugation operator to see that it satisfies

$J_{M\rtimes\Gamma}(\xi\otimes\delta_{t})=\sigma_{t^{-1}}^{0}J_{M}\xi\otimes\delta_{t^{-1}}$

for $\xi\in L^{2}M$ and $t\in\Gamma$. If we use the canonical identification
$L^{2}M\,\overline{\otimes}\,\ell^{2}\Gamma\cong\oplus_{t\in\Gamma}L^{2}M$,
then for $x\in M$ and $t\in\Gamma$ we have

(1)
$J_{M\rtimes\Gamma}\,x\,J_{M\rtimes\Gamma}=\oplus_{t\in\Gamma}J_{M}\sigma_{t}(x)J_{M},\
\ \ \ \ \ \ \ \ \
J_{M\rtimes\Gamma}\,u_{t}\,J_{M\rtimes\Gamma}=1\otimes\rho_{t},$

where $\rho:\Gamma\to\mathcal{U}(\ell^{2}\Gamma)$ denotes the right-regular
representation.

If $M$ is a von Neumann algebra, $p,q\in\mathcal{P}(M)$ are nonzero
projections and $B\subset pMp$, $Q\subset qMq$ are von Neumann subalgebras,
then we say the $B$ embeds with expectation into $Q$ inside of $M$ and write
$B\preceq M$ if there exist projections $e\in\mathcal{P}(B)$,
$f\in\mathcal{P}(Q)$, a nonzero partial isometry $v\in eMf$ and a unital
normal $*$-homomorphism $\phi:eBe\to fQf$ such that the inclusion
$\phi(eBe)\subset fQf$ is with expectation and $bv=v\phi(b)$ for all $b\in B$.
By a fundamental theorem of Popa [Pop06, Theorem 2.1], for a tracial von
Neumann algebra $M$, we have $B\not\preceq_{M}Q$ if and only if there exists a
net $\\{u_{i}\\}_{i}\subset\mathcal{U}(B)$ so that $E_{Q}(au_{i}b)\to 0$ in
the ultrastrong-∗ topology for all $a,b\in M$. Popa’s theorem has been
extended by Houdayer and Isono [HI17, Theorem 4.3] to the case when $B$ is
finite and both $B$ and $Q$ are with expectation.

##  3\. A relative topology on ${\rm C}^{*}$-bimodules

In this section we collect some needed background information, and we
introduce a generalization of Magajna’s $M$-$N$-topology on operator bimodules
[Mag00]. Many of the results here are adaptations of current techniques in the
literature, although a few results (e.g., Lemma 3.7) use new techniques.

Let $M_{0}$ and $N_{0}$ be ${\rm C}^{*}$-algebras. An (concrete) operator
$M_{0}$-$N_{0}$-bimodule consists of a concrete operator space
$X\subset{\mathbb{B}}(\mathcal{H})$, together with two faithful nondegenerate
representations $\pi:M_{0}\to{\mathbb{B}}({\mathcal{H}})$ and
$\rho:N_{0}\to{\mathbb{B}}({\mathcal{H}})$ so that $X$ is an
$\pi(M_{0})$-$\rho(N_{0})$-bimodule, whose bimodule structure is given by
composition of operators. An (concrete) operator $M_{0}$-system consists of
concrete operator system $X\subset{\mathbb{B}}({\mathcal{H}})$, together with
a faithful nondegenerate representation
$\pi:M_{0}\to{\mathbb{B}}({\mathcal{H}})$ so that $X$ is also an
$\pi(M_{0})$-bimodule. Note that we then have $M_{0}\cong\pi(M_{0})\subset X$.
We say that $X$ is a dual operator $M_{0}$-$N_{0}$-bimodule (resp. dual
operator $M_{0}$-system) if $X$ may be taken as above to be ultraweakly
closed.

If $M_{0}=M$ and $N_{0}=N$ are von Neumann algebras, then the operator
$M$-$N$-bimodule (resp. operator $M$-system) is normal if this concrete
realization can be made so that $\pi$ and $\rho$ (resp. $\pi$) is a normal
representation. More generally, we will say that an operator
$M_{0}$-$N_{0}$-bimodule (resp. operator $M_{0}$-system) is $(M_{0}\subset
M)$-$(N_{0}\subset N)$-normal (resp. $(M_{0}\subset M)$-normal) if there the
concrete realization can be made so that $\pi$ and $\rho$ (resp. $\pi$) extend
to normal representations of $M$ and $N$ (resp. $M$).

Let $X$ be a normal operator $M$-$N$-bimodule for von Neumann algebras $M$ and
$N$. For each pair of positive functionals $\omega\in M_{*}$ and $\rho\in
N_{*}$ we have an associated seminorm on $X$ given by

$s^{\rho}_{\omega}(x)=\inf\\{\rho(a^{*}a)^{1/2}\|y\|\omega(b^{*}b)^{1/2}\mid
x=a^{*}yb,a,b\in M,y\in X\\}.$

Following Magajna [Mag00], the topology induced by these seminorms will be
called the $M$-$N$-topology, or simply the $M$-topology if $M=N$. If $S\subset
X$ is a subset, then we denote the closure of $S$ in the $M$-$N$-topology by
$\overline{S}^{{}_{M-N}}$.

A linear functional $\varphi\in X^{*}$ is $M$-$N$-normal (or simply $M$-normal
if $M=N$) if for each $a\in X$ the linear functionals $M\ni
x\mapsto\varphi(xa)$ and $N\ni y\mapsto\varphi(ay)$ are normal. We let
$X^{M\sharp N}$ denote the space of $M$-$N$-normal linear functionals and we
call the resulting weak topology on $X$ the weak $M$-$N$-topology (or simply
the weak $M$-topology if $M=N$). By [Mag00, Theorem 3.7], $X^{M\sharp N}$
coincides with the space of $M$-$N$-topology continuous linear functionals on
$X$.

More generally, we may consider a similar topology that takes ultraweakly
dense ${\rm C}^{*}$-subalgebras into consideration. Let $M$ and $N$ be von
Neumann algebras, let $M_{0}\subset M$ and $N_{0}\subset N$ be ultraweakly
dense ${\rm C}^{*}$-subalgebras with $1_{M}\in M_{0}$ and $1_{N}\in N_{0}$,
and let $X$ be an operator $M_{0}$-$N_{0}$-bimodule. We may equip $M_{0}$ with
the ultraweak topology from $M$ and we denote by $M_{0}^{\sharp}\subset
M_{0}^{*}$ the space of normal functionals $M_{*}$, restricted to $M_{0}$. We
similarly define $N_{0}^{\sharp}$.

Given positive functionals $\omega\in M_{0}^{\sharp}$ and $\rho\in
N_{0}^{\sharp}$, we may consider a seminorm on $X$ given by

$s_{\omega}^{\rho}(x)=\inf\\{\rho(a^{*}a)^{1/2}\|y\|\omega(b^{*}b)^{1/2}\mid
x=a^{*}yb,a\in C_{n}(M_{0}),b\in
C_{n}(N_{0}),y\in{\mathbb{M}}_{n}(X),n\in{\mathbb{N}}\\}.$

The triangle inequality for $s_{\omega}^{\rho}$ follows from the standard
argument showing that the Haagerup norm on tensor products satisfies the
triangle inequality.

In general, if $M_{0}$ and $N_{0}$ do not necessarily contain units, we may
use the same formula above to define $s_{\omega}^{\rho}$ on the subspace
$M_{0}XN_{0}$. Note that if we set $\tilde{M}_{0}=M_{0}+\mathbb{C}1_{M}$ and
$\tilde{N}_{0}=N_{0}+\mathbb{C}1_{N}$, then the seminorm
$s_{\omega,N_{0}}^{\rho,M_{0}}$ on $M_{0}XN_{0}$ obtained by viewing this as
an operator $M_{0}$-$N_{0}$ bimodule agrees with the corresponding seminorm
$s_{\omega,\tilde{N}_{0}}^{\rho,\tilde{M}_{0}}$ obtained by viewing
$M_{0}XN_{0}$ as a operator $\tilde{M}_{0}$-$\tilde{N}_{0}$ bimodule. Indeed,
we clearly have $s_{\omega,\tilde{N}_{0}}^{\rho,\tilde{M}_{0}}\leq
s_{\omega,N_{0}}^{\rho,M_{0}}$, and if $x=a^{*}yb\in M_{0}XN_{0}$ with $a\in
C_{n}(\tilde{M}_{0})$ and $b\in R_{N}(\tilde{N}_{0})$, then letting
$\\{e_{i}\\}_{i}$ and $\\{f_{j}\\}_{j}$ denote approximate identities for
$M_{0}$ and $N_{0}$ respectively we have
$s_{\omega,N_{0}}^{\rho,M_{0}}(e_{i}a^{*}ybf_{j})\leq\rho(e_{i}a^{*}ae_{i})^{1/2}\|y\|\omega(f_{j}b^{*}bf_{j})^{1/2}$.
Since
$\lim_{i\to\infty}\rho(e_{i}a^{*}ae_{i})=\rho(a^{*}a),\lim_{j\to\infty}\omega(f_{j}b^{*}bf_{j})=\omega(b^{*}b)$,
and $\lim_{i\to\infty,j\to\infty}\|x-e_{i}xf_{j}\|=0$, it then follows that
$s_{\omega,N_{0}}^{\rho,M_{0}}\leq
s_{\omega,\tilde{N}_{0}}^{\rho,\tilde{M}_{0}}$.

For a general operator $M_{0}$-$N_{0}$ bimodule we may then define
$s_{\omega}^{\rho}$ on all of $X$ by the formula

$s_{\omega}^{\rho}(x)=\max\left\\{\inf_{z\in
M_{0}XN_{0}}\|x-z\|,s_{\omega,\tilde{N}_{0}}^{\rho,\tilde{M}_{0}}(x)\right\\}.$

We call the topology on $X$ induced by the family of seminorms
$\\{s^{\rho}_{\omega}\mid\omega\in(M_{0})^{\sharp}_{+},\rho\in(N_{0})^{\sharp}_{+}\\}$
the $(M_{0}\subset M)$-$(N_{0}\subset N)$-topology (or simply the
$(M_{0}\subset M)$-topology when $M_{0}=N_{0}$ and $M=N$). In the case when
$M_{0}=M$ and $N_{0}=N$, the following lemma is contained in [Mag98, Lemma
3.1].

######  Lemma 3.1.

Let $M$ and $N$ be von Neumann algebras, let $M_{0}\subset M$ and
$N_{0}\subset N$ be ultraweakly dense ${\rm C}^{*}$-subalgebras, let $X$ be an
operator $M_{0}$-$N_{0}$-bimodule, and $Y\subset X$ an operator
$M_{0}$-$N_{0}$-subbimodule. Then the restriction of the $(M_{0}\subset
M)$-$(N_{0}\subset N)$-topology on $X$ to $Y$ coincides with the
$(M_{0}\subset M)$-$(N_{0}\subset N)$-topology on $Y$.

###### Proof.

Given $\omega\in(M_{0})^{\sharp}_{+}$ and $\rho\in(N_{0})^{\sharp}_{+}$, we
denote by $s^{\rho}_{\omega,X}$ and $s^{\rho}_{\omega,Y}$ the above seminorms
on $X$ and $Y$, respectively. It is clear that $s^{\rho}_{\omega,X}(y)\leq
s^{\rho}_{\omega,Y}(y)$ for any $y\in Y$. To see the reverse, suppose first
that $1_{M}\in M_{0}$ and $1_{N}\in N_{0}$ and $y\in Y$ is such that we have a
decomposition $y=a^{*}xb$ with $x\in{\mathbb{M}}_{n}(X)$, $a\in C_{n}(M_{0})$,
and $b\in C_{n}(N_{0})$.

For $k\geq 1$ we let $f_{k}:[0,\infty)\to[0,\infty)$ denote the function given
by $f_{k}(t)=\begin{cases}1/k&0\leq t\leq 1/k\\\ t&t>1/k\end{cases}$, then
$y_{k}=f_{k}(|a|)^{-1}a^{*}xbf_{k}(|b|)^{-1}\in Y$ with $\|y_{k}\|\leq\|y\|$,
and so

$s^{\rho}_{\omega,Y}(y)\leq\rho(f_{k}(|a|)^{2})^{1/2}\|y\|\omega(f_{k}(|b|)^{2})^{1/2}.$

Since $f_{k}(|a|)^{2}\to|a|^{2}$ and $f_{k}(|b|)^{2}\to|b|^{2}$ in norm, we
then have $s^{\rho}_{\omega,Y}(y)\leq s^{\rho}_{\omega,X}(y)$.

In the non-unital case we notice that if $\\{e_{i}\\}_{i}$ and
$\\{f_{j}\\}_{j}$ are approximate units for $N_{0}$ and $M_{0}$, respectively,
then we have $\inf_{z\in
M_{0}XN_{0}}\|x-z\|=\lim_{i\to\infty,j\to\infty}\|x-e_{i}xf_{j}\|$ for $x\in
X$, and so if $y\in Y$, then we have $\inf_{z\in
M_{0}XN_{0}}\|y-z\|=\inf_{z\in M_{0}YN_{0}}\|y-z\|$. The general result then
follows from the unital case. ∎

######  Remark 3.2.

If $1_{M}\in M_{0}$, $1_{N}\in N_{0}$ and we are given an operator
$M_{0}$-$N_{0}$-bimodule $X$, we may also consider the function

$\tilde{s}^{\rho}_{\omega}(x)=\inf\\{\rho(a^{2})^{1/2}\|y\|\omega(b^{2})^{1/2}\mid
x=ayb,a,b\in(M_{0})_{+},y\in X\\}.$

The same argument above shows that if $Y\subset X$ is an
$M_{0}$-$N_{0}$-subbimodule and $y\in Y$, then the value of
$\tilde{s}^{\rho}_{\omega}(y)$ does not depend on whether we consider $y$ as
an element in $Y$ or $X$.

In the case when $X={\mathbb{B}}(\mathcal{H})$, we may use polar decomposition
in ${\mathbb{B}}(\mathcal{H})$ to see that
$\tilde{s}^{\rho}_{\omega}=s^{\rho}_{\omega}$. Since every operator
$M_{0}$-$N_{0}$-bimodule is a subbimodule of ${\mathbb{B}}(\mathcal{H})$ for
some Hilbert space, it then follows that
$s^{\rho}_{\omega}=\tilde{s}^{\rho}_{\omega}$ in general. When $M_{0}=N_{0}$,
$M=N$, and $\omega=\rho$, we use the notation $s_{\omega}$ in place of
$s_{\omega}^{\omega}$.

Continuing as above with $M$ and $N$ von Neumann algebras, $M_{0}\subset M$,
$N_{0}\subset N$ ultraweakly dense ${\rm C}^{*}$-subalgebras, and $X$ an
operator $M_{0}$-$N_{0}$-bimodule, we denote by $X^{M_{0}\sharp N_{0}}$ (or
simply $X^{\sharp}$ if no confusion will arise) the space of functionals
$\varphi\in X^{*}$ such that for any $x\in X$ the map $M_{0}\times
N_{0}\ni(a,b)\mapsto\varphi(axb)$ is separately ultraweak continuous. We call
the $\sigma(X,X^{M_{0}\sharp N_{0}})$-topology on $X$ the weak $(M_{0}\subset
M)$-$(N_{0}\subset N)$-topology (or simply the weak $(M_{0}\subset
M)$-topology if $M_{0}=N_{0}$ and $M=N$). The following proposition is from
[Mag00, Theorem 3.7], and we include a proof for convenience.

######  Proposition 3.3.

In the above setting, a functional $\varphi\in X^{*}$ is continuous in the
$(M_{0}\subset M)$-$(N_{0}\subset N)$-topology if and only if $\varphi\in
X^{M_{0}\sharp N_{0}}$.

###### Proof.

The backward direction is clear. To see the forward direction, note that

$M_{0}\times X\times N_{0}\ni(a,x,b)\mapsto\varphi(axb)\in{\mathbb{C}}$

is a completely bounded multilinear map. It follows from the Christensen-
Paulsen-Sinclair representation theorem (see, e.g., [SS95, Theorem 1.5.4])
that there exist $*$-representations
$\pi_{1}:M_{0}\to{\mathbb{B}}({\mathcal{K}}_{1})$,
$\pi_{2}:N_{0}\to{\mathbb{B}}({\mathcal{K}}_{2})$, a complete contraction
$\phi:X\to{\mathbb{B}}({\mathcal{H}})$, bounded operators
$T_{1}:{\mathcal{H}}\to{\mathcal{K}}_{1}$,
$T_{2}:{\mathcal{K}}_{2}\to{\mathcal{H}}$, and vectors
$\xi_{i}\in{\mathcal{K}}_{i}$ such that
$\varphi(axb)=\langle\pi_{1}(a)T_{1}\phi(x)T_{2}\pi_{2}(b)\xi_{2},\xi_{1}\rangle$.
We let $p_{1}\in M_{0}^{**}$ (resp. $p_{2}\in N_{0}^{**}$) denote the support
projection for the inclusion $M_{0}\to M$ (resp. $N_{0}\to N$). Since for any
$x\in X$ the map $M_{0}\times N_{0}\ni(a,b)\mapsto\varphi(axb)$ is separately
ultraweak continuous, it then follows that we have
$\varphi(axb)=\langle\pi_{1}(a)T_{1}\phi(x)T_{2}\pi_{2}(b)p_{2}\xi_{2},p_{1}\xi_{1}\rangle$
for $x\in X$, $a\in M_{0}$, and $b\in N_{0}$.

Then we have
$|\varphi(axb)|\leq\|T_{1}\|\|T_{2}\|\eta_{1}(aa^{*})^{1/2}\|x\|\eta_{2}(b^{*}b)^{1/2}$,
where $\eta_{i}(\cdot)=\langle\pi_{i}(\cdot)p\xi_{i},p\xi_{i}\rangle$. Since
$\eta_{1}\in M_{0}^{\sharp}$ and $\eta_{2}\in N_{0}^{\sharp}$, it is then easy
to conclude that $\varphi$ is continuous in the $(M_{0}\subset
M)$-$(N_{0}\subset N)$-topology. ∎

The following result is in the same spirit as the Noncommutative Egorov
Theorem.

######  Lemma 3.4.

Let $M$ be a von Neumann algebra, $M_{0}\subset M$ an ultraweakly dense ${\rm
C}^{*}$-subalgebra and $X$ an operator $M_{0}$-bimodule. Suppose for each
$1\leq k\leq n$, $\\{x_{i}^{k}\\}_{i\in I}\subset X$ is a net converging to
$0$ in the $(M_{0}\subset M)$-topology. Then, for any finite subset
$F=\\{\varphi_{j}\mid 1\leq j\leq m\\}\subset(M_{0}^{\sharp})_{+}$, we may
find a net of positive contractions $1-z_{i}\in M_{0}$, such that
$\|z_{i}x_{i}^{k}z_{i}\|\to 0$ and $\varphi_{j}(1-z_{i})\to 0$ for any $1\leq
j\leq m$. When $M_{0}=M$, we may choose $\\{z_{i}\\}$ to be projections.

###### Proof.

Note first that it suffices to consider the case when $\\{x_{i}^{k}\\}_{i\in
I}\subset M_{0}XM_{0}$. For each $i$, $1\leq k\leq n$ and $1\leq j\leq m$, we
may then find a decomposition $x_{i}^{k}=a_{i,j,k}^{*}y_{i,j,k}b_{i,j,k}$ with
$a_{i,j,k},b_{i,j,k}\in M_{0}$, $y_{i,j,k}\in X$ and
$\varphi_{j}(a_{i,j,k}^{*}a_{i,j,k})^{1/2}=\|y_{i,j,k}\|=\varphi_{j}(b_{i,j,k}^{*}b_{i,j,k})^{1/2}\to
0$.

We fix $N>1$ and let $f_{N}:[0,\infty)\to[0,1]$ denote a piecewise linear
continuous function such that $f_{N}(t)=1$ if $t\leq 1/N$, and $f_{N}(t)=0$ if
$t\geq 2/N$. We then define

$z_{i}=f_{N}\left(\sum_{j=1}^{m}\sum_{k=1}^{n}(a_{i,j,k}^{*}a_{i,j,k}+b_{i,j,k}^{*}b_{i,j,k})\right)\in
M_{0}.$

Note that $0\leq z_{i}\leq 1$, and

$\|z_{i}a_{i,j,k}^{*}\|^{2}\leq\|z_{i}\left(\sum_{j=1}^{m}\sum_{k=1}^{n}(a_{i,j,k}^{*}a_{i,j,k}+b_{i,j,k}^{*}b_{i,j,k})\right)z_{i}\|\leq
2/N.$

Similarly we have $\|b_{i,j,k}z_{i}\|^{2}\leq 2/N$, and it follows that
$\lim_{i\to\infty}\|z_{i}x_{i}^{k}z_{i}\|=0$ for any $1\leq k\leq n$. Also,
note that we have $1-z_{i}\leq
N\sum_{j=1}^{m}\sum_{k=1}^{n}(a_{i,j,k}^{*}a_{i,j,k}+b_{i,j,k}^{*}b_{i,j,k})$
and hence $\lim_{i\to\infty}\varphi_{j}(1-z_{i})=0$.

When $M_{0}=M$, replace $f_{N}$ with $1_{[0,1/N]}$ so that $z_{i}$ is a
projection in $M$. ∎

The next lemma will allow us to adapt the standard c.c. to u.c.p. perturbation
result to the setting of $M$-topology.

######  Lemma 3.5.

Let $M$ be a von Neumann algebra, and let $M_{0}\subset M$ be an ultraweakly
dense ${\rm C}^{*}$-subalgebra. Let $E$ be an operator $M_{0}$-system, and for
$i\in I$ suppose $E_{i}$ is an operator system and $\phi_{i}:E_{i}\to E$ is a
net of c.c. maps such that $1_{F}-\phi_{i}(1_{E_{i}})\to 0$ in the
$(M_{0}\subset M)$-topology. Then, for any finite subset of states $F\subset
M_{0}^{\sharp}$, there exist c.c. maps $\psi_{i}:E_{i}\to E$ such that
$\|1_{E}-\psi_{i}(1_{E_{i}})\|\to 0$ and

$\lim_{i}\sup_{x\in(E_{i})_{1}}s_{\omega}^{\rho}((\psi_{i}-\phi_{i})(x))=0,$

for any $\omega,\rho\in F$.

###### Proof.

Since $1_{E}-\phi_{i}(1_{E_{i}})\to 0$ in the $(M_{0}\subset M)$-topology, it
follows as in the proof of Lemma 3.4 that there exist $1-z_{i}\in M_{0}$ with
$0\leq z_{i}\leq 1$ such that $\lim_{i}\omega(1-z_{i}^{2})=0$ for any
$\omega\in F$ and such that $\|z_{i}(1_{E}-\phi_{i}(1_{E_{i}}))z_{i}\|\to 0$.

For each $i$, fix a state $\eta_{i}\in E_{i}^{*}$ and define
$\psi_{i}:E_{i}\to E$ by
$\psi_{i}(x)=z_{i}\phi_{i}(x)z_{i}+(1-z_{i}^{2})\eta_{i}(x)=\left(\begin{smallmatrix}z_{i}\\\
(1-z_{i}^{2})^{1/2}\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}\phi_{i}(x)&0\\\
0&\eta_{i}(x)\end{smallmatrix}\right)\left(\begin{smallmatrix}z_{i}\\\
(1-z_{i}^{2})^{1/2}\end{smallmatrix}\right)$.

Then, $\psi_{i}$ is c.c. and satisfies $\|1_{E}-\psi_{n}(1_{E_{i}})\|\to 0$.
Moreover, for $x\in E_{i}$ and $\omega,\rho\in F$ we have

$\displaystyle s_{\omega}^{\rho}(\phi_{i}(x)-\psi_{i}(x))$
$\displaystyle\leq\omega(1-z_{i}^{2})\|x\|+s_{\omega}^{\rho}(\phi_{i}(x)-z_{i}\phi_{i}(x)z_{i})$
$\displaystyle=\omega(1-z_{i}^{2})\|x\|+s_{\omega}^{\rho}\left(\left(\begin{smallmatrix}1\\\
-iz_{i}\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}\phi_{i}(x)&0\\\
0&\phi_{i}(x)\end{smallmatrix}\right)\left(\begin{smallmatrix}1\\\
iz_{i}\end{smallmatrix}\right)\right)$
$\displaystyle\leq\omega(1-z_{i}^{2})\|x\|+\omega(1-z_{i}^{2})^{1/2}\|x\|\rho(1-z_{i}^{2})^{1/2}.$

∎

######  Lemma 3.6.

Let $M\subset{\mathbb{B}}({\mathcal{H}})$ be a von Neumann algebra,
$M_{0}\subset M$ an ultraweakly dense ${\rm C}^{*}$-subalgebra and
$\psi_{i}:E_{i}\to{\mathbb{B}}({\mathcal{H}})$ a net c.c. maps from operator
systems $E_{i}$ to ${\mathbb{B}}({\mathcal{H}})$. Suppose
$\psi_{i}(1_{E_{i}})\to 1$ in the $(M_{0}\subset M)$-topology. Then, for any
finite set of states $F\subset M_{0}^{\sharp}$, there exists a net of u.c.p.
maps $\phi_{i}:E_{i}\to{\mathbb{B}}({\mathcal{H}})$ such that
$\lim_{i}\sup_{x\in(E_{i})_{1}}s_{\omega}^{\rho}((\phi_{i}-\psi_{i})(x))=0$
for any $\omega,\rho\in F$.

###### Proof.

By Lemma 3.5, we have c.c.
$\psi_{i}^{\prime}:E_{i}\to{\mathbb{B}}({\mathcal{H}})$ such that
$\psi_{i}(1_{E_{i}})\to 1$ in norm and
$\lim_{i}\sup_{x\in(E_{i})_{1}}s_{\omega}^{\rho}((\psi_{i}-\psi_{i}^{\prime})(x))$
for any $\omega,\rho\in F$. Replacing $\psi_{i}^{\prime}(\cdot)$ with
$(\psi_{i}^{\prime}(\cdot)+\psi_{i}^{\prime}(\cdot^{*})^{*})/2$, we may
further assume $\psi_{i}^{\prime}$ is self-adjoint. Apply [BO08, Corollary
B.9] and we obtain a u.c.p. map $\phi_{i}:E_{i}\to{\mathbb{B}}({\mathcal{H}})$
such that $\|\phi_{i}-\psi_{i}^{\prime}\|_{cb}\leq
2\|\psi_{i}^{\prime}(1_{E_{i}})-1\|$. It then follows that for any $x\in
E_{i}$,

$\displaystyle s^{\rho}_{\omega}(\phi_{i}(x)-\psi_{i}(x))$ $\displaystyle\leq
2\|\psi_{i}^{\prime}(1_{E_{i}})-1\|\|x\|+s_{\omega}^{\rho}(\psi_{i}^{\prime}(x^{*})^{*}-\psi_{i}(x))/2+s^{\rho}_{\omega}(\psi_{i}^{\prime}(x)-\psi_{i}(x))/2$
$\displaystyle\leq
2\|\psi_{i}^{\prime}(1_{E_{i}})-1\|\|x\|+\omega(1-z_{i}^{2})\|x\|+\omega(1-z_{i}^{2})^{1/2}\|x\|\rho(1-z_{i}^{2})^{1/2}.$

∎

###  3.1. $M$-${\rm C}^{*}$-algebras and normal biduals

If $M_{0}$ is a ${\rm C}^{*}$-algebra, then an $M_{0}$-${\rm C}^{*}$-algebra
consists of a ${\rm C}^{*}$-algebra $A$, together with a faithful
nondegenerate $*$-homomorphism from $M_{0}$ into the multiplier algebra
$M(A)$. If $M_{0}=M$ is a von Neumann algebra, then an $M$-${\rm
C}^{*}$-algebra $A$ is normal if $M(A)$ is a normal operator $M$-system. More
generally, if we have an ultraweakly dense ${\rm C}^{*}$-algebra $M_{0}\subset
M$, then an $M_{0}$-${\rm C}^{*}$-algebra $A$ is $(M_{0}\subset M)$-normal if
$M(A)$ is $(M_{0}\subset M)$-normal as an operator $M_{0}$-system.

If $M$ is a von Neumann algebra and $M_{0}\subset M$ is an ultraweakly dense
${\rm C}^{*}$-subalgebra, and if we have a $(M_{0}\subset M)$-normal
$M_{0}$-${\rm C}^{*}$-algebra $A$, then we let $p_{\rm nor}\in
M_{0}^{**}\subset A^{**}$ denote the projection corresponding to the support
of the identity representation $M_{0}\to M$. On the predual $(p_{\rm
nor}A^{**}p_{\rm nor})_{*}$ we may consider the restriction map to $A$, and
this gives rise to an operator space isomorphism $(p_{\rm nor}A^{**}p_{\rm
nor})_{*}\cong A^{\sharp}$ (see [DKEP22, Section 2]). The dual map then allows
us to equip $A^{\sharp*}$ with a von Neumann algebraic structure so that

$A^{\sharp*}\cong p_{\rm nor}A^{**}p_{\rm nor}.$

Note that $p_{\rm nor}$ commutes with $M_{0}\subset M(A)\subset A^{**}$, and
so this isomorphism preserves the natural $M_{0}$-bimodule structures on
$A^{\sharp*}$ and $p_{\rm nor}A^{**}p_{\rm nor}$, so we will view
$A^{\sharp*}$ as a von Neumann algebra that contains $M$ as a von Neumann
subalgebra.

It is worth noting that the canonical mapping of $A$ into $A^{\sharp*}$ is
only a complete order isomorphism, rather than a $*$-isomorphism, since
$p_{\rm nor}$ need not to be central in $A^{**}$ if $A$ is not generated as a
${\rm C}^{*}$-algebra by $M_{0}$ and $M_{0}^{\prime}\cap A$. The von Neumann
algebra $A^{\sharp*}$ may be seen as the universal version of the construction
considered in [BC15, Proposition 3.1].

If $A$ is unital and $E\subset A$ a normal operator $M_{0}$-subsystem, then we
may equip $E^{\sharp*}$ with a dual normal operator $M$-system structure, by
identifying it as an operator $M$-subsystem of $A^{\sharp*}$. We will denote
by $i_{E}:E\to E^{\sharp*}$ the canonical complete order isomorphism.

If we have an inclusion of if $(M_{0}\subset M)$-normal $M_{0}$-${\rm
C}^{*}$-algebras $J\subset A$, then we have a canonical (not necessarily
unital) embedding of von Neumann algebras $J^{**}\subset A^{**}$, and, by
considering an $M_{0}$-quasi-central approximate unit in $J$, we see that the
support projection $p_{0}$ of $J^{**}$ in $A^{**}$ commutes $p_{\rm nor}\in
A^{**}$ and the corresponding projection $p_{0}p_{\rm nor}$ gives the
corresponding normal projection in $J^{**}$. Thus, we also have a canonical
(not necessarily unital) embedding of von Neumann algebras $J^{\sharp*}\subset
A^{\sharp*}$. In particular, if $J\subset A$ is an ideal, then the support
projection $p$ for $J^{\sharp*}$ is central and gives rise to the
decomposition

$A^{\sharp*}=J^{\sharp*}\oplus p^{\perp}A^{\sharp*}.$

If $J$ is not an ideal, but rather a hereditary ${\rm C}^{*}$-subalgebra, then
$p$ is no longer central and so in this case we obtain the matrix
decomposition

$A^{\sharp*}=\begin{pmatrix}{J^{\sharp*}}&{pA^{\sharp*}p^{\perp}}\\\
{p^{\perp}A^{\sharp*}p}&{p^{\perp}A^{\sharp*}p^{\perp}}\end{pmatrix}.$

If $M$ is a von Neumann algebra and $E$ is an operator $M$-system, then the
Cauchy-Schwarz inequality easily shows that a positive linear functional
$\varphi\in E^{*}$ is contained in $E^{M\sharp M}$ if and only if
$\varphi_{|M}$ is normal. This observation can be used to give the following
useful lemma, giving a criterion for when a c.p. map is continuous in the
$M$-topology.

######  Lemma 3.7.

Let $M$ and $N$ be von Neumann algebras and let $E$ and $F$ be operator $M$
and $N$-systems, respectively. If $\phi:E\to F$ is completely positive such
that the restriction of $\phi$ to $M$ defines a normal map from $M$ to $N$,
then $\phi$ is a continuous map from $E$ with the weak $M$-topology, to $F$
with the weak $N$-topology.

###### Proof.

To prove that $\phi$ is continuous from the weak $M$-topology to the weak
$N$-topology, we need to check that if $\eta\in F^{N\sharp N}$, then
$\eta\circ\phi\in E^{M\sharp M}$. Moreover, by viewing $F$ as an operator
subsystem of $(F^{N\sharp N})^{*}$, we see that every linear functional
$\eta\in F^{N\sharp N}$ is implemented by a vector linear functional in some
$N$-bimodular c.p. representation of $F$ where $N$ is normally represented.
Hence, by the polarization identity, we may write $\eta$ as a span of states
in $F^{N\sharp N}$. Thus, it suffices to check that $\eta\circ\phi\in
E^{M\sharp M}$ whenever $\eta\in F^{N\sharp N}$ is a state. If $\eta\in
F^{N\sharp N}$ is a state, then $\eta_{|N}$ is normal and hence
$(\eta\circ\phi)_{|M}$ is normal, from which it follows that $\eta\circ\phi\in
E^{M\sharp M}$. ∎

######  Corollary 3.8.

Using the notation above, the map
${\operatorname{Ad}(e_{N})}:{\mathbb{B}}(L^{2}M)\to{\mathbb{B}}(L^{2}N)$ is
continuous from the weak $M$-topology to the weak $N$-topology, and also from
the weak $M^{\prime}$-topology to the weak $N^{\prime}$-topology.

###  3.2. The small-at-infinity boundary and boundary pieces

If $M\subset{\mathbb{B}}(\mathcal{H})$ is a von Neumann algebra, then we will
have occasion to use not only the $M$-topology, but also the
$M^{\prime}$-topology on ${\mathbb{B}}(\mathcal{H})$. We therefore introduce
the seminorms $r_{\omega}$ on ${\mathbb{B}}(L^{2}M)$ given by

$r_{\omega}(T)=\inf\\{(\omega(Ja^{*}aJ)+\omega(b^{*}b))^{1/2}\|Z\|(\omega(Jc^{*}cJ)+\omega(d^{*}d))^{1/2}\\},$

where $\omega$ is a normal state on $M$, $J$ is the modular conjugation
operator, and the infimum is taken over all decompositions
$T=\left(\begin{smallmatrix}a\\\
b\end{smallmatrix}\right)^{*}Z\left(\begin{smallmatrix}c\\\
d\end{smallmatrix}\right)$ where $a,c\in M^{\prime}$, $b,d\in M$ and
$Z\in\mathbb{M}_{2}({\mathbb{B}}(L^{2}M))$. These seminorms on bounded sets
describe the coarsest locally convex topological vector space topology that
contains both the $M^{\prime}$-topology and the $M$-topology.

If $X$ is a left operator $M$-module (i.e., an operator
$M$-$\mathbb{C}$-bimodule), then for a positive linear functional $\omega\in
M^{\sharp}$ we denote by $s_{\omega}^{\ell}$ the seminorm $s_{\omega}^{\rho}$
where $\rho:\mathbb{C}\to\mathbb{C}$ is the identity map. Given a right
operator $M_{0}$-module, we similarly denote by $s_{\omega}^{r}$ the seminorm
$s_{\rho}^{\omega}$. We will also use similar notation $r_{\omega}^{\ell}$ and
$r_{\omega}^{r}$ for the corresponding seminorms on ${\mathbb{B}}(L^{2}M)$.
For easy reference, we summarize these seminorms on an operator
$T\in{\mathbb{B}}(L^{2}M)$ here:

$s_{\omega}(T)=\inf\\{\omega(a^{*}a)\|S\|\omega(b^{*}b)\mid T=a^{*}Sb,a,b\in
M,S\in{\mathbb{B}}(L^{2}M)\\};$

$s_{\omega}^{\ell}(T)=\inf\\{\omega(a^{*}a)\|S\|\mid T=a^{*}S,a\in
M,S\in{\mathbb{B}}(L^{2}M)\\};$

$s_{\omega}^{r}(T)=\inf\\{\|S\|\omega(b^{*}b)\mid T=Sb,b\in
M,S\in{\mathbb{B}}(L^{2}M)\\};$

$\displaystyle r_{\omega}(T)=\inf\\{(\omega(Ja^{*}aJ)+$
$\displaystyle\omega(b^{*}b))^{1/2}\|Z\|(\omega(Jc^{*}cJ)+\omega(d^{*}d))^{1/2}$
$\displaystyle\mid T=\left(\begin{smallmatrix}a\\\
b\end{smallmatrix}\right)^{*}Z\left(\begin{smallmatrix}c\\\
d\end{smallmatrix}\right),a,c\in M^{\prime},b,d\in
M,Z\in\mathbb{M}_{2}({\mathbb{B}}(L^{2}M))\\};$

$r_{\omega}^{\ell}(T)=\inf\\{(\omega(Ja^{*}aJ)+\omega(b^{*}b))^{1/2}\|Z\|\mid
T=\left(\begin{smallmatrix}a\\\ b\end{smallmatrix}\right)^{*}Z,a\in
M^{\prime},b\in M,Z\in C_{2}({\mathbb{B}}(L^{2}M))\\};$

$r_{\omega}^{r}(T)=\inf\\{\|Z\|(\omega(Jc^{*}cJ)+\omega(d^{*}d))^{1/2}\mid
T=Z\left(\begin{smallmatrix}c\\\ d\end{smallmatrix}\right),c\in
M^{\prime},d\in M,Z\in R_{2}({\mathbb{B}}(L^{2}M))\\}.$

In the case when $M$ is finite with a normal faithful trace $\tau$, the norm
$r_{\tau}^{r}(T)$ is equivalent to the norm
$\|T\|_{\infty,2}=\sup_{a\in(M)_{1}}\|T\hat{a}\|$, which is nothing but the
operator norm when thinking of $T$ as an operator from $M\subset
L^{2}(M,\tau)$ with the uniform norm into $L^{2}(M,\tau)$ [Oza10]. This result
was generalized in [DKEP22, Proposition 3.1] to show that the norm
$r_{\tau}(T)$ is equivalent to the norm
$\|T\|_{\infty,1}=\sup_{a,b\in(M)_{1}}|\langle T\hat{a},\hat{b}\rangle|$ where
we think of $T$ as an operator from $M\subset L^{2}(M,\tau)$ to
$L^{1}(M,\tau)\supset L^{2}(M,\tau)$.

The relationship between the seminorms is given by the following result.

######  Proposition 3.9.

Let $M$ be a von Neumann algebra and $A$ an $M$-${\rm C}^{*}$-algebra. Then,
for $\mu$ a normal positive linear functional and $x\in A$, we have
$s_{\mu}^{\ell}(x^{*})=s_{\mu}^{r}(x)=s_{\mu}(x^{*}x)^{1/2}$. Also, if
$T\in{\mathbb{B}}(L^{2}M)$, then we have
$r_{\mu}^{\ell}(T^{*})=r_{\mu}^{r}(T)=r_{\mu}(T^{*}T)^{1/2}$.

###### Proof.

First note that the equality $s_{\mu}^{\ell}(x^{*})=s_{\mu}^{r}(x)$ is
obvious. Also, if $x=yb$ with $y\in A$ and $b\in M$, then
$x^{*}x=b^{*}y^{*}yb$, and so
$s_{\mu}(x^{*}x)^{1/2}\leq\mu(b^{*}b)^{1/2}\|y^{*}y\|^{1/2}$. Taking the
infimum over all such decompositions gives $s_{\mu}(x^{*}x)^{1/2}\leq
s_{\mu}^{r}(x)$.

To see the reverse inequality, we suppose $x^{*}x=a^{*}yb$ for some $y\in A$,
and $a,b\in M$ and set $\kappa=\omega(a^{*}a)^{1/2}\|y\|\omega(b^{*}b)^{1/2}$.
By rescaling, we will assume that $\omega(a^{*}a)=\omega(b^{*}b)$. We then
have

$x^{*}x=\frac{1}{2}(a^{*}yb+b^{*}y^{*}a)=\frac{1}{2}\left(\begin{smallmatrix}a\\\
b\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&y\\\
y^{*}&0\end{smallmatrix}\right)\left(\begin{smallmatrix}a\\\
b\end{smallmatrix}\right),$

and note that
$\frac{1}{2}\mu(a^{*}a+b^{*}b)=\mu(a^{*}a)=\mu(a^{*}a)^{1/2}\mu(b^{*}b)^{1/2}$.
We consider the polar decomposition $\left(\begin{smallmatrix}a\\\
b\end{smallmatrix}\right)=u(a^{*}a+b^{*}b)^{1/2}$, and we set
$z=u^{*}\left(\begin{smallmatrix}0&y\\\ y^{*}&0\end{smallmatrix}\right)u$ and
$a_{0}=\frac{1}{\sqrt{2}}(a^{*}a+b^{*}b)^{1/2}$ so that we have
$x^{*}x=a_{0}za_{0}$ with $\omega(a_{0}^{2})\|z\|\leq\kappa$. Note that by
replacing $z$ with $pzp$ where $p$ is the support of $a_{0}\in M$ we may
assume also that $z\geq 0$.

We then have $|x|=|z^{1/2}a_{0}|$ and so if we consider the polar
decompositions $x=v|x|$ and $z^{1/2}a_{0}=w|z^{1/2}a_{0}|=w|x|$, then we have
$x=v|x|=vw^{*}z^{1/2}a_{0}$, and hence

$s_{\mu}^{r}(x)\leq\|vw^{*}z^{1/2}\|\mu(a_{0}^{2})^{1/2}\leq\kappa^{1/2}.$

Taking the infimum over all such decompositions then gives $s_{\mu}^{r}(x)\leq
s_{\mu}(x^{*}x)^{1/2}$.

The result for the seminorms $r_{\mu}^{\ell}$, $r_{\mu}^{r}$, and $r_{\mu}$
follows similarly. ∎

######  Corollary 3.10.

Let $M$ be a von Neumann algebra, let $A$ be an $M$-${\rm C}^{*}$-algebra, and
let $\mu$ be a normal positive linear functional on $M$. Then for $x,y,z\in A$
we have

$s_{\mu}(x^{*}yz)\leq s_{\mu}(x^{*}x)^{1/2}\|y\|s_{\mu}(z^{*}z)^{1/2}.$

Also, if $x,y,z\in{\mathbb{B}}(L^{2}M)$, then we have

$r_{\mu}(x^{*}yz)\leq r_{\mu}(x^{*}x)^{1/2}\|y\|r_{\mu}(z^{*}z)^{1/2}.$

###### Proof.

We clearly have $s_{\mu}(x^{*}yz)\leq
s_{\mu}^{\ell}(x^{*})\|y\|s_{\mu}^{r}(z)$, and so the result follows directly
from Proposition 3.9. The case for $r_{\mu}$ follows similarly. ∎

Let $M$ be a von Neumann algebra. Generalizing the group setting from [BIP21],
an $M$-boundary piece was defined in [DKEP22] to be a hereditary ${\rm
C}^{*}$-subalgebra ${\mathbb{X}}\subset{\mathbb{B}}(L^{2}M)$ such that $M\cap
M({\mathbb{X}})\subset M$ and $M^{\prime}\cap M({\mathbb{X}})\subset
M^{\prime}$ are weakly dense. Denote by

${\mathbb{K}}_{\mathbb{X}}^{L}(M)=\overline{\overline{{\mathbb{B}}(L^{2}M){\mathbb{X}}}^{{}_{{\mathbb{C}}-M}}}^{{}_{{\mathbb{C}}-M^{\prime}}},$

which is a left ideal containing $M$ and $M^{\prime}$ in its space of right
multipliers. If $\mathcal{H}$ is a Hilbert space, then we set

${\mathbb{K}}_{\mathbb{X}}^{L}(M,\mathcal{H})=\overline{\overline{{\mathbb{B}}(L^{2}M,\mathcal{H}){\mathbb{X}}}^{{}_{{\mathbb{C}}-M}}}^{{}_{{\mathbb{C}}-M^{\prime}}}.$

A consequence of [Mag98, Proposition 2.2] (see [DKEP22, Proposition 2.3]) is
that an operator $T\in{\mathbb{B}}(L^{2}M)$ is contained in
${\mathbb{K}}_{\mathbb{X}}^{L}(M)$ if and only if there exist orthogonal
families of projections $\\{f_{i}\\}_{i\in I},\\{\tilde{f}_{j}\\}_{j\in
J}\subset M$ such that $TJ\tilde{f}_{i}Jf_{j}\in{\mathbb{X}}$ for each $i\in
I$, $j\in J$.

Let

${\mathbb{K}}_{\mathbb{X}}(M)=({\mathbb{K}}_{\mathbb{X}}^{L}(M))^{*}\cap{\mathbb{K}}_{\mathbb{X}}^{L}(M)=({\mathbb{K}}_{\mathbb{X}}^{L}(M))^{*}{\mathbb{K}}_{\mathbb{X}}^{L}(M)\subset{\mathbb{B}}(L^{2}M)$

be the hereditary ${\rm C}^{*}$-subalgebra associated to
${\mathbb{K}}_{\mathbb{X}}^{L}(M)$, and note that both $M$ and $M^{\prime}$
are in its multiplier algebra. We also define

${\mathbb{K}}^{\infty,1}_{\mathbb{X}}(M)=\overline{\overline{{\mathbb{K}}_{\mathbb{X}}(M)}^{{}_{M-M}}}^{{}_{M^{\prime}-M^{\prime}}},$

which agrees with the closure in the topology given by the seminorms $r_{\mu}$
defined above.

By Lemma 3.4 if $T\in{\mathbb{K}}^{\infty,1}_{\mathbb{X}}(M)_{+}$, then there
exists a net of positive contractions $z_{i}\in M$ converging ultrastrongly to
$1$ so that in uniform norm we have $d(z_{i}Tz_{i},{\mathbb{X}})\to 0$, and
hence also $d((z_{i}Tz_{i})^{1/2},{\mathbb{X}})\to 0$. By considering the
polar decomposition we then have
$d(T^{1/2}z_{i},{\mathbb{B}}(L^{2}M){\mathbb{X}})\to 0$, and hence
$T^{1/2}\in{\mathbb{K}}_{\mathbb{X}}^{L}(M)$, which shows that
$T\in({\mathbb{K}}_{\mathbb{X}}^{L}(M))^{*}\cap{\mathbb{K}}_{\mathbb{X}}^{L}(M)={\mathbb{K}}_{\mathbb{X}}(M)$.
Thus, we have the equality of positive cones

${\mathbb{K}}^{\infty,1}_{\mathbb{X}}(M)_{+}={\mathbb{K}}_{\mathbb{X}}(M)_{+}.$

In particular, for a Hilbert space $\mathcal{H}$, an operator
$T\in{\mathbb{B}}(L^{2}M,\mathcal{H})$ is contained in
${\mathbb{K}}_{\mathbb{X}}^{L}(M,\mathcal{H})$ if and only if
$|T|\in{\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M)$.

The small-at-infinity boundary of $M$ with respect to the $M$-boundary piece
${\mathbb{X}}$ is defined as

${\mathbb{S}}_{\mathbb{X}}(M)=\\{T\in{\mathbb{B}}(L^{2}M)\mid[T,x]\in{\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M),\forall
x\in M^{\prime}\\}.$

Since the operator space ${\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M)$ is
typically not a ${\rm C}^{*}$-algebra, the space
${\mathbb{S}}_{\mathbb{X}}(M)$ is also typically not a ${\rm C}^{*}$-algebra,
though it is a normal operator $M$-system.

In the case when ${\mathbb{X}}={\mathbb{K}}(L^{2}M)$, we will denote
${\mathbb{K}}_{\mathbb{X}}^{L}(M)$, ${\mathbb{K}}_{\mathbb{X}}(M)$,
${\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M)$, and ${\mathbb{S}}_{\mathbb{X}}(M)$
by ${\mathbb{K}}^{L}(M)$, ${\mathbb{K}}(M)$, ${\mathbb{K}}^{\infty,1}(M)$, and
${\mathbb{S}}(M)$, respectively.

One reason for the utility of this definition for the small-at-infinity
boundary is that if $T\in{\mathbb{B}}(L^{2}M)$ and we consider the set

$P=\\{x\in M^{\prime}\mid[T,x]\in{\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M)\\},$

then this set is always a von Neumann subalgebra of $M^{\prime}$, and hence to
show that $T\in{\mathbb{S}}_{\mathbb{X}}(M)$ it suffices to check that
$[T,x]\in{\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M)$ for $x$ in a set of
generators for $M^{\prime}$. When $M$ is a tracial von Neumann algebra this is
Lemma 6.1 in [DKEP22], but the proof works in general. Indeed, it is easy to
see that $P$ is a ${\rm C}^{*}$-algebra containing the unit, and if $x$ is in
the ultrastrong∗ closure of $P$ and $\omega\in M_{*}$ is a positive linear
functional, then taking a net $x_{i}\in P$ so that $x_{i}$ converges to $x$ in
the ultrastrong∗-topology, we have
$\omega((x-x_{i})^{*}(x-x_{i})),\omega((x-x_{i})(x-x_{i})^{*})\to 0$ and hence

$d_{r_{\omega}}([T,x],{\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M))\leq
r_{\omega}(T(x-x_{i}))+r_{\omega}((x-x_{i})T)\to 0.$

Just as we could associate to any $M$-${\rm C}^{*}$-algebra $A$ a universal
von Neumann algebra $A^{\sharp*}$ that contains $M$ as a von Neumann
subalgebra, we can also associate to ${\mathbb{B}}(L^{2}M)$ a von Neumann
algebra ${\mathbb{B}}(L^{2}M)_{J}^{\sharp*}$ containing both $M$ and $JMJ$ as
von Neumann subalgebras. Specifically, we let
${\mathbb{B}}(L^{2}M)_{J}^{\sharp*}$ be the corner $p_{\rm nor}q_{\rm
nor}{\mathbb{B}}(L^{2}M)^{**}q_{\rm nor}p_{\rm nor}$, where $p_{\rm nor}\in
M^{**}$ denotes the projection corresponding to the support of the identity
representation $M\to M$, and $q_{\rm nor}\in(JMJ)^{**}$ denotes the projection
corresponding to the support of the identity representation $JMJ\to JMJ$.

If ${\mathbb{X}}\subset{\mathbb{B}}(L^{2}M)$ is a boundary piece, then we will
also let ${\mathbb{K}}_{\mathbb{X}}(M)_{J}^{\sharp*}$ denote $p_{\rm
nor}q_{\rm nor}{\mathbb{K}}_{X}(M)^{**}q_{\rm nor}p_{\rm nor}$ so that we have
a natural identification
${\mathbb{K}}_{\mathbb{X}}(M)_{J}^{\sharp*}=q_{{\mathbb{X}}}{\mathbb{B}}(L^{2}M)_{J}^{\sharp*}q_{{\mathbb{X}}}$
where $q_{\mathbb{X}}\in{\mathbb{B}}(L^{2}M)_{J}^{\sharp*}$ denotes the
support projection of ${\mathbb{K}}_{\mathbb{X}}(M)_{J}^{\sharp*}$.

##  4\. $M$-nuclearity

Let $M$ be a von Neumann algebra, $F$ an operator system or ${\rm
C}^{*}$-algebra, and $E$ an operator $M$-system or an $M$-${\rm
C}^{*}$-algebra. We say a c.c.p. map $\phi:F\to E$ is $M$-nuclear if there
exist nets of c.c.p. maps $\phi_{i}:F\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to E$ such that
$\psi_{i}\circ\phi_{i}(x)$ converges to $\phi(x)$ in the $M$-topology for any
$x\in F$. By [Mag00, Theorem 3.7] and a standard convexity argument, this is
equivalent to the existence of such maps $\phi_{i}$ and $\psi_{i}$ so that
$\psi_{i}\circ\phi_{i}(x)$ converges to $\phi(x)$ in the weak $M$-topology for
any $x\in F$.

More generally, if $M_{0}\subset M$ is an ultraweakly dense ${\rm
C}^{*}$-subalgebra and $E$ is an $M_{0}$-system or $M_{0}$-${\rm
C}^{*}$-algebra, we then say a c.c.p. map $\phi:F\to E$ is $(M_{0}\subset
M)$-nuclear if there exist nets of c.c.p. maps
$\phi_{i}:F\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to E$ such that
$\psi_{i}\circ\phi_{i}(x)$ converges to $\phi(x)$ in the $(M_{0}\subset
M)$-topology for any $x\in F$. Equivalently, by Proposition 3.3, there exist a
net of c.c.p. maps $\phi_{i}:F\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to E$ such that
$\psi_{i}\circ\phi_{i}(x)$ converges to $\phi(x)$ in the weak $(M_{0}\subset
M)$-topology.

We remark that if $F\subset{\mathbb{B}}(\mathcal{H})$, then in the definition
of $(M_{0}\subset M)$-nuclearity we may always take the map $\phi_{i}$ to be a
compression onto a finite-dimensional subspace
$\mathcal{K}\subset\mathcal{H}$, i.e., we may take $\phi_{i}$ to be of the
form $F\ni x\mapsto
P_{\mathcal{K}}xP_{\mathcal{K}}\in{\mathbb{B}}(L^{2}\mathcal{K})\cong\mathbb{M}_{\dim(\mathcal{K})}(\mathbb{C})$.
This can be seen easily from the following general lemma.

######  Lemma 4.1.

Let $F\subset{\mathbb{B}}(\mathcal{H})$ be a ${\rm C}^{*}$-algebra or operator
system and $\phi:F\to\mathbb{M}_{k}(\mathbb{C})$ a c.c. (resp. c.p., u.c.p.)
map. There exists a net of finite-dimensional subspaces
$\mathcal{K}_{i}\subset\mathcal{H}$ and c.c. (resp. c.p., u.c.p.) maps
$\phi_{i}:{\mathbb{B}}(\mathcal{K}_{i})\to\mathbb{M}_{k}(\mathbb{C})$ so that
$(\phi_{i}\circ{\rm Ad}(P_{\mathcal{K}_{i}}))_{|F}$ converges pointwise in
norm to $\phi$.

###### Proof.

Let $\tilde{\phi}:{\mathbb{B}}(\mathcal{H})\to\mathbb{M}_{k}(\mathbb{C})$ be a
c.c. extension of $\phi$, and let
$\phi_{i}:{\mathbb{B}}(\mathcal{H})\to\mathbb{M}_{k}(\mathbb{C})$ be normal
c.c. maps so that $\phi_{i}\to\tilde{\phi}$ in the point-norm topology. Since
each $\phi_{i}$ is normal, if we consider the family of finite-dimensional
subspaces of $\mathcal{H}$ to be a net ordered by inclusion, then for each
$x\in F$ we have norm convergence
$\phi_{i}(x)=\lim_{\mathcal{K}\to\infty}\phi_{i}\circ{\rm
Ad}(P_{\mathcal{K}})(x)$, and hence
$\phi(x)=\lim_{\mathcal{K}\to\infty}\lim_{i\to\infty}\phi_{i}\circ{\rm
Ad}(P_{\mathcal{K}})(x)$. The proof in the c.p. case or the u.c.p. case, when
$F$ is an operator system, is the same. ∎

Similar to the usual notion of nuclear maps, in the unital case we may
consider u.c.p. maps instead of c.c.p. maps.

######  Lemma 4.2.

Let $M$ be a von Neumann algebra with $M_{0}\subset M$ an ultraweakly dense
${\rm C}^{*}$-subalgebra. Suppose $F$ is an operator system, $E$ is an
$M_{0}$-system, and $\phi:F\to E$ is a u.c.p. $(M_{0}\subset M)$-nuclear map.
There exists a net of u.c.p. maps $\phi_{i}:F\to\mathbb{M}_{n(i)}(\mathbb{C})$
and $\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to E$ such that
$\psi_{i}\circ\phi_{i}(x)$ converges to $\phi(x)$ in the $(M_{0}\subset
M)$-topology for any $x\in F$

###### Proof.

Let $\tilde{\phi}_{i}:F\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\tilde{\psi}_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to E$ be c.c.p. maps such that
$\tilde{\psi}_{i}\circ\tilde{\phi}_{i}(x)-\phi(x)\to 0$ in the $(M_{0}\subset
M)$-topology for any $x\in F$. By Lemma 2.2.5 in [BO08] there exist u.c.p.
maps $\phi_{i}:F\to\mathbb{M}_{n(i)}(\mathbb{C})$ so that
$\tilde{\phi}_{i}(x)=\tilde{\phi}_{i}(1)^{1/2}\phi_{i}(x)\tilde{\phi}_{i}(1)^{1/2}$.

Note that $\tilde{\psi}_{i}(\tilde{\phi}_{i}(1))$ is a net of positive
contractions that converge to $1$ in the $(M_{0}\subset M)$-topology. By Lemma
3.4 we may find a net of positive contractions $1-z_{i}\in M_{0}$, so that
$\|z_{i}(1-\tilde{\psi}_{i}(\tilde{\phi}_{i}(1)))z_{i}\|\to 0$, and $z_{i}\to
1$ ultrastrongly in $M$. It therefore follows that
$1-z_{i}\tilde{\psi}_{i}(\tilde{\phi}_{i}(1))z_{i}$ converges to $0$ in the
$(M_{0}\subset M)$-topology. Hence, if we fix states
$\eta_{i}\in\mathbb{M}_{n(i)}(\mathbb{C})$ and set

$\psi_{i}(T)=z_{i}\tilde{\psi}_{i}(\tilde{\phi}_{i}(1)^{1/2}T\tilde{\phi}_{i}(1)^{1/2})z_{i}+(1-z_{i}\tilde{\psi}_{i}(\tilde{\phi}_{i}(1))z_{i})\eta_{i}(T),$

then $\psi_{i}$ are u.c.p. and we have $\psi_{i}\circ\phi_{i}$ converges to
the $phi$ in the point-$(M_{0}\subset M)$-topology. ∎

We also remark that for a bimodular map between $M_{0}$ systems $E$ and $F$,
the convergence in the definition of $(M_{0}\subset M)$-nuclearity can be
strengthened in the following sense.

######  Lemma 4.3.

Let $M$ be a von Neumann algebra, $M_{0}\subset M$ an ultraweakly dense ${\rm
C}^{*}$-subalgebra. Suppose $M_{0}\subset E$ and $M_{0}\subset F$ are two
$M_{0}$-systems, and $\phi:E\to F$ is $M_{0}$-bimodular and u.c.p. Then $\phi$
is $(M_{0}\subset M)$-nuclear if and only if there exist nets of u.c.p. maps
$\phi_{i}:E\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to F$ so that $\psi_{i}\circ\phi_{i}$
converges in the point-$(M_{0}\subset M)$-topology to $\phi$, and such that
for each $x\in M_{0}$ there exist $T_{i}\in F$ and $a_{i}\in M_{0}$ so that
$\psi_{i}\circ\phi_{i}(x)-x=T_{i}-a_{i}$ and we have

  1. (1)

$\|T_{i}\|\to 0$.

  2. (2)

$\sup_{i}\|a_{i}\|<\infty$.

  3. (3)

$a_{i}\to 0$ ultrastrongly.

###### Proof.

We let $\phi^{0}_{i}:E\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi^{0}_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to F$ denote u.c.p. maps so that
$\psi^{0}_{i}\circ\phi_{i}^{0}$ converges to the identity in the
point-$(M_{0}\subset M)$-topology. Fix a finite set
$E=\\{x_{1},\ldots,x_{m}\\}\subset M_{0}$ and a finite set of states
$S=\\{\omega_{1},\ldots,\omega_{n}\\}\subset M_{*}$. Then, we may apply Lemma
3.4 to $\\{\psi_{i}^{0}\circ\phi_{i}^{0}(x_{j})-x_{j}\\}_{i}$ and $S$, and we
obtain positive contractions $\\{1-z_{i}\\}_{i}\subset M_{0}$ such that
$\lim_{i}\max_{1\leq k\leq n}\omega_{k}(1-z_{i})=0$ and $\lim_{i}\max_{1\leq
j\leq m}\|z_{i}(\psi_{i}^{0}\circ\phi_{i}^{0}(x_{j})-x_{j})z_{i}\|=0$.

We now fix a state $\eta\in F^{M_{0}\sharp M_{0}}$ and define the u.c.p. map
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to F$ by
$\psi_{i}(T)=z_{i}\psi_{i}^{0}(T)z_{i}+(1-z_{i}^{2})\eta(T)$. We then have
$\psi_{i}\circ\phi_{i}^{0}(x_{j})-x_{j}=z_{i}(\psi_{i}^{0}\circ\phi_{i}^{0}(x_{j})-x_{j})z_{i}+(1-z_{i}^{2})\eta(\phi_{i}^{0}(x_{j}))-(x_{j}-z_{i}x_{j}z_{i})$.

Note that
$c_{i,j}:=(1-z_{i}^{2})\eta(\phi_{i}^{0}(x_{j}))+(x_{j}-z_{i}x_{j}z_{i})\in
M_{0}$ satisfies $\|c_{i,j}\|\leq 1+2\|x_{j}\|$, and
$\lim_{i\to\infty}\omega_{k}(c_{i,j}^{*}c_{i,j})=0$ for each $1\leq j\leq m$
and $1\leq k\leq n$. Since $F$ and $S$ were arbitrary finite sets, the result
then follows. ∎

Nuclearity of a map with respect to the $(M_{0}\subset M)$-topology can be
reformulated as a weak nuclearity property by adapting the argument from
[Kir95b, Lemma 2.8(i)] and using the appropriate normal bidual.

######  Lemma 4.4.

Let $M_{0}$ be an ultraweakly dense ${\rm C}^{*}$-subalgebra of a von Neumann
algebra $M$, and let $F$ be a normal $M_{0}$-system, or a normal $M_{0}$-${\rm
C}^{*}$-algebra. If $E$ is an operator system or ${\rm C}^{*}$-algebra and
$\theta:E\to F$ is a c.c.p. map, then the following conditions are equivalent:

  1. (1)

The c.c.p. map $\theta$ is $(M_{0}\subset M)$-nuclear.

  2. (2)

The c.c.p. map $i_{F}\circ\theta:E\to(F^{M_{0}\sharp M_{0}})^{*}$ is weakly
nuclear.

  3. (3)

The c.c.p. map $i_{F}\circ\theta:E\to(F^{M_{0}\sharp M_{0}})^{*}$ is
$(M_{0}\subset M)$-nuclear.

###### Proof.

Clearly (1) $\implies$ (3) $\implies$ (2), so we need only show (2) $\implies$
(1). Notice that for each c.c.p. map $\phi:\mathbb{M}_{n}(\mathbb{C})\to
E^{\sharp*}$, there exists a net of c.c.p. maps
$\phi_{i}:\mathbb{M}_{n}(\mathbb{C})\to E$ such that $\phi_{i}\to\phi$ in the
point-weak∗ topology. Indeed, since $({\mathbb{M}}_{n}({\mathbb{C}})\otimes
E)^{**}$ is completely isometrically isomorphic to
${\mathbb{M}}_{n}({\mathbb{C}})\otimes E^{**}$ [BO08, Proposition B.16], we
have a completely isometrically isomorphism
$({\mathbb{M}}_{n}({\mathbb{C}})\otimes
E)^{\sharp*}\cong{\mathbb{M}}_{n}({\mathbb{C}})\otimes E^{\sharp*}$. Moreover,
since c.p. maps from ${\mathbb{M}}_{n}({\mathbb{C}})$ to $E$ (resp.
$E^{\sharp*}$) are one-to-one correspondent to elements in the positive cone
of ${\mathbb{M}}_{n}(E)$ (resp. ${\mathbb{M}}_{n}(E^{\sharp*})$), it then
follows from the density of
${\mathbb{M}}_{n}(E)\subset{\mathbb{M}}_{n}(E)^{\sharp*}$ that we may
approximate $\phi$ with c.c.p. maps
$\phi_{i}^{\prime}:{\mathbb{M}}_{n}({\mathbb{C}})\to E$ in the point-weak∗
topology. Since the weak∗ topology on $i_{E}(E)\subset E^{\sharp*}$ agrees
with the weak $(M_{0}\subset M)$-topology on $E$, the result then follows. ∎

######  Lemma 4.5.

Let $M$ be a von Neumann algebra and $M_{0}\subset M$ a weakly dense ${\rm
C}^{*}$-subalgebra. Let $E$ and $F$ be operator $M_{0}$-systems, and suppose
$\theta_{i}:E\to F$ are u.c.p. maps such that ${\theta_{i}}_{|M_{0}}$
converges to the identity in the point-weak $(M_{0}\subset M)$-topology, then
for each $a,b\in M_{0}$ and $x\in E$ we have
$\theta_{i}(axb)-a\theta_{i}(x)b\to 0$ in the weak $(M_{0}\subset
M)$-topology.

###### Proof.

Let $\theta:E\to(F^{{}_{M_{0}}\sharp_{M_{0}}})^{*}$ be any point-weak∗-limit
point of $\\{\theta_{i}\\}$. Since ${\theta_{i}}_{|M_{0}}$ converges to the
identity in the point-weak $(M_{0}\subset M)$-topology we have
$\theta_{|M_{0}}={\rm id}$, and since $\theta$ is u.c.p., we then have that
$\theta$ is $M_{0}$-bimodular. Thus, for all $x\in F$ and $a,b\in M_{0}$ we
have $\theta(axb)=a\theta(x)b$. Since $\theta$ was an arbitrary point-
weak∗-limit point, the result follows. ∎

The following theorem uses ideas as in the same spirit as in Section 2 of
[Kir95b], Theorem 4.7 in [EOR01], and Theorem 1 in [Oza07].

######  Theorem 4.6.

Let $M$ be a separable von Neumann algebra, $M_{0}\subset M$ a unital weakly
dense ${\rm C}^{*}$-subalgebra, and $E$ a normal $M_{0}$-system. The following
conditions are equivalent:

  1. (1)

The inclusion $M_{0}\subset E$ is $(M_{0}\subset M)$-nuclear.

  2. (2)

There exists an amenable von Neumann algebra $R$, and normal u.c.p. maps
$\Phi:M\to R$ and $\Psi:R\to(E^{M_{0}\sharp M_{0}})^{*}$ so that
$\Psi\circ\Phi(x)=i_{E}(x)$ for all $x\in M_{0}$.

###### Proof.

Since injectivity for a von Neumann algebra is equivalent to semi-
discreteness, the implication (2) $\implies$ (1) then follows from Lemma 4.4.
We now suppose that (1) holds. Since $M$ is separable, we may choose an
increasing sequence of finite-dimensional operator systems $\\{E_{n}\\}_{n}$
inside $M_{0}$ so that $M_{00}:=\cup E_{n}\subset M_{0}$ is ultraweakly dense.
By $(M_{0}\subset M)$-nuclearity of the inclusion map $j_{M_{0}}:M_{0}\to E$,
we have nets of u.c.p. maps
$\phi_{i}:M_{0}\to{\mathbb{M}}_{k(i)}({\mathbb{C}})$ and
$\psi_{i}:{\mathbb{M}}_{k(i)}({\mathbb{C}})\to E$ so that
$\psi_{i}\circ\phi_{i}\to j_{M_{0}}$ in the point-$(M_{0}\subset M)$-topology.
By Arveson’s Extension Theorem we may assume that each $\phi_{i}$ is defined
on $E$.

Denote by $\mu$ a normal faithful state on $M$ and $\mu_{0}$ the restriction
of $\mu$ to $M_{0}$. For each $n\geq 1$, we inductively choose u.c.p. maps
$\phi_{n}:E\to{\mathbb{M}}_{k(n)}({\mathbb{C}})$ and
$\psi_{n}:{\mathbb{M}}_{k(n)}({\mathbb{C}})\to E$ so that setting
$\theta_{n}=\psi_{n}\circ\phi_{n}$, we have

(2) $s_{\mu_{0}}(\theta_{n}(a)-a)<2^{-n+3}\|a\|,a\in E_{n},$

and

(3)
$s_{\mu_{0}}(\theta_{n}\circ\cdots\circ\theta_{n-j}(\theta_{n-j-1}(a)-a))<2^{-n-j}\|a\|,a\in
E_{n-j-1},0\leq j\leq n-2.$

Indeed, (2) follows directly from $(M_{0}\subset M)$-nuclearity. We now verify
(3) in the case of $j=0$, the general case follows similarly. Notice that
since $s_{\mu_{0}}(\theta_{n-1}(a)-a)<2^{-n+2}\|a\|$ for $a\in E_{n-1}$, we
may find $a_{1},a_{2}\in A$ and $b\in E$ such that $a=a_{1}^{*}ba_{2}$ and
$\mu(a_{1}^{*}a_{1})^{1/2}\|b\|\mu(a_{2}^{*}a_{2})^{1/2}<2^{-n+1}\|a\|$. Thus
by Lemma 4.5 and Proposition 3.3, we may, after passing to convex
combinations, choose $\theta_{n}$ that satisfies
$s_{\mu_{0}}(\theta_{n}(a_{1}^{*}ba_{2})-a_{1}^{*}\theta_{n}(b)a_{2})<2^{-n-1}\|a\|$
for all $a\in E_{n-1}$, and it follows that

$\displaystyle s_{\mu_{0}}(\theta_{n}(\theta_{n-1}(a)-a))$ $\displaystyle\leq
s_{\mu_{0}}(\theta_{n}(a_{1}^{*}ba_{2})-a_{1}^{*}\theta_{n}(b)a_{2})+s_{\mu_{0}}(a_{1}^{*}\theta_{n}(b)a_{2})$
$\displaystyle\leq
2^{-n-1}\|a\|+\mu(a_{1}^{*}a_{1})^{1/2}\|b\|\mu(a_{2}^{*}a_{2})^{1/2}\leq
2^{-n}\|a\|.$

Therefore, we have the following commutative diagram:

${M_{0}}$${E}$${E}$${\cdots}$${\mathbb{M}_{k(1)}(\mathbb{C})}$${\mathbb{M}_{k(2)}(\mathbb{C})}$${\mathbb{M}_{k(3)}(\mathbb{C})}$${\cdots}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\theta_{1}}$$\scriptstyle{\phi_{2}}$$\scriptstyle{\theta_{2}}$$\scriptstyle{\theta_{3}}$$\scriptstyle{\phi_{3}}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{\psi_{2}}$$\scriptstyle{\sigma_{2}}$$\scriptstyle{\sigma_{3}}$$\scriptstyle{\psi_{3}}$

Let $S$ be the operator system inductive limit of
$(\mathbb{M}_{k(n)}(\mathbb{C}),\sigma_{n})$ in the sense of [Kir95b, Section
2]. More precisely, if $\tilde{S}\subset\prod_{n\geq
1}{\mathbb{M}}_{k(n)}({\mathbb{C}})$ is the norm closure of

$\left\\{(x_{n})_{n}\in\prod_{n\geq
1}{\mathbb{M}}_{k(n)}({\mathbb{C}})\mid{\rm there\ exists\ }m\geq 1{\rm\ s.t.\
}x_{n+1}=\sigma_{n}(x_{n}){\rm\ for\ all\ }n\geq m\right\\},$

then $S=Q(\tilde{S})$, where

$Q:\prod_{n\geq 1}{\mathbb{M}}_{k(n)}({\mathbb{C}})\to\prod_{n\geq
1}{\mathbb{M}}_{k(n)}({\mathbb{C}})/\sum_{n\geq
1}{\mathbb{M}}_{k(n)}({\mathbb{C}})$

is the quotient map.

It is easy to see that $\tilde{S}$ is nuclear. As operator spaces, we may
identify $S^{**}=p^{\perp}\tilde{S}^{**}$ where $p$ is the central projection
in the bidual of $\prod_{n\geq 1}{\mathbb{M}}_{k(n)}({\mathbb{C}})$ given by
the support of the ideal $\sum_{n\geq 1}{\mathbb{M}}_{k(n)}({\mathbb{C}})$. We
therefore have that $S^{**}$ is unitally completely order isomorphic to an
amenable von Neumann algebra [EOR01, Theorem 4.5], [HP11, Theorem 3.5].

Define $\Phi:E\to S^{**}$ to be a point-weak∗ limit point of the sequence

$\phi_{m}:E\ni x\mapsto\phi_{m}(x)\in S\subset S^{**},$

where we view $\phi_{m}(x)\in{\mathbb{M}}_{k(m)}({\mathbb{C}})\to S$ is the
canonical map.

Let $\tilde{\Psi}:\tilde{S}\to(E^{M_{0}\sharp M_{0}})^{*}$ be the point-weak∗
limit point of $\tilde{\psi}_{m}:\tilde{S}\ni(x_{n})_{n}\to\psi_{m}(x_{m})\in
E\subset(E^{M_{0}\sharp M_{0}})^{*}$. It is not hard to see that
$\tilde{\Psi}=\Psi\circ Q$ for some u.c.p. map $\Psi:S\to(E^{M_{0}\sharp
M_{0}})^{*}$. We still denote by $\Psi$ the unique weak∗ continuous extension
$\Psi:S^{**}\to(E^{M_{0}\sharp M_{0}})^{*}$.

Then for any $a\in M_{00}$ with norm $1$ and $\varphi\in(E^{M_{0}\sharp
M_{0}})^{*}$, we have that

$\displaystyle|\varphi(a-\Psi\circ\Phi(a))|$
$\displaystyle=|\varphi(a-\lim_{m}\lim_{n}\tilde{\psi}_{n}\circ\phi_{m}(a))|$
$\displaystyle=\lim_{m}\lim_{n}|\varphi(a-(\theta_{n}\circ\cdots\theta_{m})(a))|$
$\displaystyle\leq\lim_{m}\lim_{n}\sum_{j=1}^{n-m}|\varphi((\theta_{n}\circ\cdots\circ\theta_{n-j+1})(a-\theta_{n-j}(a)))|+|\varphi(a-\theta_{n}(a))|.$

Since $\varphi\in E^{M_{0}\sharp M_{0}}$, there exists some $\kappa>0$ such
that $|\varphi(x)|\leq\kappa s_{\mu_{0}}(x)$ for $x\in(E)_{2}$. It then
follows that for $m$ and $n$ large enough, we have

$|\varphi((\theta_{n}\circ\cdots\circ\theta_{n-j+1})(a-\theta_{n-j}(a)))|\leq
2^{-n-j}\kappa,$

and hence

$|\varphi(a-\Psi\circ\Phi(a))|\leq\lim_{m}\lim_{n}\sum_{j=0}^{n-m}2^{-n-j}\kappa\leq\lim_{m}2^{-m+1}\kappa=0.$

From the above argument, we obtain a u.c.p map $\Phi_{M_{00}}:M_{00}\to
S^{**}$ and a weak∗-continuous u.c.p. map $\Psi:S^{**}\to(E^{M_{0}\sharp
M_{0}})^{*}$ such that $\Psi\circ\Phi_{\mid{M_{00}}}=i_{M_{00}}$, where
$i_{M_{00}}:M_{00}\to E^{\sharp*}$ is the canonical embedding.

Denote by $\bar{\Phi}:(M_{00})^{**}\to S^{**}$ the unique weak∗ continuous
extension of $\Phi_{\mid M_{00}}$, and realize $M=p(M_{00})^{**}$, where
$p\in(M_{00})^{**}$ is the support projection of the identity representation
from $M_{00}$ into $M$. Let $\\{e_{i}\\}_{i}\subset M_{00}$ be a net that
converges to $p$ in the weak∗ topology. Then for any linear functional
$\varphi\in E^{M_{0}\sharp M_{0}}$, we have
$\varphi(\Psi\circ\bar{\Phi}(p))=\varphi(\Psi(\lim_{i}\Phi(e_{i})))=\lim_{i}\varphi(e_{i})=\varphi(1)$,
and hence $\Psi\circ\bar{\Phi}(p)=1$. Thus, ${\bar{\Phi}_{|M}}:M\to S^{**}$ is
a normal u.c.p. map such that $\Psi\circ\bar{\Phi}(x)=i_{E}(x)$ for all $x\in
M_{0}$. ∎

To move beyond the separable case, we will use the following technical version
of the previous theorem. The proof is essentially the same and so we will
leave it to the reader.

######  Theorem 4.7.

Let $M$ be a von Neumann algebra, $M_{0}\subset M$ an ultraweakly dense ${\rm
C}^{*}$-subalgebra, and $E$ a normal $M_{0}$-system. If the inclusion
$M_{0}\subset E$ is $(M_{0}\subset M)$-nuclear, then for every countably
generated ${\rm C}^{*}$-subalgebra $M_{00}\subset M_{0}$, and $\sigma$-finite
projection $p\in\mathcal{P}(M)$, there exists an amenable von Neumann algebra
$R$, and normal c.c.p. maps $\Phi:M\to R$ and $\Psi:R\to(E^{M_{0}\sharp
M_{0}})^{*}$ so that $\Psi\circ\Phi(p)=p$ and $p\Psi\circ\Phi(x)=pi_{E}(x)p$
for all $x\in M_{0}$.

######  Proposition 4.8.

Let $M$ be a von Neumann algebra, $M_{0},M_{1}\subset M$ weakly dense ${\rm
C}^{*}$-subalgebras with $M_{0}\subset M_{1}$, and let $E$ be a normal
$M_{1}$-system. Consider the following conditions:

  1. (1)

The inclusion $M_{0}\subset E$ is $(M_{0}\subset M)$-nuclear.

  2. (2)

The inclusion $M_{1}\subset E$ is $(M_{1}\subset M)$-nuclear.

  3. (3)

The inclusion $M_{0}\subset E$ is $(M_{1}\subset M)$-nuclear.

Then the implications (1) $\implies$ (2) $\implies$ (3) hold.

Moreover, if $M_{0}$ is locally reflexive, then conditions (2) and (3) are
equivalent.

###### Proof.

If $M$ is separable, and if condition (1) holds, then by Theorem 4.6 the
canonical map $i_{0}:M\to(E^{M_{0}\sharp M_{0}})^{*}$ is weakly nuclear. Since
$M_{0}\subset M_{1}$, the identity map ${\rm id}:E\to E$ induces a normal
completely positive contraction $i_{1,0}:(E^{M_{0}\sharp
M_{0}})^{*}\to(E^{M_{1}\sharp M_{1}})^{*}$ so that the canonical embedding
$i_{1}:M\to(E^{M_{1}\sharp M_{1}})^{*}$ is given by the composition
$i_{1}=i_{1,0}\circ i_{0}$. Hence, the map $i_{1}$ is weakly nuclear and it
follows from Lemma 4.4 that condition (2) holds.

For the general case, fix $x_{1},\ldots,x_{n}\in M_{1}$, $\varepsilon>0$, and
$\eta_{1},\ldots,\eta_{m}\in E^{M_{1}\sharp M_{1}}\subset E^{M_{0}\sharp
M_{0}}$. Since each $\eta_{i}$ may be implemented by vector linear functional
in some representation of $E$ where $M$ is normally represented, we may use
the polarization identity to assume that each $\eta_{i}$ is a state. Let $p$
denote the supremum of the supports of ${\eta_{i}}_{|M}$, so that $p$ is a
$\sigma$-finite. Since $M_{0}$ is ultraweakly dense in $M$ and since $p$ is
$\sigma$-finite, we may choose for each $1\leq i\leq n$ a sequence $a_{i,k}\in
M_{0}$ so that $pa_{i,k}p\to px_{i}p$ in the weak operator topology.

We let $M_{00}$ denote the ${\rm C}^{*}$-algebra generated by
$\\{a_{i,k}\\}_{1\leq i\leq n,k\geq 1}$, and by Theorem 4.7 there exists an
amenable von Neumann algebra $R$, and normal c.c.p. maps $\Phi:M\to R$ and
$\Psi:R\to(E^{M_{0}\sharp M_{0}})^{*}$ so that $\Psi\circ\Phi(p)=p$ and
$p\Psi\circ\Phi(x)=pi_{E}(x)p$ for all $x\in M_{0}$. Since $p$ is in the
multiplicative domain of the normal c.c.p. map $\Psi\circ\Phi$, we then have,
after compressing down to $(E^{M_{1}\sharp M_{1}})^{*}$, a weak operator
topology limit

$p\Psi\circ\Phi(x_{i})p=p\Psi\circ\Phi(px_{i}p)p=\lim_{k\to\infty}p\Psi\circ\Phi(a_{i,k})p=px_{i}p\in(E^{M_{1}\sharp
M_{1}})^{*}.$

In particular, for each $1\leq i\leq n$ and $1\leq j\leq m$ we have
$\eta_{i}(\Psi\circ\Phi(x_{i})-x_{i})=\eta_{i}(p(\Psi\circ\Phi(x_{i})-x_{i})p)=0$.
Since $R$ is semi-discrete, there then exist c.c.p. maps
$\phi_{0}:R\to\mathbb{M}_{d}(\mathbb{C})$ and
$\psi_{0}:\mathbb{M}_{d}(\mathbb{C})\to R$ so that for each $1\leq i\leq n$
and $1\leq j\leq m$ we have
$\eta_{j}(\Psi\circ\psi_{0}\circ\phi_{0}\circ\Phi(x_{i})-x_{i})<\varepsilon$.
Hence $M_{1}\subset E$ is $(M_{1}\subset M)$-nuclear.

The implication (2) $\implies$ (3) is trivial. Since weak nuclearity of the
inclusion $i_{1}:M\to(E^{M_{1}\sharp M_{1}})^{*}$ is a local property and
since $i_{1}$ is normal, if $M_{0}$ is locally reflexive then we see that weak
nuclearity of the inclusion $M_{0}$ into $(E^{M_{1}\sharp M_{1}})^{*}$ implies
that $i_{1}$ is weakly nuclear and hence another application of Lemma 4.4
shows that (2) and (3) are equivalent in this case. ∎

The following special case of Proposition 4.8 will be used a number of times
in the sequel, and so we explicitly state it as a corollary.

######  Corollary 4.9.

Let $M$ be a von Neumann algebra with a dense ${\rm C}^{*}$-algebra
$M_{0}\subset M$. If $E$ is a normal $M$-system and if the inclusion
$M_{0}\subset E$ is $(M_{0}\subset M)$-nuclear, then the inclusion $M\subset
E$ is $M$-nuclear.

######  Example 4.10.

Let $A$ be a ${\rm C}^{*}$-algebra, and let $E$ be an $A$-system. Then the
weak $(A\subset A^{**})$-topology on $E$ is simply the weak topology on $E$,
and hence the inclusion $A\subset E$ is $(A\subset A^{**})$-nuclear if and
only if the inclusion $A\subset E$ is nuclear.

In particular, if we consider a universal representation
$\pi:A\to{\mathbb{B}}(\mathcal{H})$, then the inclusion
$\pi:A\to{\mathbb{B}}(\mathcal{H})$ is $(A\subset A^{**})$-nuclear if and only
if $A$ is exact. However, we will see from Proposition 5.5 below that $A^{**}$
is weakly exact if and only if the identity map from
$A^{**}=\pi(A)^{\prime\prime}$ to ${\mathbb{B}}(\mathcal{H})$ is
$A^{**}$-nuclear. Hence, the converse of Corollary 4.9 does not hold in
general (see, e.g., Remark 14.1.3 in [BO08]).

While the converse of Corollary 4.9 does not hold in general, we do have the
following partial converse, which should be compared with Corollary 3.1.6 in
[Iso13].

######  Proposition 4.11.

Let $M$ be a separable von Neumann algebra and $E$ a normal $M$-system. If
$M\subset E$ is $M$-nuclear, then for any separable ${\rm C}^{*}$-subalgebra
$M_{0}\subset M$ there exists a separable ${\rm C}^{*}$-subalgebra
$M_{1}\subset M$ containing $M_{0}$ such that $M_{1}\subset E$ is
$(M_{1}\subset M)$-nuclear.

###### Proof.

Choose $\\{a_{n}\\}_{n}\subset M_{0}$ a countable norm dense subset. Since
$M\subset E$ is $M$-nuclear, there exist sequences of u.c.p. maps
$\phi_{k}:M_{0}\to{\mathbb{M}}_{n(k)}({\mathbb{C}})$ and
$\psi_{k}:{\mathbb{M}}_{n(k)}({\mathbb{C}})\to E$ such that
$\psi_{k}\circ\phi_{k}(a_{n})-a_{n}=x_{n,k}T_{n,k}y_{n,k}$ with
$x_{n,k},y_{n,k}\in M_{+}$ and $T_{n,k}\in E$, such that
$\lim_{k}s_{\rho}^{\omega}(x_{n,k}T_{n,k}y_{n,k})=0$ for any $\rho,\omega\in
M_{*,+}$.

Set $A_{1}=C^{*}(M_{0},\\{x_{n,k},y_{n,k}\mid
n\in{\mathbb{N}},k\in{\mathbb{N}}\\})$ and note that $M_{0}\subset E$ is
$(A_{1}\subset M)$-nuclear, implemented by $\phi_{k}$ and $\psi_{k}$.
Repeating this process inductively, we then have a sequence of separable ${\rm
C}^{*}$-subalgebras $M_{0}\subset A_{1}\subset A_{2}\cdots$, such that
$A_{n}\subset E$ is $(A_{n+1}\subset M)$-nuclear. Let
$M_{1}=\overline{\cup_{n}A_{n}}^{\|\cdot\|}$ and one checks that $M_{1}\subset
E$ is $(M_{1}\subset M)$-nuclear. ∎

We end this section with a result showing the relationship between
$M$-nuclearity and topologically amenable actions. To describe this we recall
that if $E\subset{\mathbb{B}}(\mathcal{H})$ is an operator space and $I$ is a
set, then the operator space
$\mathbb{M}_{I}(E)\subset\mathcal{B}(\mathcal{H}\,\overline{\otimes}\,\ell^{2}I)$
is the operator system that we may identify as those $I\times I$ matrices with
entries in $E$ that correspond to bounded operators. This operator space is
independent of the representation $E\subset{\mathbb{B}}(\mathcal{H})$.
Moreover, if $\Gamma{\,\curvearrowright\,}^{\sigma}E$ by completely isometric
isomorphisms, then we may choose a covariant representation
$E\subset{\mathbb{B}}(\mathcal{H})$ so that the equation
$\sigma_{t}(x)=\pi_{t}x\pi_{t}^{*}$ holds for $x\in E$ and $t\in\Gamma$ for
some unitary representation $\pi$. Then conjugation by
$\pi_{t}\otimes\rho_{t}$ gives an action of $\Gamma$ on
$\mathbb{M}_{\Gamma}(E)$, and we let $\mathbb{M}_{\Gamma}(E)^{\Gamma}$ denote
the fixed points under this action. This is then a normal operator
$L\Gamma$-bimodule, and if $E$ is an operator system, then
$\mathbb{M}_{\Gamma}(E)^{\Gamma}$ will be a normal operator $L\Gamma$-system.
Note however that $\mathbb{M}_{\Gamma}(E)^{\Gamma}$ need not be a ${\rm
C}^{*}$-algebra, even when $E$ is a ${\rm C}^{*}$-algebra.

######  Theorem 4.12.

Let $\Gamma$ be a group and suppose $\Gamma{\,\curvearrowright\,}K$ is an
action by homeomorphisms on a compact Hausdorff space. The action
$\Gamma{\,\curvearrowright\,}K$ is topologically amenable if and only if the
inclusion $L\Gamma\subset\mathbb{M}_{\Gamma}(C(K))^{\Gamma}$ is
$L\Gamma$-nuclear.

###### Proof.

Note that by [BO08, Section 4.1] we have an operator space embedding
$C(K)\rtimes_{r}\Gamma\subset\mathbb{M}_{\Gamma}(C(K))^{\Gamma}$ that takes
$C^{*}_{\lambda}\Gamma$ canonically onto $C^{*}_{\lambda}\Gamma\subset
L\Gamma$ and takes $C(K)$ to diagonal matrices $C(K)\ni
f\mapsto\oplus_{t\in\Gamma}\sigma_{t^{-1}}(f)\in\mathbb{M}_{\Gamma}(C(K))^{\Gamma}$,
where $\sigma:\Gamma\to{\rm Aut}(C(K))$ is the corresponding action
$\sigma_{s}(f)=f\circ s^{-1}$. If the action $\Gamma{\,\curvearrowright\,}K$
is amenable, then the inclusion $C^{*}_{\lambda}\Gamma\subset
C(K)\rtimes_{r}\Gamma$ is nuclear [BO08, Section 4.3], and hence by Corollary
4.9 the inclusion $L\Gamma\subset\mathbb{M}_{\Gamma}(C(K))^{\Gamma}$ is
$L\Gamma$-nuclear.

For the converse, note that we have a $\Gamma$-equivariant conditional
expectation onto the diagonal matrices
$E_{\ell^{\infty}\Gamma}:\mathbb{M}_{\Gamma}(C(K))^{\Gamma}\to\ell^{\infty}(\Gamma;C(K))^{\Gamma}$
where the later space we may identify, in a $\Gamma$-equivariant way, with
$C(K)$ by evaluating at the identity. Also note that $E_{\ell^{\infty}\Gamma}$
restricts to the trace on $L\Gamma$, and so if $a,b\in L\Gamma$,
$T\in\mathbb{M}_{\Gamma}(C(K))^{\Gamma}$, and $\varphi\in C(K)^{*}$ is a
state, then we have $|\varphi\circ
E_{\ell^{\infty}\Gamma}(a^{*}Tb)|\leq\|T\|(\varphi\circ
E_{\ell^{\infty}\Gamma}(a^{*}a))^{1/2}(\varphi\circ
E_{\ell^{\infty}\Gamma}(b^{*}b))^{1/2}=\|T\|\tau(a^{*}a)^{1/2}\tau(b^{*}b)^{1/2}$.
It therefore follows that
$E_{\ell^{\infty}\Gamma}:\mathbb{M}_{\Gamma}(C(K))^{\Gamma}\to C(K)$ is
continuous from the $L\Gamma$-topology into the norm topology.

If $L\Gamma\subset\mathbb{M}_{\Gamma}(C(K))^{\Gamma}$ is $L\Gamma$-nuclear,
$F\subset\Gamma$ is finite, and $\varepsilon>0$, then using Lemma 4.1 and the
observations above there exists a finite subset $E\subset\Gamma$ and a u.c.p.
map $\psi:\mathbb{M}_{E}(\mathbb{C})\to\mathbb{M}_{\Gamma}(C(K))^{\Gamma}$ so
that setting $h(t)=E_{\ell^{\infty}\Gamma}(\psi\circ{\rm
Ad}(P_{\ell^{2}E})(\lambda_{t})\lambda_{t}^{*})\in C(K)$ we have
$\|1-h(t)\|_{\infty}<\varepsilon$ for each $t\in F$. The function $\Gamma\ni
t\mapsto h(t)\in C(K)$ is easily seen to be of positive type (see the
implication (4) $\implies$ (2) in [BO08, Theorem 4.4.3]), and hence it follows
that the action $\Gamma{\,\curvearrowright\,}K$ is amenable. ∎

##  5\. Weakly exact von Neumann algebras

Recall from [Iso13, Definition 3.1.1] that for a von Neumann algebra $M$ and
an ultraweakly dense ${\rm C}^{*}$-algebra $M_{0}\subset M$, the ${\rm
C}^{*}$-algebra $M_{0}$ is said to be weakly exact in $M$ if for any unital
${\rm C}^{*}$-algebra $B$ with a closed two-sided ideal $J$ and any
representation $\pi:M_{0}\otimes B\to{\mathbb{B}}({\mathcal{K}})$ with
$M_{0}\otimes J\subset\ker\pi$ that is ultraweakly continuous on
$M\otimes{\mathbb{C}}$, the induced representation $\tilde{\pi}:M_{0}\odot
B/J\to\mathbb{B}({\mathcal{H}})$ is min-continuous. This notion generalizes
weak exactness for von Neumann algebras and exactness for ${\rm
C}^{*}$-algebras simultaneously: if $M_{0}=M$ is a von Neumann algebra, then
$M$ is weakly exact in $M$ if and only if $M$ is weakly exact in the sense of
Kirchberg ([Kir95a], see also [BO08, Chapter 14]); at the other extreme, if
$M=(M_{0})^{**}$, then $M_{0}$ is weakly exact in $(M_{0})^{**}$ if and only
if $M_{0}$ is exact.

In this section, we characterize weak exactness via $(M_{0}\subset
M)$-nuclearity, analogous to the case of exactness for ${\rm C}^{*}$-algebras.

######  Theorem 5.1.

Let $M\subset{\mathbb{B}}(\mathcal{H})$ be a von Neumann algebra and
$M_{0}\subset M$ an ultraweakly dense ${\rm C}^{*}$-subalgebra. The following
conditions are equivalent.

  1. (1)

The ${\rm C}^{*}$-algebra $M_{0}$ is weakly exact in $M$.

  2. (2)

There exist nets of c.c. maps
$\phi_{i}:M_{0}\to{\mathbb{M}}_{n(i)}({\mathbb{C}})$ and
$\psi_{i}:{\mathbb{M}}_{n(i)}({\mathbb{C}})\to{\mathbb{B}}({\mathcal{H}})$
such that $\psi_{i}\circ\phi_{i}(a)\to a$ in the $(M_{0}\subset M)$-topology
for any $a\in M_{0}$.

  3. (3)

The inclusion $M_{0}\subset{\mathbb{B}}({\mathcal{H}})$ is $(M_{0}\subset
M)$-nuclear, i.e., there exist nets of c.c.p. maps
$\phi_{i}:M_{0}\to{\mathbb{M}}_{n(i)}({\mathbb{C}})$ and
$\psi_{i}:{\mathbb{M}}_{n(i)}({\mathbb{C}})\to{\mathbb{B}}({\mathcal{H}})$
such that $\psi_{i}\circ\phi_{i}(a)\to a$ in the $(M_{0}\subset M)$-topology
for any $a\in M_{0}$.

Before proceeding to the proof, we collect a few lemmas.

######  Lemma 5.2.

Let $M$ be a von Neumann algebra and $M_{0}\subset M$ an ultraweakly dense
${\rm C}^{*}$-subalgebra. Then $M_{0}$ is weakly exact in $M$ if and only if
for any closed two-sided ideal $J\lhd B$ and any $\pi:M_{0}\otimes
B\to{\mathbb{B}}({\mathcal{K}})$ such that $\pi_{\mid
M_{0}\otimes{\mathbb{C}}}$ is ultraweakly continuous and $M_{0}\otimes
J\subset\ker\pi$, we have $\ker q\subset\ker\pi$, where $q$ is the quotient
map $q:M_{0}\otimes B\to M_{0}\otimes B/J$.

###### Proof.

It is clear that $\ker q\subset\ker\pi$ implies that $\tilde{\pi}:A\odot
B/J\to{\mathbb{B}}({\mathcal{K}})$ is min-continuous. Conversely, suppose
there is some $x\in\ker q\setminus\ker\pi$. Then, we may find a sequence
$x_{n}\in A\odot B$ such that $x_{n}\to x$ in norm. Note that $\|q(x_{n})\|\to
0$ while $\liminf_{n}\|\pi(x_{n})\|\geq\delta$ for some $\delta>0$. However,
since $x_{n}\in A\odot B$ and $A\otimes J\subset\ker\pi$, we have
$\tilde{\pi}(q(x_{n}))=\pi(x_{n})$, which is a contradiction. ∎

######  Lemma 5.3.

Let $M$ be a von Neumann algebra and $M_{0}\subset M$ an ultraweakly dense
${\rm C}^{*}$-subalgebra with $1\not\in M_{0}$. Let
$\tilde{M}_{0}=M_{0}+{\mathbb{C}}1\subset M$. If $M_{0}$ is weakly exact in
$M$, then $\tilde{M}_{0}$ is also weakly exact in $M$.

###### Proof.

Let $\pi:\tilde{M}_{0}\otimes B\to{\mathbb{B}}({\mathcal{K}})$ and $J\lhd B$
be given as in the definition of weak exactness. Denote by
$q:\tilde{M}_{0}\otimes B\to\tilde{M}_{0}\otimes B/J$ the quotient map. By
Lemma 5.2, it suffices to show that $\ker q\subset\ker\pi$. For $x\in\ker q$,
consider the following diagram:

${0}$${0}$${0}$${0}$${M_{0}\otimes J}$${M_{0}\otimes B}$${M_{0}\otimes
B/J}$${0}$${0}$${\tilde{M}_{0}\otimes J}$${\tilde{M}_{0}\otimes
B}$${\tilde{M}_{0}\otimes B/J}$${0}$${0}$${{\mathbb{C}}\otimes
J}$${{\mathbb{C}}\otimes B}$${{\mathbb{C}}\otimes B/J}$${0}$${0}$${0}$${0.}$

The $3\times 3$ lemma shows that we may find $y\in\tilde{M}_{0}\otimes J$ such
that $x-y\in\ker q_{\mid M_{0}\otimes B}$. Since
$M_{0}\subset{\mathbb{B}}({\mathcal{H}})$ is weakly exact, we have $\ker
q_{\mid M_{0}\otimes B}\subset\ker\pi_{\mid M_{0}\otimes B}$, and it follows
that $x\in\ker\pi$. ∎

######  Proposition 5.4.

Let $M\subset{\mathbb{B}}(\mathcal{H})$ be a von Neumann algebra and
$M_{0}\subset M$ an ultraweakly dense ${\rm C}^{*}$-subalgebra. Then, $M_{0}$
is weakly exact in $M$ if and only if there exist nets of c.c. maps
$\phi_{i}:M_{0}\to{\mathbb{M}}_{n(i)}({\mathbb{C}})$ and
$\psi_{i}:{\mathbb{M}}_{n(i)}({\mathbb{C}})\to{\mathbb{B}}({\mathcal{H}})$
such that $\psi_{i}\circ\phi_{i}(a)\to a$ in the $(M_{0}\subset M)$-topology
for any $a\in M_{0}$.

###### Proof.

The backward direction is a direct consequence of [Iso13, Theorem 3.1.3, (1),
(v)], Wittstock’s extension theorem, and the fact that the restriction of the
weak $(M_{0}\subset M)$-topology on $M_{0}$ coincides with the ultraweak
topology on $M_{0}$ endowed from $M$. By a standard convexity argument we
obtain that the convergence is in the $(M_{0}\subset M)$-topology instead of
the weak $(M_{0}\subset M)$-topology.

For the forward direction, we follow the idea of [BO08, Proposition 3.7.8].
Let $\pi$ and $J\lhd B$ be given as in the definition. Denote by
$q:M_{0}\otimes B\to M_{0}\otimes B/J$ the quotient map, and it suffices to
show $\ker q\subset\ker\pi$.

For any contraction $x\in\ker q$, consider the following diagram:

${0}$${M_{0}\otimes J}$${M_{0}\otimes B}$${M_{0}\otimes
B/J}$${0}$${0}$${D_{i}\otimes J}$${D_{i}\otimes B}$${D_{i}\otimes
B/J}$${0}$${0}$${{\mathbb{B}}({\mathcal{H}})\otimes
J}$${{\mathbb{B}}({\mathcal{H}})\otimes
B}$${{\mathbb{B}}({\mathcal{H}})\otimes
B/J}$${0,}$$\scriptstyle{\phi_{i}\otimes\operatorname{id}}$$\scriptstyle{\phi_{i}\otimes\operatorname{id}}$$\scriptstyle{\phi_{i}\otimes\operatorname{id}}$$\scriptstyle{\psi_{i}\otimes\operatorname{id}}$$\scriptstyle{\psi_{i}\otimes\operatorname{id}}$$\scriptstyle{\psi_{i}\otimes\operatorname{id}}$

where $D_{i}={\mathbb{M}}_{n(i)}({\mathbb{C}})$. Since $D_{i}$ is exact, we
have
$x_{i}:=\big{(}(\psi_{i}\circ\phi_{i})\otimes\operatorname{id}\big{)}(x)\in{\mathbb{B}}({\mathcal{H}})\otimes
J$.

Let $\\{e_{n}\\}\in J$ be an approximate unit. Fix a unit vector
$\xi\in{\mathcal{H}}$ and $\varepsilon>0$. By Lemma 3.4, we may find
contractions $z_{i}\in(M_{0})_{+}$ such that $\|(z_{i}\otimes
1)(x_{i}-x)(z_{i}\otimes 1)\|\to 0$ and $\langle\pi((1-z_{i})\otimes
1)\xi,\xi\rangle\to 0$. We may then choose $e_{n}$ and $z_{j}$ such that

$\|(z_{j}\otimes 1)(x_{j}-x)(z_{j}\otimes 1)\|<\varepsilon/8,\ \ \ \ \
\langle\pi((1-z_{j})\otimes 1)\xi,\xi\rangle<\varepsilon/8$

and

$\|(1\otimes 1-1\otimes e_{n})((z_{j}\otimes 1)x_{j}(z_{j}\otimes
1))\|<\varepsilon/8.$

Note that $(1\otimes e_{n})(z_{j}\otimes 1)x(z_{j}\otimes 1)\in M_{0}\otimes
J$ and

$\|(1\otimes e_{n})(z_{j}\otimes 1)x(z_{j}\otimes 1)-(z_{j}\otimes
1)x(z_{j}\otimes 1)\|<\varepsilon/2.$

It follows that

$|\langle\pi(x)\xi,\xi\rangle|=|\langle\pi(((1-z_{j})\otimes 1)x+(z_{j}\otimes
1)x((1-z_{j})\otimes 1)+(z_{j}\otimes 1)x(z_{j}\otimes
1))\xi,\xi\rangle|\leq\varepsilon.$

Since $\xi$ and $\varepsilon$ are both arbitrary, by polarization we conclude
$x\in\ker\pi$. ∎

We next argue that in the unital case, one may take $\phi_{i}$ and $\psi_{i}$
in the above proposition to be u.c.p. maps.

######  Proposition 5.5.

Let $M\subset{\mathbb{B}}(\mathcal{H})$ be a von Neumann algebra and
$M_{0}\subset M$ an ultraweakly dense ${\rm C}^{*}$-subalgebra with $1\in
M_{0}$. Then, $M_{0}$ is weakly exact in $M$ if and only if the inclusion
$M_{0}\subset{\mathbb{B}}({\mathcal{H}})$ is $(M_{0}\subset M)$-nuclear.

###### Proof.

The forward direction follows from Proposition 5.4.

To see the converse, take c.c. maps $\psi_{i}$ and $\phi_{i}$ as in
Proposition 5.4. First notice that by Lemma 4.1 we may assume
$\phi_{i}=\operatorname{Ad}(P_{{\mathcal{H}}_{i}})$, where
$P_{{\mathcal{H}}_{i}}$ is the orthogonal projection onto a finite-dimensional
subspace ${\mathcal{H}}_{i}\subset{\mathcal{H}}$, by replacing
$\\{\psi_{i}\\}$ with another net of complete contractions. One then checks
that $\psi_{i}\circ\phi_{i}\to\operatorname{id}_{M_{0}}$ in the
point-$(M_{0}\subset M)$-topology after the replacement.

For any finite set of states $F\subset M_{0}^{\sharp}$, since
$\psi_{i}\circ\phi_{i}(1)=\psi_{i}(1_{{\mathbb{M}}_{n(i)}({\mathbb{C}})})\to
1$ in the $(M_{0}\subset M)$-topology, by Lemma 3.6 we obtain a net of u.c.p.
maps
$\psi_{i,F}:{\mathbb{M}}_{n(i)}({\mathbb{C}})\to{\mathbb{B}}({\mathcal{H}})$
such that for any $a\in M_{0}$,
$\lim_{i}s_{\omega}^{\rho}(\psi_{i,F}\circ\phi_{i}(a)-a)\leq\lim_{i}s_{\omega}^{\rho}\big{(}(\psi_{i}-\psi_{i,F})(\phi_{i}(a))\big{)}+\lim_{i}s_{\omega}^{\rho}(\psi_{i}\circ\phi_{i}(a)-a)=0$,
for any $\omega,\rho\in F$. By ordering the collection of finite subsets of
states in $M_{0}^{\sharp}$ by inclusion, we may then produce nets of u.c.p.
maps that show the inclusion $M_{0}\subset{\mathbb{B}}(\mathcal{H})$ is
$(M_{0}\subset M)$-nuclear. ∎

###### Proof of Theorem 5.1.

Clearly (3)$\implies$(2), and (2)$\implies$(1) by Proposition 5.4.

To see (1)$\implies$(3), note that by Lemma 5.3 we have
$\tilde{M}_{0}=M_{0}+{\mathbb{C}}1$ is weakly exact in $M$ and hence
Proposition 5.5 shows the inclusion
$\tilde{M}_{0}\subset{\mathbb{B}}(\mathcal{H})$ is $(\tilde{M}_{0}\subset
M)$-nuclear.

By Lemma 4.3 there then exist nets of u.c.p. maps
$\phi_{i}:\tilde{M}_{0}\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to{\mathbb{B}}(\mathcal{H})$ so that
for each $x\in M_{0}$ there exist $T_{i}\in{\mathbb{B}}(\mathcal{H})$ and
$a_{i}\in\tilde{M}_{0}$ with $\psi_{i}\circ\phi_{i}(x)-x=T_{i}-a_{i}$ where
$\|T_{i}\|\to 0$, and $\\{a_{i}\\}_{i}$ is uniformly bounded with $a_{i}\to 0$
ultrastrongly.

We fix an approximate unit $\\{e_{n}\\}_{n}\subset M_{0}$ and notice that for
a fixed $i$ and $x$ we have
$\operatorname{Ad}(e_{n})\circ\psi_{i}\circ\phi_{i}(x)-x$ is asymptotically
close in the uniform norm to $e_{n}T_{i}e_{n}-e_{n}a_{i}e_{n}$. Since we have
an ultrastrong limit $\lim_{n\to\infty}e_{n}a_{i}e_{n}=a_{i}$ and since
$\operatorname{Ad}(e_{n})\circ\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to M_{0}$
are complete contractions, this then shows that
$M_{0}\subset{\mathbb{B}}(\mathcal{H})$ is $(M_{0}\subset M)$-nuclear. ∎

In particular, we obtain the following characterization of weakly exact von
Neumann algebras.

######  Corollary 5.6.

Let $M\subset{\mathbb{B}}(\mathcal{H})$ be a von Neumann algebra. Then $M$ is
weakly exact if and only if the inclusion $M\subset{\mathbb{B}}(\mathcal{H})$
is $M$-nuclear.

######  Corollary 5.7.

Let $M$ be a von Neumann algebra and $M_{0}\subset M$ a weakly dense ${\rm
C}^{*}$-subalgebra such that $M_{0}$ is weakly exact in $M$. Suppose $E$ is a
dual $M_{0}$-system. The inclusion $M_{0}\subset E$ is $(M_{0}\subset
M)$-nuclear if and only if the inclusion $M_{0}\subset E$ is weakly nuclear.

###### Proof.

Suppose $M_{0}\subset E$ is weakly nuclear, then let
$\phi_{i}:M_{0}\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to E$ denote the c.c.p. maps realizing
weak nuclearity for the inclusion. We extend $\phi_{i}$ to c.c.p. maps
$\tilde{\phi}_{i}:{\mathbb{B}}(L^{2}M)\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
let $\Phi:{\mathbb{B}}(L^{2}M)\to E$ denote a point-ultraweak limit of the
maps $\psi_{i}\circ\tilde{\phi}_{i}$. Then $\Phi$ is $M_{0}$-bimodular, and
hence is continuous with respect to the $(M_{0}\subset M)$-topologies. Since
$M_{0}$ is weakly exact in $M$, we have that the inclusion
$M_{0}\subset{\mathbb{B}}(L^{2}M)$ is $(M_{0}\subset M)$-nuclear, and
composing the corresponding maps with $\Phi$ shows then that the inclusion
$M_{0}\subset E$ is $(M_{0}\subset M)$-nuclear. ∎

The following result answers Problem 10.4.3 from [BO08].

######  Theorem 5.8.

Let $M$ be a von Neumann algebra, then $M$ is weakly exact if and only if for
every normal faithful representation $M\subset{\mathbb{B}}(\mathcal{H})$, and
for any intermediate von Neumann algebra $M\subset
N\subset{\mathbb{B}}(\mathcal{H})$ such that there exists an $M$-bimodular
u.c.p. map $\phi:{\mathbb{B}}(\mathcal{H})\to N$, then the inclusion $M\subset
N$ is weakly nuclear.

###### Proof.

Suppose $M$ is faithfully and normally represented on some Hilbert space
$\mathcal{K}$. We represent $({\mathbb{B}}(\mathcal{K}))^{\sharp*}$ faithfully
and normally on a Hilbert space $\mathcal{H}$, and notice that since
${\mathbb{B}}(\mathcal{K})$ is injective (as an operator system) and we have
an $M$-system embedding of ${\mathbb{B}}(\mathcal{K})$ into
${\mathbb{B}}(\mathcal{K})^{\sharp*}$, there then exists an $M$-bimodular
u.c.p. map from ${\mathbb{B}}(\mathcal{H})$ into
${\mathbb{B}}(\mathcal{K})\subset{\mathbb{B}}(\mathcal{K})^{\sharp*}$. From
Corollary 5.6 and Lemma 4.4 we see that if $M$ is not weakly exact, then the
inclusion of von Neumann algebras
$M\subset{\mathbb{B}}(\mathcal{K})^{\sharp*}$ is not weakly nuclear.

The converse is Exercise 14.1.4 in [BO08]. Or, just notice that when $M$ is
weakly exact we have that $M\subset{\mathbb{B}}(\mathcal{H})$ is $M$-nuclear
by Corollary 5.6, and since $M$-bimodular u.c.p. maps are continuous in the
$M$-topology, if we are given an $M$-bimodular u.c.p. map
$\phi:{\mathbb{B}}(\mathcal{H})\to N$, it follows that the inclusion $M\subset
N$ is $M$-nuclear and hence is also weakly nuclear. ∎

###  5.1. Weak exactness and free products

######  Lemma 5.9.

Let $M\subset{\mathbb{B}}(\mathcal{H})$ be a von Neumann algebra and let
$M_{0}\subset M$ be an ultraweakly dense ${\rm C}^{*}$-subalgebra. Suppose
$E\subset{\mathbb{B}}(\mathcal{H})$ is a normal $M_{0}$-system such that
${\mathbb{K}}(\mathcal{H})\subset E$. If $M_{0}\subset E$ is $(M_{0}\subset
M)$-nuclear, then for each vector state
$\varphi(\cdot)=\langle\cdot\xi,\xi\rangle$ there exist nets of u.c.p. maps
$\phi_{i}:M_{0}\to\mathbb{M}_{n(i)}(\mathbb{C})$ and
$\psi_{i}:\mathbb{M}_{n(i)}(\mathbb{C})\to E$, and pure states $\mu_{i}$ on
$\mathbb{M}_{n(i)}(\mathbb{C})$ such that $\mu_{i}\circ\phi_{i}=\varphi$ and
$\varphi\circ\psi_{i}=\mu_{i}$, and for each $x\in M_{0}$ we have
$\psi_{i}\circ\phi_{i}(x)-x=T_{i}+a_{i}$, where $\|T_{i}\|\to 0$, and
$a_{i}\in M_{0}$ is uniformly bounded with $\varphi(a_{i}^{*}a_{i})\to 0$.

###### Proof.

Let $\mathcal{H}_{1}=\mathcal{H}\,\overline{\otimes}\,\ell^{2}\mathbb{N}$ and
let
$E_{1}=E\otimes\mathbb{C}+{\mathbb{K}}(\mathcal{H}_{1})\subset{\mathbb{B}}(\mathcal{H}_{1})$.
Then, the inclusion map $M_{0}=M_{0}\otimes\mathbb{C}\subset E_{1}$ is also
$(M_{0}\subset M)$-nuclear and we also have
$M_{0}\cap{\mathbb{K}}(\mathcal{H}_{1})=\\{0\\}$.

Note that $E_{1}$ is a ${\mathbb{K}}(\mathcal{H}_{1})$-system, and we have
$(E_{1}^{M_{0}\sharp M_{0}})^{*}=({\mathbb{K}}(\mathcal{H}_{1})^{M_{0}\sharp
M_{0}})^{*}\oplus q_{{\mathbb{K}}(\mathcal{H}_{1})}^{\perp}(E_{1}^{M_{0}\sharp
M_{0}})^{*}$ where $q_{{\mathbb{K}}(\mathcal{H}_{1})}\in E_{1}^{**}$ is the
support projection for ${\mathbb{K}}(\mathcal{H}_{1})$. Since $M_{0}\subset
E_{1}$ is $(M_{0}\subset M)$-nuclear, we have that the inclusion map from
$M_{0}$ into $(E_{1}^{M_{0}\sharp M_{0}})^{*}$ is weakly nuclear. Since
$({\mathbb{K}}(\mathcal{H}_{1})^{M_{0}\sharp
M_{0}})^{*}={\mathbb{K}}(\mathcal{H}_{1})^{**}\cong{\mathbb{B}}(\mathcal{H}_{1})$
and since $M_{0}\cap{\mathbb{K}}(\mathcal{H}_{1})=\\{0\\}$, it then follows
that the inclusion map from $M_{0}+{\mathbb{K}}(\mathcal{H}_{1})$ into
$(E_{1}^{M_{0}\sharp M_{0}})^{*}$ is also weakly nuclear, and hence the
inclusion $M_{0}+{\mathbb{K}}(\mathcal{H}_{1})\subset E_{1}$ is $(M_{0}\subset
M)$-nuclear.

By Lemmas 4.1 and 4.3, there exist finite-dimensional subspaces
$\mathcal{K}_{i}\subset\mathcal{H}_{1}$, and u.c.p. maps
$\psi_{i}:{\mathbb{B}}(\mathcal{K}_{i})\to E_{1}$ so that setting
$\phi_{i}(x)=P_{\mathcal{K}_{i}}xP_{\mathcal{K}_{i}}$ we have that
$\psi_{i}\circ\phi_{i}$ converges to the identity in the point-$(M_{0}\subset
M)$-topology on $M_{0}+{\mathbb{K}}(\mathcal{H}_{1})$, and such that for each
$x\in M_{0}$ we have $\psi_{i}\circ\phi_{i}(x)-x=T_{i}+a_{i}$, where
$\|T_{i}\|\to 0$, and $a_{i}\in M_{0}$ is uniformly bounded with
$\varphi(a_{i}^{*}a_{i})\to 0$. If we fix $\xi\in\mathcal{H}$, then we may
assume that $\xi_{1}:=\xi\otimes\delta_{1}\in\mathcal{K}_{i}$ and we define a
pure state $\mu_{i}$ on ${\mathbb{B}}(\mathcal{K}_{i})$ to be the vector state
associated to $\xi_{1}$. Note that we have $\mu_{i}\circ\phi_{i}=\varphi$.

We let $P_{\xi_{1}}$ denote the rank-one projection onto $\mathbb{C}\xi_{1}$,
and note that we have $\psi_{i}(P_{\xi_{1}})-P_{\xi_{1}}\to 0$ in the
$(M_{0}\subset M)$-topology, and hence as in the proof of Lemma 3.4 there
exists a net of positive contractions $z_{i}\in M_{0}$ so that
$\|z_{i}(\psi_{i}(P_{\xi_{1}})-P_{\xi_{1}})z_{i}\|\to 0$ and $z_{i}\to 1$
ultrastrongly. Since $z_{i}\to 1$ ultrastrongly, we have that
$\|z_{i}P_{\xi_{1}}z_{i}-P_{\xi_{1}}\|\to 0$, and hence if we fix a state
$\eta\in(E_{1})^{\sharp}$ such that $\eta$ vanishes on
${\mathbb{K}}(\mathcal{H}_{1})$ and define
$\psi_{i}^{\prime}(T)=z_{i}\psi_{i}(T)z_{i}+(1-z_{i}^{2})\eta(T)$, then we
have $\|\psi_{i}^{\prime}(P_{\xi_{1}})-P_{\xi_{1}}\|\to 0$, and
$\psi_{i}^{\prime}\circ\phi_{i}$ still satisfies the pointwise convergence
described above. Since $P_{\xi_{1}}$ is a rank-one projection, a standard
additional perturbation in norm gives u.c.p. maps
$\psi_{i}^{\prime\prime}:\mathbb{M}_{n(i)}(\mathbb{C})\to E_{1}$ so that
$\psi_{i}^{\prime\prime}(P_{\xi_{1}})=P_{\xi_{1}}$. (Note that $E_{1}$ is an
operator ${\mathbb{K}}(\mathcal{H}_{1})$-system, so such a perturbation still
maps into $E_{1}$).

Since $P_{\xi_{1}}$ is then in the multiplicative domain of
$\psi_{i}^{\prime\prime}$, we have that
$\varphi\circ\psi_{i}^{\prime\prime}(T)=P_{\xi_{1}}\psi_{i}^{\prime\prime}(T)P_{\xi_{1}}=\psi_{i}^{\prime\prime}(P_{\xi_{1}}TP_{\xi_{1}})=\mu_{i}(T)$
for all $T\in\mathbb{M}_{n(i)}(\mathbb{C})$. We may then compose
$\psi_{i}^{\prime\prime}$ with the compression map given by ${\rm
id}\otimes{\rm Ad}(P_{\delta_{1}}):E_{1}\to E$ (which is $M$-bimodular) to
finish the proof. ∎

In order to set notation, we briefly recall the free product construction. We
refer the reader to [VDN92] for a more detailed discussion. If
$(\mathcal{H}_{i},\xi_{i})$ are Hilbert spaces with distinguished unit
vectors, for $i=1,2$, then we set
$\mathcal{H}_{i}^{o}=\mathcal{H}_{i}\ominus\mathbb{C}\xi_{i}$. The Hilbert
space free product is $(\mathcal{H},\xi)$ given by

$\mathcal{H}=\mathbb{C}\xi\oplus\bigoplus_{n\geq
1}\left(\bigoplus_{\iota_{1}\not=\iota_{2}\not=\cdots\not=\iota_{n}}\mathcal{H}_{\iota_{1}}^{o}\,\overline{\otimes}\,\cdots\,\overline{\otimes}\,\mathcal{H}_{\iota_{n}}^{o}\right).$

We also consider

$\mathcal{H}(i)=\mathbb{C}\xi\oplus\bigoplus_{n\geq
1}\left(\bigoplus_{\iota_{1}\not=\iota_{2}\not=\cdots\not=\iota_{n}\atop\iota_{1}\not=i}\mathcal{H}_{\iota_{1}}^{o}\,\overline{\otimes}\,\cdots\,\overline{\otimes}\,\mathcal{H}_{\iota_{n}}^{o}\right),$

and note that we have a canonical isomorphism
$\mathcal{H}\cong\mathcal{H}_{i}\,\overline{\otimes}\,\mathcal{H}(i)$. Via
this isomorphism we obtain a representation
$\lambda:{\mathbb{B}}(\mathcal{H}_{i})\to{\mathbb{B}}(\mathcal{H})$. We may
similarly consider the Hilbert space

$\mathcal{H}(r,i)=\mathbb{C}\xi\oplus\bigoplus_{n\geq
1}\left(\bigoplus_{\iota_{1}\not=\iota_{2}\not=\cdots\not=\iota_{n}\atop\iota_{n}\not=i}\mathcal{H}_{\iota_{1}}^{o}\,\overline{\otimes}\,\cdots\,\overline{\otimes}\,\mathcal{H}_{\iota_{n}}^{o}\right),$

and note that we again have a canonical isomorphism
$\mathcal{H}\cong\mathcal{H}(r,i)\,\overline{\otimes}\,\mathcal{H}_{i}$, so
that we obtain another representation
$\rho:{\mathbb{B}}(\mathcal{H}_{i})\to{\mathbb{B}}(\mathcal{H})$.

We let $\omega_{i}$ denote the vector state on ${\mathbb{B}}(\mathcal{H}_{i})$
given by $\xi_{i}$. If $A_{i}\subset{\mathbb{B}}(\mathcal{H}_{i})$ are ${\rm
C}^{*}$-algebras, then the reduced free product ${\rm C}^{*}$-algebra
$(A_{1},\omega_{1})*_{r}(A_{2},\omega_{2})$ is the ${\rm C}^{*}$-subalgebra of
${\mathbb{B}}(\mathcal{H})$ generated by $\lambda_{1}(A_{1})$ and
$\lambda_{2}(A_{2})$. If $M_{i}\subset{\mathbb{B}}(\mathcal{H}_{i})$ are von
Neumann algebras, then the free product von Neumann algebra
$(M_{1},\omega_{1})*(M_{2},\omega_{2})$ is the von Neumann algebra generate by
$\lambda(M_{1})$ and $\lambda(M_{2})$. The representations $\lambda_{i}$ give
rise to a representation of the unital algebraic free product
${\mathbb{B}}(\mathcal{H}_{1})*_{\mathbb{C}}{\mathbb{B}}(\mathcal{H}_{1})$ and
if $E_{i}\subset{\mathbb{B}}(\mathcal{H}_{i})$ are operator systems, then we
denote by $(E_{1},\omega_{1})*_{r}(E_{2},\omega_{2})$ the operator subsystem
of
$({\mathbb{B}}(\mathcal{H}_{1}),\omega_{1})*_{r}({\mathbb{B}}(\mathcal{H}_{2}),\omega_{2})$
generated as the closed span of applying the representation $\lambda$ to all
reduced words that appear in $E_{1}$ and $E_{2}$.

If $A_{i},B_{i}\subset{\mathbb{B}}(\mathcal{H}_{i})$ are ${\rm
C}^{*}$-subalgebras, and we have u.c.p. maps $\phi_{i}:A_{i}\to B_{i}$ that
preserve the state $\omega_{i}$, then there is a unique u.c.p. map
$(\phi_{1}*\phi_{2}):A_{1}*_{r}A_{2}\to B_{1}*_{r}B_{2}$ such that
$(\phi_{1}*\phi_{2})(a_{1}b_{1}a_{2}\cdots
a_{n}b_{n})=\phi_{1}(a_{1})\phi_{2}(b_{1})\cdots\phi_{1}(a_{n})\phi_{2}(b_{n})$,
whenever $a_{i}\in A_{1}$ with $\omega_{1}(a_{i})=0$ for $i>1$, and $b_{i}\in
A_{2}$ with $\omega_{2}(b_{i})=0$ for $i<n$.

The following is adapted from [Oza06b, Lemma 2.4].

######  Theorem 5.10.

For each $i=1,2$, let $M_{i}\subset{\mathbb{B}}(\mathcal{H}_{i})$ be a von
Neumann algebra with cyclic vector $\xi_{i}\in\mathcal{H}_{i}$, and suppose
$A_{i}\subset M_{i}$ is an ultraweakly dense ${\rm C}^{*}$-algebra. Suppose
$E_{i}\subset{\mathbb{B}}(\mathcal{H}_{i})$ is an operator $A_{i}$-system such
that ${\mathbb{K}}(\mathcal{H}_{i})\subset E_{i}$. If for each $i=1,2$, the
inclusion $A_{i}\subset E_{i}$ is $(A_{i}\subset M_{i})$-nuclear, then the
inclusion
$(A_{1},\omega_{1})*_{r}(A_{2},\omega_{2})\subset(E_{1},\omega_{1})*_{r}(E_{2},\omega_{2})$
is
$((A_{1},\omega_{1})*_{r}(A_{2},\omega_{2})\subset(M_{1},\omega_{1})*(M_{2},\omega_{2}))$-nuclear.

###### Proof.

By Lemma 5.9 for each $i=1,2$, there exist two nets of state preserving u.c.p.
maps
$\phi_{k}^{i}:(M_{i},\omega_{i})\to({\mathbb{M}}_{n(k,i)}({\mathbb{C}}),\mu_{k}^{i})$,
and $\psi_{k}^{i}:({\mathbb{M}}_{n(k,i)}({\mathbb{C}}),\mu_{k}^{i})\to E_{i}$
such that if we set $\theta_{k}^{i}=\psi_{k}^{i}\circ\phi_{k}^{i}$, then for
any $x\in A_{i}$, $\theta_{k}^{i}(a)-a$ is of the form

(4) $\theta_{k}^{i}(a)-a=T^{i}_{k}+a_{k}$

with $a_{k}\in A_{i}$ satisfying $\omega_{i}(a_{k}^{*}a_{k})\to 0$, and such
that $T^{i}_{k}\in E_{i}$ is converging to $0$ in norm, where $\mu_{k}^{i}$ is
a pure state on ${\mathbb{M}}_{n(k,i)}({\mathbb{C}})$.

We consider

$\phi_{(j,k)}:\phi_{j}^{1}\ast\phi_{k}^{2}:(A_{1},\omega_{1})\ast_{r}(A_{2},\omega_{2})\to({\mathbb{M}}_{n(j,1)}({\mathbb{C}}),\mu_{j}^{1})\ast_{r}({\mathbb{M}}_{n(k,2)}({\mathbb{C}}),\mu_{k}^{2})$

and

$\psi_{(j,k)}:\psi_{j}^{1}\ast\psi_{k}^{2}:({\mathbb{M}}_{n(k,1)}({\mathbb{C}}),\mu_{j}^{1})\ast_{r}({\mathbb{M}}_{n(k,2)}({\mathbb{C}}),\mu_{k}^{2})\to(E_{1},\omega_{1})\ast_{r}(E_{2},\omega_{2}).$

Notice that for $x=a_{1}b_{1}a_{2}b_{2}\cdots a_{n}b_{n}\in
A_{1}\ast_{r}A_{2}$, with $a_{i}\in A_{1}$ with $\omega_{1}(a_{i})=0$ for
$i>1$, and $b_{i}\in A_{2}$ with $\omega_{2}(b_{i})=0$ for $i<m$, we have
$\psi_{(j,k)}\circ\phi_{(j,k)}(x)\to x$ in the point-weak-$(M_{0}\subset
M)$-topology, where $M_{0}=M_{1}\ast_{r}M_{2}$ and $M=M_{1}\ast M_{2}$.
Indeed, for any $\varphi\in(E_{1}*_{r}E_{2})^{{}_{M_{0}}\sharp_{M_{0}}}$, it
follows from (4) that for each $1\leq m\leq n$ we have

$|\varphi(\theta_{j}^{1}(a_{1})\theta_{k}^{2}(b_{1})\cdots\theta_{j}^{1}(a_{m})(\theta^{2}_{k}(b_{m})-b_{m})a_{m+1}\cdots
a_{n}b_{n})|\to 0,$

and similarly we also have

$|\varphi(\theta_{j}^{1}(a_{1})\theta_{k}^{2}(b_{1})\cdots\theta^{2}_{k}(b_{m-1})(\theta_{j}^{1}(a_{m})-a_{m})b_{m}\cdots
a_{n}b_{n})|\to 0.$

Taking closed spans then shows that $\theta_{(j,k)}$ converges pointwise in
the weak-$(M_{0}\subset M)$-topology. Since
$({\mathbb{M}}_{n(j,1)}({\mathbb{C}}),\mu_{j}^{1})\ast_{r}({\mathbb{M}}_{n(k,2)}({\mathbb{C}}),\mu_{k}^{2})$
is nuclear [Oza02], the result then follows. ∎

We note that the assumption that each $E_{i}$ contains the compact operators
is necessary even in the case when each $M_{i}$ is finite-dimensional, since
otherwise by taking $E_{i}=M_{i}$ the previous result would falsely claim that
reduced free products of finite-dimensional ${\rm C}^{*}$-algebras are
nuclear.

We recall that a state on a ${\rm C}^{*}$-algebra is said to be nondegenerate
if the corresponding GNS-representation is faithful . Some special cases of
the following corollary were previously obtained by Isono [Iso13].

######  Corollary 5.11.

For $i=1,2$, let $M_{i}$ be von Neumann algebras with nondegenerate normal
states $\omega_{i}$. If each $M_{i}$ is weakly exact, then
$M=(M_{1},\omega_{1})*(M_{2},\omega_{2})$ is weakly exact.

###### Proof.

Using Corollary 5.6, we may set $E_{i}={\mathbb{B}}(L^{2}(M_{i},\omega_{i}))$
and then apply Theorem 5.10 and Corollary 4.9. ∎

Note that an increasing union of weakly exact von Neumann algebras is again
weakly exact. When the union is separable, this is [Iso13, Proposition 4.1.2].
The general case also follows from [Iso13, Proposition 4.1.2], but one needs
to then apply Corollary 4.9 and use Corollary 5.6. It therefore follows that
the free product of an arbitrary family of weakly exact von Neumann algebras
with nondegenerate states is again weakly exact.

##  6\. Biexact von Neumann algebras

We recall from Section 3.2 that if $M$ is a von Neumann algebra and
${\mathbb{X}}\subset{\mathbb{B}}(L^{2}M)$ is a boundary piece, then the small-
at-infinity boundary relative to ${\mathbb{X}}$ is the normal operator
$M$-system

${\mathbb{S}}_{\mathbb{X}}(M)=\\{T\in{\mathbb{B}}(L^{2}M)\mid[T,x]\in{\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M),\forall
x\in M^{\prime}\\}.$

######  Definition 6.1.

Let $M$ be a von Neumann algebra with an $M$-boundary piece
${\mathbb{X}}\subset{\mathbb{B}}(L^{2}M)$. We say $M$ is biexact relative to
${\mathbb{X}}$ if the inclusion $M\subset{\mathbb{S}}_{{\mathbb{X}}}(M)$ is
$M$-nuclear.

Given a discrete group $\Gamma$, a boundary piece $I$ is a
$\Gamma\times\Gamma$ invariant closed ideal such that $c_{0}\Gamma\subset
I\subset\ell^{\infty}\Gamma$ [BIP21]. The small at infinity compactification
of $\Gamma$ relative to $I$ is the spectrum of the ${\rm C}^{*}$-algebra
${\mathbb{S}}_{I}(\Gamma)=\\{f\in\ell^{\infty}\Gamma\mid f-R_{t}f\in I,{\rm\
for\ any\ }t\in\Gamma\\}$. Recall that $\Gamma$ is biexact relative to $X$ if
$\Gamma{\,\curvearrowright\,}{\mathbb{S}}_{I}(\Gamma)/I$ is topologically
amenable [Oza04, BO08, BIP21]111Actually, we are abusing terminology somewhat
here. In [BO08, Definition 15.1.2] biexactness is defined via an a priori more
restrictive formulation, which is only shown to be equivalent to this
formulation for countable groups.. We remark that this is equivalent to
amenability of $\Gamma{\,\curvearrowright\,}{\mathbb{S}}_{I}(\Gamma)$. Indeed,
we may embed $\ell^{\infty}\Gamma\hookrightarrow I^{**}$ in a
$\Gamma$-equivariant way by taking $\\{e_{i}\\}_{i}$ a $\Gamma$-asymptotically
invariant approximate identity for $I$ and then letting
$\phi:\ell^{\infty}\Gamma\to I^{**}$ be a point-weak∗ cluster point of the
u.c.p. maps $\phi_{i}:\ell^{\infty}\Gamma\to I\subset I^{**}$ given by
$\phi_{i}(f)=e_{i}f$. Since $\Gamma$ is exact, we then have
$\Gamma{\,\curvearrowright\,}I^{**}\oplus({\mathbb{S}}_{I}(\Gamma)/I)^{**}={\mathbb{S}}_{I}(\Gamma)^{**}$
is amenable, and it follows that
$\Gamma{\,\curvearrowright\,}{\mathbb{S}}_{I}(\Gamma)$ is an amenable action
[BEW20, Proposition 2.7].

######  Theorem 6.2.

Let $\Gamma$ be a discrete group with a boundary piece $I$, and ${\mathbb{X}}$
an $L\Gamma$-boundary piece. Let $I_{\mathbb{X}}\subset\ell^{\infty}\Gamma$
denote the closed ideal generated by $E({\mathbb{X}})$, where
$E:{\mathbb{B}}(\ell^{2}\Gamma)\to\ell^{\infty}\Gamma$ is canonical
conditional expectation, and let ${\mathbb{X}}_{I}$ denote the
$L\Gamma$-boundary piece
$\overline{I{\mathbb{B}}(\ell^{2}\Gamma)I}^{\|\cdot\|}$ generated by $I$. The
following statements are true:

  1. (1)

If $L\Gamma$ is biexact relative to ${\mathbb{X}}$, then $\Gamma$ is biexact
relative to $I_{\mathbb{X}}$.

  2. (2)

$\Gamma$ is biexact relative to $I$ if and only if $L\Gamma$ is biexact
relative to ${\mathbb{X}}_{I}$.

###### Proof.

(1) Suppose $L\Gamma$ is biexact relative to ${\mathbb{X}}$. By the argument
in [DKEP22, Theorem 6.4], we have
$E:{\mathbb{S}}_{\mathbb{X}}(L\Gamma)\to{\mathbb{S}}_{I_{\mathbb{X}}}(\Gamma)$
where $E$ is the canonical conditional expectation from
${\mathbb{B}}(\ell^{2}\Gamma)$ to the diagonal subalgebra
$\ell^{\infty}\Gamma$. Now for each $n\in\mathbb{N}$, consider
$h_{n}:\Gamma\to{\mathbb{S}}_{I_{\mathbb{X}}}(\Gamma)$ given by
$h_{n}(\gamma)=E(\theta_{n}(\lambda_{\gamma})\lambda_{\gamma}^{*})$, where
$\theta_{n}=\psi_{n}\circ\phi_{n}:L\Gamma\to{\mathbb{S}}_{\mathbb{X}}(L\Gamma)$
are u.c.p. maps that factor through matrix algebras coming from the
$M$-nuclear embedding. Note that we may assume $\theta_{n}(\lambda_{s})=0$ for
all but finitely many $s\in\Gamma$, since by Lemma 4.1 we may take the first
u.c.p. map $\phi_{n}$ from $M$-nuclearity to be a compression down to
${\mathbb{B}}(\ell^{2}E_{n})$ for some finite subsets $E_{n}\subset\Gamma$.
Viewing $h_{n}$ as an element in
$C_{c}(\Gamma,{\mathbb{S}}_{I_{\mathbb{X}}}(\Gamma))\subset{\mathbb{S}}_{I_{\mathbb{X}}}(\Gamma)\rtimes_{r}\Gamma$,
we now check that $h_{n}$ is positive. For a finite subset
$\\{s_{i}\\}_{i=1}^{m}\subset\Gamma$, observe that

$\displaystyle[\lambda_{s_{i}}^{-1}h_{n}(s_{i}s_{j}^{-1})]_{i,j}$
$\displaystyle=[E(\lambda_{s_{i}}\theta_{n}(\lambda_{s_{i}s_{j}^{-1}})\lambda_{s_{j}^{-1}})]_{i,j}$
$\displaystyle=E(\operatorname{diag}(\lambda_{s_{1}},\dots,\lambda_{s_{m}})^{*}\theta_{n}^{(m)}([\lambda_{s_{i}}\lambda_{s_{j}}^{*}]_{i,j})\operatorname{diag}(\lambda_{s_{1}},\dots,\lambda_{s_{m}}))\geq
0.$

The same argument as in the proof of Theorem 4.12 shows that $E$ is continuous
from the $L\Gamma$-topology into the norm topology, and so for any
$s\in\Gamma$, we have
$\|h_{n}(s)-1\|_{{\mathbb{S}}_{I_{\mathbb{X}}}(\Gamma)}=\|E((\lambda_{s}-\theta_{n}(\lambda_{s}))\lambda_{s}^{*})\|\to
0$. We therefore conclude that
$\Gamma{\,\curvearrowright\,}{\mathbb{S}}_{I_{\mathbb{X}}}(\Gamma)$ is
amenable [BO08, Theorem 4.4.3].

(2) As
${\mathbb{S}}_{I}(\Gamma)\subset{\mathbb{S}}_{{\mathbb{X}}_{I}}(L\Gamma)$, we
have an embedding
$\iota:{\mathbb{S}}_{I}(\Gamma)\rtimes_{r}\Gamma\to{\mathbb{S}}_{{\mathbb{X}}_{I}}(L\Gamma)$
given by $\iota(f)=M_{f}$ and $\iota(u_{\gamma})=\lambda_{\gamma}$ for any
$f\in{\mathbb{S}}_{I}(\Gamma)$ and $\gamma\in\Gamma$ by [BO08, Proposition
5.1.3]. Since ${\mathbb{S}}_{I}(\Gamma)\rtimes_{r}\Gamma$ is nuclear, in
particular, the inclusion

$C^{*}_{\lambda}\Gamma\subset{\mathbb{S}}_{I}(\Gamma)\rtimes_{r}\Gamma\subset{\mathbb{S}}_{{\mathbb{X}}_{I}}(L\Gamma)$

is $(C^{*}_{\lambda}\Gamma\subset L\Gamma)$-nuclear. Corollary 4.9 then gives
that $L\Gamma\subset{\mathbb{S}}_{{\mathbb{X}}_{I}}(L\Gamma)$ is
$L\Gamma$-nuclear.

The “only if” part follows from the first statement upon noticing that
$E({\mathbb{X}}_{I})=I$. ∎

######  Corollary 6.3.

Let $\Gamma$ and $\Lambda$ be two discrete groups. If $\Gamma$ is biexact and
$L\Gamma\cong L\Lambda$, then $\Lambda$ is also biexact.

###  6.1. Properties of biexact von Neumann algebras

Next we collect some basic properties of biexact von Neumann algebras,
generalizing some results from the group von Neumann algebra setting.

Given $M$ a von Neumann algebra with a normal faithful representation
$\pi:M\to{\mathbb{B}}({\mathcal{H}})$, we may consider the following normal
$M$-system, which is a natural variant of ${\mathbb{S}}(M)$,

${\mathbb{S}}(M;{\mathcal{H}})=\\{T\in{\mathbb{B}}({\mathcal{H}})\mid[T,x]\in{\mathbb{K}}(M;{\mathcal{H}}),\forall
x\in M^{\prime}\\},$

where ${\mathbb{K}}(M;{\mathcal{H}})$ is the $M$-$M$ and
$M^{\prime}$-$M^{\prime}$ closure of ${\mathbb{K}}({\mathcal{H}})$. We say $M$
is biexact with respect to the normal faithful representation
$\pi:M\to{\mathbb{B}}({\mathcal{H}})$ if the inclusion
$M\subset{\mathbb{S}}(M;{\mathcal{H}})$ is $M$-nuclear. Since every normal
faithful representation of $M$ is a reduction of an amplification of the
standard form of $M$, the next lemma shows that biexactness is independent of
representations of $M$.

######  Lemma 6.4.

Let $M\subset{\mathbb{B}}({\mathcal{H}})$ be a von Neumann algebra.

  1. (1)

If $M\subset{\mathbb{S}}(M;{\mathcal{H}})$ is $M$-nuclear, then
$M\subset{\mathbb{S}}(M;{\mathcal{H}}\otimes\ell^{2}S)$ is $M$-nuclear for any
set $S$.

  2. (2)

If $M\subset{\mathbb{S}}(M;{\mathcal{H}})$ is $M$-nuclear,
$p\in\mathcal{P}(M)$, and $q\in\mathcal{P}(M^{\prime})$ with $pq\not=0$, then
$pMpq\subset{\mathbb{S}}(pMpq;pq{\mathcal{H}})$ is $pMpq$-nuclear.

###### Proof.

(1) It suffices to show
${\mathbb{S}}(M;{\mathcal{H}})\otimes\operatorname{id}_{\ell^{2}S}\subset{\mathbb{S}}(M;{\mathcal{H}}\otimes\ell^{2}S)$.
For any $T\in{\mathbb{S}}(M;{\mathcal{H}})$, $x\in
M^{\prime}\subset{\mathbb{B}}({\mathcal{H}})$ and $i,j\in S$, we denote by
$e_{i,j}=\langle\cdot,\delta_{i}\rangle\delta_{j}$ and compute
$[T\otimes\operatorname{id}_{\ell^{2}S},x\otimes e_{i,j}]=[T,x]\otimes
e_{i,j}\in{\mathbb{K}}(M;{\mathcal{H}}\otimes\ell^{2}S)$. Since the span of
$\\{x\otimes e_{i,j}\mid x\in M^{\prime},i,j\in S\\}\subset
M^{\prime}\cap{\mathbb{B}}({\mathcal{H}}\otimes\ell^{2}S)$ is ultraweakly
dense, we conclude that
${\mathbb{S}}(M;{\mathcal{H}})\otimes\operatorname{id}_{\ell^{2}S}\subset{\mathbb{S}}(M;{\mathcal{H}}\otimes\ell^{2}S)$.

(2) If $S\in{\mathbb{B}}({\mathcal{H}})$ and $a,b\in M$, then for a non-zero
projection $p\in\mathcal{P}(M)$ we have
$pa^{*}Sbp=(pa^{*}ap)^{1/2}v^{*}Sw(pb^{*}bp)^{1/2}$, where $pa=v|pa|$ and
$pb=w|pb|$ are the polar decompositions. In particular, if $\varphi$ is a
normal state on $pMp$, and we denote by $\tilde{\phi}$ the extension to $M$
given by $\tilde{\varphi}(x)=\varphi(pxp)$, then we have
$s_{\varphi}(pa^{*}Sbp)\leq\varphi(pa^{*}ap)^{1/2}\|S\|\varphi(pb^{*}bp)^{1/2}=\tilde{\varphi}(a^{*}a)^{1/2}\|S\|\tilde{\varphi}(b^{*}b)^{1/2}$.
Taking infimums shows that $s_{\varphi}(pTp)\leq s_{\tilde{\varphi}}(T)$ for
each $T\in{\mathbb{B}}({\mathcal{H}})$, and so conjugation by $p$ is
continuous from the $M$-topology on ${\mathbb{B}}({\mathcal{H}})$ to the
$pMp$-topology on ${\mathbb{B}}(p{\mathcal{H}})$. Note that conjugation by $p$
is also continuous from the $M^{\prime}$-topology on
${\mathbb{B}}({\mathcal{H}})$ to the $pMp^{\prime}$-topology on
${\mathbb{B}}(p{\mathcal{H}})$, and we have similar continuity properties for
conjugation by $q\in\mathcal{P}(M^{\prime})$.

Hence, we have that
$pq{\mathbb{K}}(M;{\mathcal{H}})pq\subset{\mathbb{K}}(pMpq;pq{\mathcal{H}})$,
and so for $T\in{\mathbb{S}}(M;{\mathcal{H}})$ and $x\in
M^{\prime}\subset{\mathbb{B}}({\mathcal{H}})$ we have
$[pqTpq,qxqp]=pq[T,qxq]pq\in{\mathbb{K}}(pMpq;pq{\mathcal{H}})$. Therefore if
we consider the maps associated to the $M$-nuclear embedding
$M\subset{\mathbb{S}}(M;{\mathcal{H}})$, then conjugation by $pq$ gives rise
to an $pMpq$-nuclear embedding of $pMpq$ in to
${\mathbb{S}}(pMpq;pq{\mathcal{H}})$. ∎

######  Proposition 6.5 (cf. [HI17, Proposition 2.10]).

Let $M$ be a biexact von Neumann algebra. Then
$M\,\overline{\otimes}\,{\mathbb{B}}({\mathcal{K}})$ and $pMp$ are also
biexact, for any Hilbert space ${\mathcal{K}}$ and nonzero projection
$p\in{\mathcal{P}}(M)$.

###### Proof.

Suppose $M$ is represented on ${\mathcal{H}}$. To see that
$N:=M\,\overline{\otimes}\,{\mathbb{B}}({\mathcal{K}})$ is bi-exact, first we
note that
${\mathbb{K}}(M;{\mathcal{H}})\otimes{\mathbb{B}}({\mathcal{K}})\subset{\mathbb{K}}(N;{\mathcal{H}}\otimes{\mathcal{K}})$.
Indeed, it is easy to see that we have
${\mathbb{K}}(M;{\mathcal{H}})\otimes{\mathbb{K}}({\mathcal{K}})\subset{\mathbb{K}}(N;{\mathcal{H}}\otimes{\mathcal{K}})$
and ${\mathbb{K}}({\mathcal{K}})$ is dense in ${\mathbb{B}}({\mathcal{K}})$ in
the ultrastrong topology, and hence also in the
${\mathbb{B}}({\mathcal{K}})$-topology. It then follows that
${\mathbb{S}}(M;{\mathcal{H}})\otimes{\mathbb{B}}({\mathcal{K}})\subset{\mathbb{S}}(N;{\mathcal{H}}\otimes{\mathcal{K}})$,
since $[T\otimes S,a\otimes\operatorname{id}_{{\mathcal{K}}}]=[T,a]\otimes
S\in{\mathbb{K}}(N;{\mathcal{H}}\otimes{\mathcal{K}})$ for any $a\in
M^{\prime}\cap{\mathbb{B}}({\mathcal{H}})$,
$T\in{\mathbb{S}}(M;{\mathcal{H}})$, and $S\in{\mathbb{B}}({\mathcal{K}})$.

Since the identity map on ${\mathbb{B}}({\mathcal{K}})$ is weakly nuclear, by
tensoring with the maps coming from the $M$-nuclearity of the embedding
$M\subset{\mathbb{S}}(M;{\mathcal{H}})$ we may obtain u.c.p. maps
$\phi_{i}:M\otimes{\mathbb{B}}({\mathcal{K}})\to{\mathbb{M}}_{n(i)}({\mathbb{C}})$
and
$\psi_{i}:{\mathbb{M}}_{n(i)}({\mathbb{C}})\to{\mathbb{S}}(M;{\mathcal{H}})\otimes{\mathbb{B}}({\mathcal{K}})$
such that
$\psi_{i}\circ\phi_{i}\to\operatorname{id}_{M\otimes{\mathbb{B}}({\mathcal{K}})}$
in the $(M\otimes{\mathbb{B}}({\mathcal{K}})\subset
M\,\overline{\otimes}\,{\mathbb{B}}({\mathcal{K}}))$-topology. Corollary 4.9
then gives that the embedding
$N\subset{\mathbb{S}}(N;{\mathcal{H}}\,\overline{\otimes}\,{\mathcal{K}})$ is
$N$-nuclear.

That $pMp$ is biexact when $M$ is already follows from Lemma 6.4. ∎

Let $M$ be a von Neumann algebra, and
${\mathbb{X}}\subset{\mathbb{B}}(L^{2}M)$ an $M$-boundary piece. A net of
unitaries $\\{u_{i}\\}\subset{\mathcal{U}}(M)$ is said to converge to $0$ over
${\mathbb{X}}$ if we have $\operatorname{Ad}(u_{i})(T)\to 0$ ultraweakly for
each $T\in{\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M)$. If $N\subset M$ is a von
Neumann subalgebra, then we say that $N$ is weak mixing over ${\mathbb{X}}$ if
there exists a net $\\{u_{i}\\}\subset{\mathcal{U}}(N)$ such that
$\\{u_{i}\\}$ converges to $0$ over ${\mathbb{X}}$. If $M$ is finite, then it
is not difficult to see that $\\{u_{i}\\}$ converges to $0$ over
${\mathbb{X}}$ if and only if $\operatorname{Ad}(u_{i})(T)\to 0$ ultraweakly
for each $T\in{\mathbb{X}}$ (see, e.g., [DKEP22, Lemma 4.2]). For instance,
when $M$ is finite, a net $\\{u_{i}\\}$ converges to $0$ over
${\mathbb{K}}(L^{2}M)$ if and only if the net $\\{u_{i}\\}$ converges to $0$
ultraweakly. This does not hold for general von Neumann algebra, e.g., if
$M={\mathbb{B}}({\mathcal{H}})$, then
${\mathbb{K}}_{{\mathbb{K}}({\mathcal{H}})}(M)={\mathbb{B}}({\mathcal{H}})$,
and so there is no net $\\{u_{i}\\}$ that converges to $0$ over
${\mathbb{K}}({\mathcal{H}})$ in this case. The following lemma gives a
general condition for concluding that unitaries converge to $0$ over
${\mathbb{X}}$.

######  Lemma 6.6.

Let $M$ be a $\sigma$-finite von Neumann algebra with a normal faithful state
$\omega$, and let ${\mathbb{X}}\subset{\mathbb{B}}(L^{2}(M,\omega))$ be an
$M$-boundary piece. Suppose $\\{u_{i}\\}\subset{\mathcal{U}}(M)$ is such that
the weak∗-limit $\lim_{i\to\infty}\omega\circ{\rm Ad}(u_{i})$ exists and is in
$M_{*}\subset M^{*}$. If $u_{i}Tu_{i}^{*}\to 0$ ultraweakly for each
$T\in{\mathbb{X}}$, then $\\{u_{i}\\}$ converges to $0$ over ${\mathbb{X}}$.

###### Proof.

If $a,b\in M$, $S\in{\mathbb{B}}(L^{2}(M,\omega))$, and $x,y\in M^{\prime}$,
then by Cauchy-Schwarz we have

$|\langle
u_{i}^{*}a^{*}Sbu_{i}x\omega^{1/2},y\omega^{1/2}|\leq\omega(u_{i}^{*}a^{*}au_{i})^{1/2}\|y^{*}\|\|S\|\|x\|\omega(u_{i}^{*}b^{*}bu_{i})^{1/2}.$

Taking infimums over all decompositions of the type $T=a^{*}Sb$, we then have

$\limsup_{i\to\infty}|\langle
u_{i}^{*}Tu_{i}x\omega^{1/2},y\omega^{1/2}\rangle|\leq\|x\|\|y\|s_{\tilde{\omega}}(T),$

where $\tilde{\omega}$ is the weak∗-limit of $\omega\circ{\rm Ad}(u_{i})$.

Thus, if $u_{i}^{*}Tu_{i}\to 0$ ultraweakly for each $T\in{\mathbb{X}}$, then
we also have $u_{i}^{*}Tu_{i}\to 0$ ultraweakly for each
$T\in\overline{{\mathbb{X}}}^{{}_{M-M}}$. Since conjugation by $u_{i}$ is
$M^{\prime}$-bimodular, we then also have that $u_{i}^{*}Tu_{i}\to 0$
ultraweakly for each
$T\in\overline{\overline{{\mathbb{X}}}^{{}_{M-M}}}^{{}_{M^{\prime}-M^{\prime}}}={\mathbb{K}}_{\mathbb{X}}^{\infty,1}(M)$.
∎

If $M$ is a $\sigma$-finite von Neumann algebra and $\mathcal{U}\in\beta
N\setminus N$ is a nonprincipal ultrafilter, then we define

$\mathcal{I}_{\mathcal{U}}=\\{(x_{n})_{n}\in\ell^{\infty}(M)\mid x_{n}\to 0\
*-{\rm strongly\ as\ }n\to\mathcal{U}\\};$
$\mathcal{M}^{\mathcal{U}}(M)=\\{(x_{n})_{n}\in\ell^{\infty}(M)\mid(x_{n})_{n}\mathcal{I}_{\mathcal{U}}\subset\mathcal{I}_{\mathcal{U}}{\rm\
and\
}\mathcal{I}_{\mathcal{U}}(x_{n})_{n}\subset\mathcal{I}_{\mathcal{U}}\\}.$

The multiplier algebra $\mathcal{M}^{\mathcal{U}}(M)$ is a ${\rm
C}^{*}$-algebra containing $\mathcal{I}_{\mathcal{U}}$ as a norm closed two-
sided ideal. The Ocneanu ultraproduct $M^{\mathcal{U}}$ is the quotient
$\mathcal{M}^{\mathcal{U}}/\mathcal{I}_{\mathcal{U}}$, which is a von Neumann
algebra [Ocn85]. If $(x_{n})_{n}\in\mathcal{M}^{\mathcal{U}}$, then we denote
by $(x_{n})^{\mathcal{U}}$ the image of $(x_{n})_{n}$ in $M^{\mathcal{U}}$. We
may view $M$ as a von Neumann subalgebra of $M^{\mathcal{U}}$ by considering
the constant sequences. We have a canonical normal conditional expectation
$E_{M}:M^{\mathcal{U}}\to M$ by considering the weak∗-limit
$E_{M}((x_{n})^{\omega})=\lim_{n\to\mathcal{U}}x_{n}$.

If $Q\subset M$ is a von Neumann subalgebra with expectation, then given
$(x_{n})_{n}\in\ell^{\infty}(Q)$ we have that
$(x_{n})_{n}\in\mathcal{M}^{\mathcal{U}}(Q)$ if and only if
$(x_{n})_{n}\in\mathcal{M}^{\mathcal{U}}(M)$, and this gives rise to a
canonical embedding $Q^{\mathcal{U}}\subset M^{\mathcal{U}}$. Moreover, the
map $M^{\mathcal{U}}\ni(x_{n})^{\omega}\mapsto(E(x_{n}))^{\omega}\in
Q^{\mathcal{U}}$ gives a normal faithful conditional expectation onto
$Q^{\mathcal{U}}$ that satisfies

$E^{M^{\mathcal{U}}}_{M}E^{M^{\mathcal{U}}}_{Q^{\mathcal{U}}}=E^{M^{\mathcal{U}}}_{Q^{\mathcal{U}}}E^{M^{\mathcal{U}}}_{M}=E^{M}_{Q}E^{M^{\mathcal{U}}}_{M}=E^{Q^{\mathcal{U}}}_{Q}E^{M^{\mathcal{U}}}_{Q^{\mathcal{U}}}.$

We refer to [AH14] for more information on the Ocneanu ultraproduct.

######  Theorem 6.7.

Let $M$ be a $\sigma$-finite biexact von Neumann algebra, and let
$\mathcal{U}\in\beta{\mathbb{N}}\setminus{\mathbb{N}}$ be a nonprincipal
ultrafilter. If $A\subset M$ is a von Neumann subalgebra such that there
exists $u\in\mathcal{U}(A^{\prime}\cap M^{\mathcal{U}})$ with $E_{M}(u)=0$,
then the inclusion $A\subset M$ is weakly nuclear.

###### Proof.

Fix a normal faithful state $\varphi$ on $M$. Suppose $A\subset M$ is a von
Neumann subalgebra and $u\in\mathcal{U}(A^{\prime}\cap M^{\mathcal{U}})$ with
$E_{M}(u)=0$. We may then choose a sequence
$\\{u_{n}\\}\subset{\mathcal{U}}(M)$ such that
$u=(u_{n})_{n}\in\mathcal{U}(M^{\mathcal{U}})$ [HI16, Lemma 2.1], i.e., we
have $\lim_{n\to\mathcal{U}}u_{n}=0$ ultraweakly,
$\lim_{n\to\mathcal{U}}[u_{n},a]=0$ $*$-strongly for any $a\in A$, and
$\tilde{\varphi}:=\lim_{n\to\mathcal{U}}\varphi\circ{\rm
Ad}(u_{n})=\varphi\circ E_{M}\circ{\rm Ad}(u)_{|M}$ is contained in $M_{*}$.

We let
$\theta:{\mathbb{B}}(L^{2}(M,\varphi))\to{\mathbb{B}}(L^{2}(M,\varphi))$ be
the point-ultraweak limit $\theta=\lim_{n\to\mathcal{U}}{\rm Ad}(u_{n})$. By
Lemma 6.6 we have $\theta_{|{\mathbb{K}}^{\infty,1}(M)}=0$. From the
definition of ${\mathbb{S}}(M)$, we therefore have
$\theta({\mathbb{S}}(M))\subset M$.

Note that since $\lim_{n\to\mathcal{U}}[u_{n},a]\to 0$ $*$-strongly for any
$a\in A$, we also have $\theta_{|A}={\rm id}$ and $\theta_{|M^{\prime}}={\rm
id}$. Also, note that, by Kadison’s Inequality, if $x\in M$ and $a\in
M^{\prime}$, we have
$\|\theta(x)a\varphi^{1/2}\|^{2}\leq\|a\|^{2}\varphi\circ\theta(x^{*}x)=\|a\|^{2}\tilde{\varphi}(x^{*}x)$,
and from this it follows that $\theta_{|M}$ defines a normal map from $M$ to
$M$.

Now consider

$A\subset M\subset{\mathbb{S}}(M;{\mathcal{H}})\xrightarrow{\theta}M.$

Since $M\subset{\mathbb{S}}(M)$ is $M$-nuclear, and since by Lemma 3.7