# Amenability and vanishing of $L^{2}$-Betti numbers:
an operator algebraic approach

Vadim Alekseev  Vadim Alekseev, Mathematisches Institut, Georg-August-
Universität Göttingen, Bunsenstraße 3-5, D-37073 Göttingen, Germany.
<EMAIL_ADDRESS>and  David Kyed  David Kyed, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium.
<EMAIL_ADDRESS>
###### Abstract.

We introduce a Følner condition for dense subalgebras in finite von Neumann
algebras and prove that it implies dimension flatness of the inclusion in
question. It is furthermore proved that the Følner condition naturally
generalizes the existing notions of amenability and that the ambient von
Neumann algebra of a Følner algebra is automatically injective. As an
application, we show how our techniques unify previously known results
concerning vanishing of $L^{2}$-Betti numbers for amenable groups, quantum
groups and groupoids and moreover provide a large class of new examples of
algebras with vanishing $L^{2}$-Betti numbers.

###### Key words and phrases:

Amenability, $L^{2}$-Betti numbers, operator algebras

###### 2010 Mathematics Subject Classification:

46L10, 43A07, 18G15

The research of the second named author is supported by The Danish Council for
Independent Research $|$ Natural Sciences and the ERC Starting Grant
VNALG-200749

##  1\. Introduction

$L^{2}$-Betti numbers originated from topology, but have later unveiled an
interesting connection to operator algebras. Indeed, while the original
definition of $L^{2}$-Betti numbers for groups involved a geometric
construction [Ati76, CG86], it was later shown by Lück (see also [Far98]) that
they can also be described in terms of certain $\operatorname{Tor}$ modules —
thus using only homological algebra and modules over von Neumann algebras.
This has led to several generalizations of $L^{2}$-Betti numbers: they have
been defined for discrete measured groupoids [Sau05], quantum groups [Kye08b]
and general subalgebras of finite von Neumann algebras [CS05]. Similarly,
$L^{2}$-Betti numbers were introduced in the setting of equivalence relations
in [Gab02] following the original geometric approach, but as shown in [Sau05]
these can also be expressed in terms of homological algebra.

There are two properties which attract attention in all these situations.
Firstly, in each case the definition involves certain operator algebras
canonically associated with the situation at hand. Secondly, for groups,
quantum groups and groupoids it is well known that the $L^{2}$-Betti numbers
vanish in the presence of amenability. In view of this, it is natural to seek
a common operator algebraic reason for this to happen. In doing so, one
firstly observes that the actual reason for the vanishing of $L^{2}$-Betti
numbers for amenable groups and quantum groups is a certain dimension-flatness
property of an inclusion $\mathcal{A}\subseteq M$, where $\mathcal{A}$ is a
strongly dense $\ast$-subalgebra of a finite von Neumann algebra $M$.
Secondly, the key to the proof of this dimension-flatness result is a Følner
condition for the notion of amenability in question. In the present paper we
introduce a Følner type condition for a general weakly dense $*$-subalgebra
$\mathcal{A}$ in a tracial von Neumann algebra $(M,\tau)$ and show how this
leads to dimension-flatness of the inclusion $\mathcal{A}\subseteq M$ and
subsequently to the vanishing of the operator algebraic $L^{2}$-Betti numbers
$\beta_{p}^{(2)}(\mathcal{A},\tau)$. This approach unifies the above mentioned
vanishing results and furthermore provides a large class of new examples of
algebras with vanishing $L^{2}$-Betti numbers (see Section 6). More precisely,
we prove the following:

######  Theorem (see Theorem 4.4 & Corollary 6.5).

If $\mathcal{A}\subseteq M$ satisfies the Følner condition then for any left
$\mathcal{A}$-module $X$ and any $p\geqslant 1$ we have

$\dim_{M}\operatorname{Tor}_{p}^{\mathcal{A}}(M,X)=0,$

and the Connes-Shlyakhtenko $L^{2}$-Betti numbers of $\mathcal{A}$ vanish in
positive degrees.

Secondly, we link our Følner condition to the classical operator algebraic
notion of amenability by proving (a slightly more general version of) the
following:

######  Theorem (see Theorem 5.1).

If $\mathcal{A}\subseteq M$ satisfies the Følner condition $M$ is injective.

The Følner condition consists of two requirements: an almost invariance
property and a trace approximation property. The almost invariance property
requires that the left action of $\mathcal{A}$ on $L^{2}(M)$ admits almost
invariant subspaces, the almost invariance being measured by a dimension
function. The naive approach would be to use the usual dimension over
$\mathbb{C}$ and require these subspaces to be finite-dimensional; however, it
turns out that the dimension theory over a von Neumann subalgebra
$N\subseteq\mathcal{A}$ can be efficiently applied in order to allow for
almost invariant subspaces which are finitely generated projective $N$-modules
(thus usually having infinite dimension over $\mathbb{C}$). For this reason,
parts of the paper deals with results related to dimension theory for modules
over von Neumann algebras; the reader not familiar with these notions may,
without loosing the essential ideas, think of the case $N={\mathbb{C}}$, which
reduces most arguments to finite dimensional linear algebra.

_Structure:_ The paper is organized as follows. In the second section we
recapitulate Lück’s dimension theory for modules over von Neumann algebras and
prove a few results concerning a relative dimension function in this context.
In the third section we introduce the operator-algebraic Følner condition and
draw some easy consequences, preparing for the proof of the main theorem which
is given in the fourth section. The fifth section is devoted to the discussion
concerning the relationship between the Følner condition for an algebra and
injectivity of its enveloping von Neumann algebra. In the sixth section we
discuss examples and show how the Følner condition implies vanishing of
$L^{2}$-Betti numbers in a variety of different instances.

_Assumptions:_ Throughout the paper, all generic von Neumann algebras are
assumed to have separable predual. Our main focus will be on finite von
Neumann algebras, for which separability of the predual is equivalent to
separability of the von Neumann algebra itself for the strong, weak and ultra-
weak topology. These topologies are furthermore all metrizable on bounded
subsets of $M$. In order to be consistent with the general Hilbert
$C^{*}$-module terminology (see below) inner products on Hilbert spaces are
assumed to be linear in the second variable and conjugate linear in the first.
Algebraic tensor products are denoted “$\odot$”, tensor products of Hilbert
spaces by “$\otimes$” and tensor products of von Neumann algebras by
$``\bar{\otimes}$”. Our main setup will consist of an inclusion $N\subseteq M$
of finite von Neumann algebras with a fixed normal tracial state $\tau$ on
$M$. The von Neumann algebra $N$ should be thought of as a “coefficient
algebra” and will in many applications be equal to ${\mathbb{C}}$. We denote
by $H$ the GNS-space $L^{2}(M,\tau)$, by $J\colon H\to H$ the modular
conjugation arising from $\tau$ and by $E\colon M\to N$ the $\tau$-preserving
conditional expectation. Note that $H$ automatically carries a normal right
$N$-action given by $\xi\cdot x:=Jx^{*}J\xi$ which extends the natural right
$N$-module structure on $M$.

_Acknowledgements:_ The authors would like to thank Thomas Schick for a
crucial remark at an early stage in the project and Stefaan Vaes for
suggesting an improvement of Theorem 5.1. Moreover, thanks are due to Étienne
Blanchard, Henrik D. Petersen and the anonymous referee for numerous valuable
corrections and suggestions.

##  2\. Dimension theory

In this section we recapitulate parts of the von Neumann dimension theory for
modules over a finite von Neumann algebra and introduce a relative version of
this notion of dimension. The relative dimension function will play a
prominent role in the definition of the Følner condition given in Section 3.

###  2.1. Dimension of modules and the trace on endomorphisms

Consider a finite von Neumann algebra $N$ endowed with a normal, faithful
tracial state $\tau$ and denote by $L^{2}(N)$ the GNS-space arising from
$\tau$ and by $N^{{\operatorname{{op}}}}$ the opposite von Neumann algebra.
Denoting by $\Lambda$ the natural embedding of $N$ into $L^{2}(N)$ the inner
product in $L^{2}(N)$ is therefore given by
$\left\langle\Lambda(x),\Lambda(y)\right\rangle=\tau(x^{*}y)$ for $x,y\in N$.
In what follows, we will often suppress the map $\Lambda$ and simply consider
$N$ as a subspace of $L^{2}(N)$. A Hilbert space $F$ endowed with a normal
$*$-representation of $N^{{\operatorname{{op}}}}$ is called a _finitely
generated normal (right) $N$-module_ if there exists an isometric
$N^{{\operatorname{{op}}}}$-equivariant embedding of $F$ into $L^{2}(N)^{k}$
for some finite integer $k$, where $L^{2}(N)^{k}$ is considered with the
standard $N^{{\operatorname{{op}}}}$-action given by

$a^{{\operatorname{{op}}}}\colon(\xi_{1},\dots,\xi_{n})\mapsto(Ja^{*}J\xi_{1},\dots,Ja^{*}J\xi_{n}).$

Thus, any finitely generated normal $N$-module $F$ is isomorphic to one of the
form $pL^{2}(N)^{k}$ for a projection $p\in{\mathbb{M}}_{k}(N)$ and _the von
Neumann dimension_ of $F$ is then defined as

$\dim_{N}F:=\sum_{i=1}^{k}\tau(p_{ii}).$

The number $\dim_{N}F$ is independent of the choice of $p$ and $k$ (see e.g.
[Lüc02, Section 1.1.2]) and thus depends only on the isomorphism class of $F$.

######  Remark 2.1.

In most of the existing literature, finitely generated normal right modules
over $N$ are called _finitely generated Hilbert $N$-modules_, but since we are
going to consider finitely generated Hilbert $C^{*}$-modules over $N$ (in the
operator algebraic sense [Lan95]) as well, we have chosen the term “normal
module” in order to avoid unnecessary confusion. We advise the reader not
familiar with these notions to consult the article [Fra01] where a detailed
comparison is carried out.

Next we recall the construction of the trace on the endomorphism von Neumann
algebra $\operatorname{End}_{N}(F)$ associated with a finitely generated
normal $N$-module $F$; here, and in what follows, $\operatorname{End}_{N}(F)$
denotes the von Neumann algebra of all bounded operators on $F$ commuting with
the $N$-action. The trace on $\operatorname{End}_{N}(F)$ was previously
considered by Lück [Lüc02, Section 1.1] and also by Farber [Far98] in the more
general context of von Neumann categories.

######  Lemma 2.2 ([Lüc02, Section 1.1.3]).

Let $F$ be a finitely generated normal $N$-module and consider an operator
$T\in\operatorname{End}_{N}(F)$. Upon choosing an isomorphism $F\simeq
pL^{2}(N)^{k}$ we obtain an induced $*$-isomorphism
$\operatorname{End}_{N}(F)\simeq p{\mathbb{M}}_{k}(N)p$ and we denote the
matrix representing $T$ by $(T_{ij})_{i,j=1}^{k}$. Then the number
$\operatorname{Tr}_{N}(T):=\sum_{i=1}^{k}\tau(T_{ii})$ does not depend on the
choice of isomorphism and defines a normal, faithful, positive trace on
$\operatorname{End}_{N}(F)$ with
$\operatorname{Tr}_{N}(\operatorname{id}_{F})=\dim_{N}F$.

######  Remark 2.3.

In the case $N={\mathbb{C}}$ the normal $N$-module $F$ is simply a finite
dimensional vector space and hence $\operatorname{Tr}_{N}(-)$ is just the
standard, non-normalized trace on $B(F)$.

The first step towards extending the notion of dimension to arbitrary
$N$-modules is to pass from normal modules to Hilbert $C^{*}$-modules over
$N$.

######  Definition 2.4.

A (right) Hilbert $C^{*}$-module over $N$ consists of an algebraic right
$N$-module $\mathcal{X}$ together with a map
$\left\langle\cdot,\cdot\right\rangle_{N}\colon\mathcal{X}\times\mathcal{X}\to
N$ such that

  * (i)

For all $x\in\mathcal{X}$ we have ${\left\langle
x,x\right\rangle}_{N}\geqslant 0$ and ${\left\langle x,x\right\rangle}_{N}=0$
only for $x=0$.

  * (ii)

For all $x,y\in\mathcal{X}$ we have $\left\langle
x,y\right\rangle_{N}=\left\langle y,x\right\rangle_{N}^{*}$.

  * (iii)

For all $x,y,z\in\mathcal{X}$ and $a,b\in N$ we have $\left\langle
z,xa+yb\right\rangle_{N}=\left\langle z,x\right\rangle_{N}a+\left\langle
z,y\right\rangle_{N}b$.

  * (iv)

The space $\mathcal{X}$ is complete with respect to the norm
$\|x\|:=\|\left\langle x,x\right\rangle\|^{\frac{1}{2}}$.

A Hilbert $C^{*}$-module over $N$ is called _finitely generated_ if it is
algebraically finitely generated over $N$ and _projective_ if it is projective
as a right $N$-module. Moreover, the terms “Hilbert module over $N$”, “Hilbert
$C^{*}$-module over $N$” and “Hilbert $N$-module” will be used synonymously in
the sequel.

Note that the finitely generated free $N$-modules $N^{k}$ become Hilbert
$N$-modules when endowed with the $N$-valued inner product $\left\langle
x,y\right\rangle_{{\operatorname{st}}}:=\sum_{i=1}^{k}x_{i}^{*}y_{i}$; we
refer to this as the _standard_ inner product on $N^{k}$. If
$(\mathcal{P},\left\langle\cdot,\cdot\right\rangle_{N})$ is a finitely
generated projective Hilbert $N$-module then
$\tau\circ\left\langle\cdot,\cdot\right\rangle_{N}$ defines a positive
definite ${\mathbb{C}}$-valued inner product on $\mathcal{P}$ and we denote
the Hilbert space completion of $\mathcal{P}$ with respect to this inner
product by $L^{2}(\mathcal{P})$. It was shown by Lück that we get a finitely
generated normal $N$-module in this way and that this construction yields an
equivalence of categories:

######  Theorem 2.5 (Lück, [Lüc02, Lemma 6.23 & Theorem 6.24]).

Every finitely generated projective $N$-module allows an $N$-valued inner
product turning it into a Hilbert $N$-module and the inner product is unique
up to unitary isomorphism. Furthermore, Hilbert space completion constitutes
an equivalence of categories between the category of finitely generated
projective Hilbert $N$-modules and the category of finitely generated normal
$N$-modules.

We remark that a finitely generated projective Hilbert $N$-module
$(\mathcal{P},\left\langle\cdot,\cdot\right\rangle_{N})$ is automatically
self-dual; i.e. $\operatorname{Hom}_{N}(\mathcal{P},N)=\\{\left\langle
x,\cdot\right\rangle_{N}\mid x\in\mathcal{P}\\}$. This follows easily from the
uniqueness of the inner product and the obvious self-duality of the finitely
generated free Hilbert modules
$(N^{k},\left\langle\cdot,\cdot\right\rangle_{{\operatorname{st}}})$.
Furthermore, by [MT05, Lemma 2.3.7] every finitely generated Hilbert submodule
in $\mathcal{P}$ splits off orthogonally as a direct summand and is therefore,
in particular, itself finitely generated and projective. Due to Theorem 2.5,
it makes sense, for a finitely generated projective $N$-module $\mathcal{P}$,
to define the von Neumann dimension of $\mathcal{P}$ as
$\dim_{N}\mathcal{P}=\dim_{N}L^{2}(\mathcal{P})$ where $L^{2}(\mathcal{P})$ is
the Hilbert space completion relative to some choice of Hilbert $N$-module
structure on $\mathcal{P}$. Moreover, for an arbitrary (algebraic) $N$-module
$\mathcal{X}$ its von Neumann dimension is defined as

$\dim_{N}\mathcal{X}=\sup\\{\dim_{N}\mathcal{P}\mid\mathcal{P}\subseteq\mathcal{X},\mathcal{P}\
\text{finitely generated projective}\\}\in[0,\infty].$

This dimension function is no longer faithful (i.e. non-zero modules might be
zero-dimensional) but it still has remarkably good properties; one of its
prime features being additivity with respect to short exact sequences [Lüc02,
Theorem 6.7]. For a submodule $\mathcal{E}$ in an $N$-module $\mathcal{F}$ the
_algebraic closure_ $\overline{\mathcal{E}}^{{\operatorname{alg}}}$ of
$\mathcal{E}$ inside $\mathcal{F}$ is defined as the intersection of the
kernels of those homomorphisms in the dual module
$\operatorname{Hom}_{N}(\mathcal{F},N)$ that vanish on $\mathcal{E}$, and if
$\mathcal{F}$ is finitely generated we have that
$\dim_{N}\mathcal{E}=\dim_{N}\overline{\mathcal{E}}^{{\operatorname{alg}}}$
[Lüc02, Theorem 6.7].

######  Lemma 2.6.

If $\mathcal{P}$ is a finitely generated projective Hilbert $N$-module and
$\mathcal{E}\subseteq\mathcal{P}$ a submodule then
${\mathcal{E}}^{\perp\perp}=\overline{\mathcal{E}}^{{\operatorname{alg}}}$ and
$\mathcal{E}^{\perp\perp}$ splits off orthogonally as a direct summand in
$\mathcal{P}$. Furthermore, the Hilbert space closures of $\mathcal{E}$ and
$\mathcal{E}^{\perp\perp}$ in $L^{2}(\mathcal{P})$ coincide.

###### Proof.

Since $\mathcal{P}$ is selfdual, the equality
$\mathcal{E}^{\perp\perp}=\overline{\mathcal{E}}^{\operatorname{alg}}$ follows
directly from the definition of the algebraic closure. Since $\mathcal{P}$ is
finitely generated and projective it now follows from [Lüc02, Theorem 6.7]
that the same is true for $\mathcal{E}^{\perp\perp}$ and by [MT05, Lemma
2.3.7] it therefore follows that $\mathcal{E}^{\perp\perp}$ splits off as an
orthogonal direct summand. Since
$\dim_{N}(\mathcal{E}^{\perp\perp}/\mathcal{E})=0$, Sauer’s local criterion
[Sau05, Theorem 2.4] implies that for every $x\in\mathcal{E}^{\perp\perp}$
there exists a sequence of projections $p_{i}\in N$ such that
$\lim_{i}\tau(p_{i})=1$ and $xp_{i}\in\mathcal{E}$. Thus

$\displaystyle\|x-xp_{i}\|_{2}^{2}$ $\displaystyle=\tau\left(\left\langle
x(1-p_{i}),x(1-p_{i})\right\rangle_{N}\right)$
$\displaystyle=\tau\left(\left\langle
x,x\right\rangle_{N}(1-p_{i})^{2}\right)$
$\displaystyle\leqslant\tau\left(\left\langle
x,x\right\rangle_{N}^{2}\right)^{\frac{1}{2}}\tau\left(1-p_{i}\right)^{\frac{1}{2}}\underset{i\to\infty}{\longrightarrow}0.$

Hence
$\mathcal{E}^{\perp\perp}\subseteq\overline{\mathcal{E}}^{\|\cdot\|_{2}}$ and
since $\mathcal{E}\subseteq\mathcal{E}^{\perp\perp}$ this proves that the two
closures coincide.

∎

If $(\mathcal{F},\left\langle\cdot,\cdot\right\rangle_{N})$ is a finitely
generated projective Hilbert $N$-module we will often identify the algebraic
endomorphism ring $\operatorname{End}_{N}(\mathcal{F})$ with
$\operatorname{End}_{N}(L^{2}(\mathcal{F}))$ and thus also consider
$\operatorname{Tr}_{N}(-)$ as defined on
$\operatorname{End}_{N}(\mathcal{F})$. Recall, e.g. from [KW00, Lemma 1.7],
that every finitely generated projective Hilbert $N$-module
$(\mathcal{F},\left\langle\cdot,\cdot\right\rangle_{N})$ has a _basis_ ; that
is there exist $u_{1},\dots,u_{n}\in\mathcal{F}$ such that
$x=\sum_{i=1}^{n}u_{i}\left\langle u_{i},x\right\rangle_{N}$ for each
$x\in\mathcal{F}$.111Note that in general the elements $u_{1},\dots,u_{n}$ are
not linearly independent over $N$ and thus not a basis in the standard sense
of homological algebra. Furthermore, the matrix $p=(\left\langle
u_{i},u_{j}\right\rangle_{N})_{i,j=1}^{n}$ is a projection in
${\mathbb{M}}_{n}(N)$ and the map
$\alpha\colon(\mathcal{F},\left\langle\cdot,\cdot\right\rangle_{N})\to(pN^{n},\left\langle\cdot,\cdot\right\rangle_{\operatorname{st}})$
given by $\alpha(x)=(\left\langle u_{i},x\right\rangle_{N})_{i=1}^{n}$ is a
unitary isomorphism of Hilbert $N$-modules with $u_{i}=\alpha^{-1}(pe_{i})$,
where $e_{1},\dots,e_{n}$ is the standard basis in $N^{n}$. From this we
obtain the following result.

######  Lemma 2.7.

For a finitely generated projective Hilbert $N$-module
$(\mathcal{F},\left\langle\cdot,\cdot\right\rangle_{N})$ and an endomorphism
$T\in\operatorname{End}_{N}(\mathcal{F})$ we have
$\operatorname{Tr}_{N}(T)=\sum_{i=1}^{n}\tau(\left\langle
u_{i},Tu_{i}\right\rangle_{N})$ for any basis $u_{1},\dots,u_{n}$ in
$\mathcal{F}$.

###  2.2. The relative dimension function

Next we consider a trace-preserving inclusion of $N$ into a bigger finite von
Neumann algebra $(M,\tau)$ acting on its GNS-space $H=L^{2}(M,\tau)$. Note
that any $N$-submodule $\mathcal{F}\subseteq M$ acquires an $N$-valued inner
product arising from the trace-preserving conditional expectation $E\colon
M\to N$ by setting $\left\langle a,b\right\rangle_{N}:=E(a^{*}b)$. Consider
now a finitely generated, projective $N$-submodule $\mathcal{F}\subseteq M$
which is complete with respect to this pre-Hilbert module structure. In other
words, the relation $\left\langle a,b\right\rangle_{N}:=E(a^{*}b)$ defines a
Hilbert $N$-module structure on $\mathcal{F}$ and hence $L^{2}(\mathcal{F})$
can be obtained by completion with respect to the ${\mathbb{C}}$-valued inner
product $\tau\circ\left\langle\cdot,\cdot\right\rangle_{N}$. Since $E$ is
trace-preserving, this completion is exactly the closure of $\mathcal{F}$ in
$H$.

######  Definition 2.8.

An $N$-submodule $\mathcal{F}\subseteq M$ which is a Hilbert $N$-module with
respect to the $N$-valued inner product arising from the conditional
expectation $E\colon M\to N$ is called complete.

Let $\mathcal{F}\subseteq M$ be a complete, non-zero, finitely generated
projective $N$-submodule and denote by $P_{F}\in B(H)$ the projection onto its
closure $F$ in $H$. Note that $F$ is a finitely generated normal $N$-module.
Any operator $T\in(JNJ)^{\prime}\subseteq B(H)$ gives rise to an
$N$-equivariant operator $P_{F}TP_{F}|_{F}$ on $F$ and we may therefore define
a normal state $\varphi_{\mathcal{F}}$ on $(JNJ)^{\prime}$ by setting

(2.1)
$\displaystyle\varphi_{\mathcal{F}}(T)=\frac{\operatorname{Tr}_{N}(P_{F}TP_{F}|_{F})}{\dim_{N}(\mathcal{F})}.$

By choosing a basis $u_{1},\dots,u_{k}\in\mathcal{F}$ for the Hilbert
$N$-module structure arising from $E$, the state $\varphi_{\mathcal{F}}$ can
be computed as
$\varphi_{\mathcal{F}}(T)=(\dim_{N}\mathcal{F})^{-1}\sum_{i=1}^{k}\left\langle\Lambda(u_{i}),T\Lambda(u_{i})\right\rangle$
where the inner product is taken in $H$. This is due to the fact that
$P_{F}TP_{F}|_{\mathcal{F}}\in\operatorname{End}_{N}(\mathcal{F})$ and hence
by Lemma 2.7

$\displaystyle\varphi_{\mathcal{F}}(T)$
$\displaystyle=\frac{\sum_{i=1}^{k}\tau(\left\langle
u_{i},P_{F}TP_{F}u_{i}\right\rangle_{N})}{\dim_{N}\mathcal{F}}$
$\displaystyle=\frac{\sum_{i=1}^{k}\tau(E(u_{i}^{*}(P_{F}TP_{F}u_{i}))}{\dim_{N}\mathcal{F}}$
$\displaystyle=\frac{\sum_{i=1}^{k}\left\langle\Lambda(u_{i}),P_{F}TP_{F}\Lambda(u_{i})\right\rangle}{\dim_{N}\mathcal{F}}$
$\displaystyle=\frac{\sum_{i=1}^{k}\left\langle\Lambda(u_{i}),T\Lambda(u_{i})\right\rangle}{\dim_{N}\mathcal{F}}.$

In what follows we will also consider operators acting on a $k$-fold
amplification $H^{k}=\bigoplus_{1}^{k}H$ of the Hilbert space $H$; these
amplifications are always implicitly assumed equipped with the diagonal normal
right action of $N$. For a subspace $L\subseteq H$ we denote by
$L^{k}\subseteq H^{k}$ its amplification.

######  Definition 2.9.

Let $k\in{\mathbb{N}}$ and a complete, non-zero, finitely generated,
projective right $N$-submodule $\mathcal{F}$ in $M$ be given. For an
$N$-submodule $\mathcal{E}\subseteq H^{k}$ we define _the dimension of
$\mathcal{E}$ relative to $\mathcal{F}$_ as

$\dim_{\mathcal{F}}(\mathcal{E})=\frac{\operatorname{Tr}_{N}(P_{F^{k}}P_{E}P_{F^{k}}|_{F^{k}})}{\dim_{N}\mathcal{F}},$

where $F$ and $E$ are the closures in $H^{k}$ of $\mathcal{F}$ and
$\mathcal{E}$, respectively.

Note that this is well-defined since $P_{E}$ commutes with the right action of
$N$ so that
$P_{F^{k}}P_{E}P_{F^{k}}|_{F^{k}}\in\operatorname{End}_{N}(F^{k})$. Note also
the trivial fact that if two submodules have the same closure in $H^{k}$ then
their relative dimensions agree.

######  Proposition 2.10.

The relative dimension function $\dim_{\mathcal{F}}(-)$ has the following
properties.

  * (i)

If $\mathcal{E}_{1}\subseteq\mathcal{E}_{2}\subseteq H^{k}$ are two
$N$-submodules then
$\dim_{\mathcal{F}}(\mathcal{E}_{1})\leqslant\dim_{\mathcal{F}}(\mathcal{E}_{2})$.

  * (ii)

If $\mathcal{E}\subseteq\mathcal{F}^{k}$ then
$\dim_{\mathcal{F}}(\mathcal{E})=\frac{\dim_{N}\mathcal{E}}{\dim_{N}\mathcal{F}}$.
In other words, $\dim_{N}\mathcal{E}=\operatorname{Tr}_{N}(P_{E}|_{F^{k}})$
where $P_{E}$ is the projection onto the closure $E$ of $\mathcal{E}$ in
$H^{k}$.

###### Proof.

For the sake of notational convenience we restrict attention to the case
$k=1$, but the same proof goes through in higher dimensions. The first claim
follows directly from positivity of the state $\varphi_{\mathcal{F}}$. To
prove the second claim, first note that $\mathcal{F}$ splits orthogonally as
$\mathcal{E}^{\perp\perp}\oplus\mathcal{G}$ for some Hilbert $N$-submodule
$\mathcal{G}$. Since $\mathcal{F}$ is assumed to be complete, its
$L^{2}$-completion coincides with its closure $F$ inside $H$ and hence

$F=\overline{\mathcal{E}^{\perp\perp}}\oplus\overline{\mathcal{G}}$

as an orthogonal direct sum inside $H$ and
$L^{2}(\mathcal{E}^{\perp\perp})=\overline{\mathcal{E}^{\perp\perp}}$. Denote
by $P_{\mathcal{E}^{\perp\perp}}\in\operatorname{End}_{N}(\mathcal{F})$ the
projection onto the summand ${\mathcal{E}^{\perp\perp}}$ and by
$P_{\mathcal{E}^{\perp\perp}}^{(2)}\in\operatorname{End}_{N}(F)$ its extension
to $F$. Then clearly $P_{\mathcal{E}^{\perp\perp}}^{(2)}$ projects onto the
summand $\overline{\mathcal{E}^{\perp\perp}}$ which by Lemma 2.6 coincides
with $\overline{\mathcal{E}}$. Thus
$P_{F}P_{E}P_{F}|_{F}=P_{\mathcal{E}^{\perp\perp}}^{(2)}$. By choosing a basis
$u_{1},\dots u_{n}$ for $\mathcal{E}^{\perp\perp}$ and a basis
$v_{1},\dots,v_{l}$ for $\mathcal{G}$ we get a basis for $\mathcal{F}$ and
using this basis to compute the trace we get

$\displaystyle\dim_{\mathcal{F}}\mathcal{E}$
$\displaystyle=\frac{\sum_{i=1}^{n}\left\langle
u_{i},P_{\mathcal{E}^{\perp\perp}}^{(2)}u_{i}\right\rangle+\sum_{j=1}^{l}\left\langle
v_{j},P_{\mathcal{E}^{\perp\perp}}^{(2)}v_{j}\right\rangle}{\dim_{N}\mathcal{F}}$
$\displaystyle=\frac{\sum_{i=1}^{n}\left\langle
u_{i},u_{i}\right\rangle}{\dim_{N}\mathcal{F}}$
$\displaystyle=\frac{\dim_{N}\mathcal{E}^{\perp\perp}}{\dim_{N}\mathcal{F}}$
$\displaystyle=\frac{\dim_{N}\mathcal{E}}{\dim_{N}\mathcal{F}},$

where the last equality follows from Lemma 2.6.

∎

Relative dimension functions were originally introduced by Eckmann [Eck99] in
a topological setting (see also [Ele03]) and the relative dimension function
$\dim_{\mathcal{F}}(-)$ introduced above can be seen as an operator algebraic
analogue of this construction. A similar construction for quantum groups was
considered in [KT09]. We end this section with a small lemma which will turn
out useful in the following.

######  Lemma 2.11.

Let $\mathcal{F}$ be a non-zero, finitely generated, projective $N$-module and
let $\mathcal{S}_{1},\dots,\mathcal{S}_{n}$ be a family of submodules in
$\mathcal{F}$. Suppose that

$\forall
i\in\\{1,\dots,n\\}:\quad\frac{\dim_{N}\mathcal{S}_{i}}{\dim_{N}\mathcal{F}}>1-\varepsilon_{i}$

for some $\varepsilon_{1},\dots,\varepsilon_{n}>0$. Then

$\frac{\dim_{N}\left(\bigcap_{i=1}^{n}\mathcal{S}_{i}\right)}{\dim_{N}\mathcal{F}}>1-\sum_{i=1}^{n}\varepsilon_{i}.$

###### Proof.

By induction it is sufficient to consider the case $n=2$. In this case we have
an exact sequence of $N$-modules

$0\longrightarrow\mathcal{S}_{1}\cap\mathcal{S}_{2}\longrightarrow\mathcal{S}_{1}\oplus\mathcal{S}_{2}\longrightarrow{\mathcal{S}_{1}+\mathcal{S}_{2}}\longrightarrow
0,$

and by [Lüc02, Theorem 6.7] we have
$\dim_{N}(\mathcal{S}_{1}\oplus\mathcal{S}_{2})=\dim_{N}(\mathcal{S}_{1}\cap\mathcal{S}_{2})+\dim_{N}({\mathcal{S}_{1}+\mathcal{S}_{2}})$.
Thus

$\frac{\dim_{N}(\mathcal{S}_{1}\cap\mathcal{S}_{2})}{\dim_{N}\mathcal{F}}=\frac{\dim_{N}\mathcal{S}_{1}+\dim_{N}\mathcal{S}_{2}-\dim_{N}({\mathcal{S}_{1}+\mathcal{S}_{2}})}{\dim_{N}\mathcal{F}}>1-\varepsilon_{1}-\varepsilon_{2}.$

∎

Of course the whole theory has a mirror counterpart for _left_ modules over
$N$ and we will use both theories without further specification throughout the
paper. Moreover, we will not make any notational distinction between the
corresponding dimension functions; so when $\mathcal{X}$ is a _left_
$N$-module $\dim_{N}\mathcal{X}$ will also denote the von Neumann dimension of
$\mathcal{X}$ as a left module and similarly with the relative dimension
functions.

##  3\. The Følner property

Consider again a trace-preserving inclusion of finite von Neumann algebras
$N\subseteq M$ and let $\mathcal{A}$ be an intermediate $\ast$-algebra (i.e.
$N\subseteq\mathcal{A}\subseteq M$) which is strongly dense in $M$. We will
keep this setup fixed throughout the present section and the symbols $N$,
$\mathcal{A}$ and $M$ will therefore refer to objects specified above.
Moreover, $\tau$ will denote the common trace on $N$ and $M$ and $H$ the GNS-
space $L^{2}(M,\tau)$. The aim of this section is to introduce a Følner type
condition for $\mathcal{A}$, relative to the chain
$N\subseteq\mathcal{A}\subseteq M$, and study its basic properties. As already
mentioned, the von Neumann algebra $N$ will be thought of as a “coefficient
algebra”, and we advise the reader to keep the example $N=\mathbb{C}$ in mind
in order to get a more intuitive picture. Recall from Section 2, that an
$N$-submodule $\mathcal{F}$ in $M$ is called _complete_ if it is a Hilbert
module with respect to the $N$-valued inner product arising from the trace-
preserving conditional expectation $E\colon M\to N$.

######  Definition 3.1.

The algebra $\mathcal{A}$ is said to have the strong Følner property with
respect to $N$ if for every finite set $T_{1},\dots,T_{r}\in\mathcal{A}$ and
every $\varepsilon>0$ there exists a complete, non-zero, finitely generated
projective $N$-submodule $\mathcal{F}\subseteq\mathcal{A}$ such that

$\frac{\dim_{N}(T_{i}^{-1}(\mathcal{F})\cap\mathcal{F})}{\dim_{N}(F)}>1-\varepsilon\quad\text{
and }\quad\|\varphi_{\mathcal{F}}-\tau\|_{M_{*}}<\varepsilon$

for all $i\in\\{1,\dots,r\\}$. Here $T_{i}^{-1}(\mathcal{F})$ denotes the
preimage of $\mathcal{F}$ under the left multiplication operator given by
$T_{i}$.

This definition is an operator algebraic analogue of the Følner condition for
discrete groups, where the almost invariant finite subset of the group is
replaced by an almost invariant “finite-dimensional” $N$-submodule
$\mathcal{F}$ in $\mathcal{A}$. In fact, putting $N={\mathbb{C}}$ one can
easily check that the linear span of a subset $F$ in a group $\Gamma$ which is
$\varepsilon$-invariant under the action of another set $S$ gives rise to an
almost invariant submodule $\mathcal{F}$ in the above sense (see Corollary 6.1
for details) for any set of operators in ${\mathbb{C}}\Gamma$ not supported
outside of $S$. The condition regarding the trace approximation is trivially
fulfilled in this case as $\varphi_{\mathcal{F}}(T)=\tau(T)$ for each $T\in
L\Gamma$, a fact due to ${\mathbb{C}}\Gamma$ being spanned by a multiplicative
set of orthogonal unitaries. Since this need not be the case for a general
$\mathcal{A}$, we have to include the approximation property in order to
compare the dimension of an $N$-submodule with the relative dimensions of its
“finite-dimensional approximations”; see Proposition 4.1 for the precise
statement.

######  Remark 3.2.

Strictly speaking, the norm estimate in Definition 3.1 should read
$\|\varphi_{\mathcal{F}}|_{M}-\tau\|_{M_{*}}<\varepsilon$ but for notational
convenience we will consistently suppress the restriction in the sequel. This
should not lead to any confusion as the algebra on which the states are
considered can always be read off the subscript on the norm. Moreover, the
strong Følner property should, more precisely, be called the _right_ strong
Følner property since we have chosen to use _right_ modules in the definition.
However, if ${\mathcal{A}}$ has the right Følner property then it also has the
corresponding left Følner property and vice versa. This can be seen by noting
that if $T_{1},\dots,T_{r}\in{\mathcal{A}}$ and $\varepsilon>0$ are given and
$\mathcal{F}\subseteq\mathcal{A}$ is a right $N$-module satisfying the
conditions in Definition 3.1, then $\mathcal{F}^{*}$ (the adjoint taken inside
$\mathcal{A}$) is a complete, finitely generated projective _left_ Hilbert
$N$-submodule in $M$ (the latter endowed with the inner product $\left\langle
x,y\right\rangle_{N}=E(xy^{*})$) which satisfies the conditions in the left
version of Definition 3.1 for the set of operators
$T_{1}^{*},\dots,T_{r}^{*}$. Note, in particular, that this implies that the
strong Følner property is stable under passing to opposite algebras; i.e. if
the tower $N\subseteq\mathcal{A}\subseteq M$ has the strong Følner property
then the same is true for the tower
$N^{\operatorname{{op}}}\subseteq\mathcal{A}^{\operatorname{{op}}}\subseteq
M^{\operatorname{{op}}}$.

For our purposes the following slight reformulation of the Følner property
will turn out convenient.

######  Proposition 3.3.

The algebra ${\mathcal{A}}$ has the strong Følner property with respect to $N$
iff for any finite set $T_{1},\dots,T_{r}\in\mathcal{A}$ there exists a
sequence of complete, non-zero, finitely generated, projective $N$-modules
$\mathcal{P}_{n}\subseteq\mathcal{A}$ with submodules
$\mathcal{S}_{n}\subseteq\mathcal{P}_{n}$ such that

  1. (i)

$T_{i}(\mathcal{S}_{n})\subseteq\mathcal{P}_{n}$ for each
$i\in\\{1,\dots,r\\}$ and each $n\in\mathbb{N}$;

  2. (ii)

$\displaystyle\lim_{n\to\infty}\frac{\dim_{N}\mathcal{S}_{n}}{\dim_{N}\mathcal{P}_{n}}=1$;

  3. (iii)

The sequence $\varphi_{\mathcal{P}_{n}}$ (restricted to $M$) converges
uniformly to the trace $\tau$; i.e.
$\lim_{n\to\infty}\|\varphi_{\mathcal{P}_{n}}-\tau\|_{M_{*}}=0$.

###### Proof.

Clearly (i)-(iii) imply the Følner property. Conversely, if $\mathcal{A}$ has
the Følner property and $T_{1},\dots,T_{r}\in\mathcal{A}$ is given then for
each $n\in{\mathbb{N}}$ we get a complete, non-zero, finitely generated
projective module $\mathcal{F}_{n}\subseteq\mathcal{A}$ such that

$\frac{\dim_{N}(T_{i}^{-1}(\mathcal{F}_{n})\cap\mathcal{F}_{n})}{\dim_{N}\mathcal{F}_{n}}>1-\frac{1}{rn}\quad\text{
and }\quad\|\varphi_{\mathcal{F}_{n}}-\tau\|_{M_{*}}<\frac{1}{n}.$

Putting $\mathcal{P}_{n}:=\mathcal{F}_{n}$ and
$\mathcal{S}_{n}:=\cap_{i=1}^{r}T_{i}^{-1}(\mathcal{F}_{n})\cap\mathcal{F}_{n}$
we clearly have (i) and (iii) satisfied and (ii) follows from Lemma 2.11. ∎

######  Definition 3.4.

If $\mathcal{A}$ has the strong Følner property and $\\{T_{1},\dots,T_{r}\\}$
is a finite subset in $\mathcal{A}$, then we call a sequence
$(\mathcal{P}_{n},\mathcal{S}_{n})$ with the properties in Proposition 3.3 a
_strong Følner sequence_ for the given set of operators.

The final result in this section shows that the von Neumann dimension can be
approximated by relative dimensions in the presence of a strong Følner
sequence.

######  Proposition 3.5.

Let ${\mathcal{A}}$ have the strong Følner property relative to $N$ and let
$(\mathcal{P}_{n},\mathcal{S}_{n})$ be a strong Følner sequence for an
arbitrary finite set in $\mathcal{A}$. If $K\subseteq H^{k}$ is a closed
subspace which is invariant under the diagonal right action of $M$ then
$\dim_{\mathcal{P}_{n}}K\underset{n\to\infty}{\longrightarrow}\dim_{M}K$.

###### Proof.

Denote by $P_{K}\in{\mathbb{M}}_{k}(M)\subseteq B(H^{k})$ the projection onto
$K$. Fix an $n\in{\mathbb{N}}$ and choose a basis
$u_{1},\dots,u_{l}\in\mathcal{P}_{n}$. Then the set

$\\{u_{i}\otimes e_{j}\mid 1\leqslant i\leqslant l,1\leqslant j\leqslant k\\}$

is a basis for the amplification
$\mathcal{P}_{n}^{k}=\mathcal{P}_{n}\otimes{\mathbb{C}}^{k}$; here
$e_{1},\dots,e_{k}$ denotes the standard basis in ${\mathbb{C}}^{k}$. By
computing the trace in this basis one easily gets

$\dim_{\mathcal{P}_{n}}K=\sum_{j=1}^{k}\varphi_{\mathcal{P}_{n}}((P_{K})_{jj})\underset{n\to\infty}{\longrightarrow}\sum_{j=1}^{k}\tau((P_{K})_{jj})=\dim_{M}K,$

where the convergence follows from Proposition 3.3. ∎

##  4\. Dimension flatness

Throughout this section we consider again the setup consisting of a trace-
preserving inclusion $N\subseteq M$ of tracial von Neumann algebras together
with an intermediate $*$-algebra $N\subseteq\mathcal{A}\subseteq M$ which is
weakly dense in $M$. We will also consider a $k$-fold amplification $H^{k}$ of
the GNS-space $H:=L^{2}(M,\tau)$ and the natural left action of
${\mathbb{M}}_{k}(\mathcal{A})\subseteq{\mathbb{M}}_{k}(M)$ thereon. Our aim
is Theorem 4.4 which roughly says that when $\mathcal{A}$ is strongly Følner
then the ring inclusion $\mathcal{A}\subseteq M$ is flat from the point of
view of dimension theory. Before reaching our goal we need a few preparatory
results.

######  Proposition 4.1.

Assume that $\mathcal{A}$ has the strong Følner property relative to $N$ and
let $T=(T_{ij})\in{\mathbb{M}}_{k}(\mathcal{A})$ be given. If
$(\mathcal{P}_{n},\mathcal{S}_{n})$ is a strong Følner sequence for the set of
matrix entries of $T$, then

$\dim_{M}\ker(T)=\lim_{n\to\infty}\dim_{\mathcal{P}_{n}}\ker(T|_{\mathcal{S}_{n}^{k}}).$

The proof is an extension of the corresponding argument in [Ele03].

###### Proof.

By Lück’s dimension theorem [Lüc02, Theorem 6.7] we have

$\dim_{N}\ker(T|_{\mathcal{S}_{n}^{k}})+\dim_{N}{\operatorname{rg\hskip
1.13791pt}}(T|_{\mathcal{S}_{n}^{k}})=\dim_{N}\mathcal{S}_{n}^{k}=k\dim_{N}\mathcal{S}_{n}.$

Since $T_{ij}\mathcal{S}_{n}\subseteq\mathcal{P}_{n}$ we have
${{\operatorname{rg\hskip
1.13791pt}}(T|_{\mathcal{S}_{n}^{k}})}\subseteq\mathcal{P}_{n}^{k}$ and
$\ker(T|_{\mathcal{S}_{n}^{k}})\subseteq\mathcal{P}_{n}^{k}$ and by the basic
properties of the relative dimension function (Proposition 2.10) we now get

(4.1)
$\displaystyle\dim_{\mathcal{P}_{n}}\ker(T|_{\mathcal{S}_{n}^{k}})+\dim_{\mathcal{P}_{n}}{{\operatorname{rg\hskip
1.13791pt}}(T|_{\mathcal{S}_{n}^{k}}})=k\frac{\dim_{N}\mathcal{S}_{n}}{\dim_{N}\mathcal{P}_{n}}.$

Denote by $P$ the kernel projection of $T\colon H^{k}\to H^{k}$ and by $R$ the
projection onto the closure of its range. By Proposition 3.5, for any
$\varepsilon>0$ we can find $n_{0}\in{\mathbb{N}}$ such that for all
$n\geqslant n_{0}$ we have

(4.2) $\displaystyle a_{n}$
$\displaystyle:=\dim_{\mathcal{P}_{n}}\ker(T|_{\mathcal{S}_{n}^{k}})\leqslant\dim_{\mathcal{P}_{n}}\ker(T)\leqslant\dim_{M}\ker(T)+\varepsilon;$
(4.3) $\displaystyle b_{n}$
$\displaystyle:=\dim_{\mathcal{P}_{n}}{{\operatorname{rg\hskip
1.13791pt}}(T|_{\mathcal{S}_{n}^{k}})}\leqslant\dim_{\mathcal{P}_{n}}\overline{{\operatorname{rg\hskip
1.13791pt}}(T)}\leqslant\dim_{M}\overline{{\operatorname{rg\hskip
1.13791pt}}(T)}+\varepsilon.$

By (4.1) we have $\lim_{n}(a_{n}+b_{n})=k$ and by [Lüc02, Theorem 1.12 (2)]

$\dim_{M}\ker(T)+\dim_{M}\overline{{\operatorname{rg\hskip 1.13791pt}}(T)}=k.$

Our task is to prove that $a_{n}$ converges to $\dim_{M}\ker(T)$. If this were
not the case, by passing to a subsequence we can assume that there exists
$\varepsilon_{0}>0$ and an $n_{1}\in{\mathbb{N}}$ such that
$a_{n}\notin[\dim_{M}\ker(T)-\varepsilon_{0},\dim_{M}\ker(T)+\varepsilon_{0}]$
for $n\geqslant n_{1}$. The estimates (4.2) and (4.3) imply the existence of
an $n_{2}\in{\mathbb{N}}$ such that

$\displaystyle
a_{n}\leqslant\dim_{M}\ker(T)+\frac{\varepsilon_{0}}{2}\quad\text{ and }\quad
b_{n}\leqslant\dim_{M}\overline{{\operatorname{rg\hskip
1.13791pt}}(T)}+\frac{\varepsilon_{0}}{2}$

for $n\geqslant n_{2}$; hence we must have
$a_{n}\leqslant\dim_{M}\ker(T)-\varepsilon_{0}$ for
$n\geqslant{\operatorname{max}}\\{n_{1},n_{2}\\}$. But then from this point on

$a_{n}+b_{n}\leqslant\dim_{M}\ker(T)-\varepsilon_{0}+\dim_{M}\overline{{\operatorname{rg\hskip
1.13791pt}}(T)}+\frac{\varepsilon_{0}}{2}=k-\frac{\varepsilon_{0}}{2},$

contradicting the fact that $\lim_{n}(a_{n}+b_{n})=k$. ∎

Consider again the operator $T\in{\mathbb{M}}_{k}(\mathcal{A})\subseteq
B(H^{k})$ as well as its restriction
$T_{0}\colon\mathcal{A}^{k}\to\mathcal{A}^{k}$.

######  Lemma 4.2.

If $\mathcal{A}$ has the strong Følner property relative to $N$ then the
closure of ${\ker(T_{0})}$ in $H^{k}$ coincides with $\ker(T)$.

###### Proof.

Let $(\mathcal{P}_{n},\mathcal{S}_{n})$ be a strong Følner sequence for the
matrix coefficients of $T$. Denote by $P$ the kernel projection of $T$ and let
$Q$ be the projection onto ${\overline{\ker(T_{0})}}^{\perp}\cap\ker(T)$. We
need to prove that $Q=0$. One easily checks that ${\operatorname{rg\hskip
1.13791pt}}(Q)$ is a finitely generated, normal, right $M$-module and it is
therefore enough to prove that $\dim_{M}{\operatorname{rg\hskip
1.13791pt}}(Q)=0$. Denote by $R$ the projection onto the space
$\overline{\ker(T_{0})}$. Given $\varepsilon>0$, Proposition 4.1 provides an
$n_{0}\in{\mathbb{N}}$ such that for $n\geqslant n_{0}$ we have

$\dim_{M}{\operatorname{rg\hskip
1.13791pt}}(P)=\dim_{M}\ker(T)\leqslant\dim_{\mathcal{P}_{n}}\ker(T|_{\mathcal{S}_{n}^{k}})+\varepsilon\leqslant\dim_{\mathcal{P}_{n}}{\operatorname{rg\hskip
1.13791pt}}(R)+\varepsilon,$

simply because
$\ker(T|_{\mathcal{S}_{n}^{k}})\subseteq\ker(T_{0})\subseteq\overline{\ker(T_{0})}={\operatorname{rg\hskip
1.13791pt}}(R)$. Since $P=R+Q$, Proposition 3.5 implies that we eventually
have

$\displaystyle\dim_{\mathcal{P}_{n}}{\operatorname{rg\hskip
1.13791pt}}(R)+\dim_{\mathcal{P}_{n}}{\operatorname{rg\hskip 1.13791pt}}(Q)$
$\displaystyle=\dim_{\mathcal{P}_{n}}{\operatorname{rg\hskip 1.13791pt}}(P)$
$\displaystyle\leqslant\dim_{M}{\operatorname{rg\hskip
1.13791pt}}(P)+\varepsilon$
$\displaystyle\leqslant\dim_{\mathcal{P}_{n}}{\operatorname{rg\hskip
1.13791pt}}(R)+2\varepsilon$

and hence $\dim_{\mathcal{P}_{n}}{\operatorname{rg\hskip
1.13791pt}}(Q)\leqslant 2\varepsilon$ from a certain point on. Thus
$\dim_{M}{\operatorname{rg\hskip
1.13791pt}}(Q)=\lim_{n}\dim_{\mathcal{P}_{n}}{\operatorname{rg\hskip
1.13791pt}}(Q)=0$. ∎

######  Remark 4.3.

If instead $T_{0}$ is given by _right_ multiplication with a matrix from
${\mathbb{M}}_{k}(\mathcal{A})$ then
$T\in\operatorname{diag}(M)^{\prime}={\mathbb{M}}_{k}(M^{\prime})$ and hence
commutes with the diagonal action of $N$ from the left. By using the obvious
variations of the above results for left modules we therefore also obtain
$\overline{\ker(T_{0})}=\ker(T)$ if $\mathcal{A}$ has the Følner property (see
e.g. Remark 3.2). We will use this variant of the result in the proof of
Theorem 4.4 which is formulated using dimension-theory for _left_ modules over
$M$; this is done in order be consistent with the majority of the references
(e.g. [Lüc02, CS05, Sau05]) on $L^{2}$-Betti numbers in the homological
algebraic context.

We are now ready to state and prove the main theorem of this section. Recall,
that $N\subseteq M$ is a trace-preserving inclusion of finite von Neumann
algebras and that $\mathcal{A}$ denotes an intermediate $*$-algebra which is
weakly dense in $M$.

######  Theorem 4.4 (Dimension flatness).

If $\mathcal{A}$ has the strong Følner property relative to $N$ then the
inclusion $\mathcal{A}\subseteq M$ is dimension flat; that is

$\dim_{M}\operatorname{Tor}_{p}^{\mathcal{A}}(M,X)=0$

for any $p\geqslant 1$ and any left $\mathcal{A}$-module $X$.

Note that if the ring inclusion $\mathcal{A}\subseteq M$ actually were flat
(in the standard sense of homological algebra) then we would have
$\operatorname{Tor}_{p}^{\mathcal{A}}(M,X)=0$ for every left
$\mathcal{A}$-module $X$ and every $p\geqslant 1$. This need not be the case
in our setup222For example, the tower
${\mathbb{C}}\subseteq{\mathbb{C}}[{\mathbb{Z}}\times{\mathbb{Z}}]\subseteq
L({\mathbb{Z}}\times{\mathbb{Z}})$ has the strong Følner property (Corollary
6.1) but is not flat as
$\operatorname{Tor}_{1}^{{\mathbb{C}}[{\mathbb{Z}}\times{\mathbb{Z}}]}(L({\mathbb{Z}}\times{\mathbb{Z}}),{\mathbb{C}})\neq\\{0\\}$.
See e.g. [LRS99, Theorem 3.7]., but from the point of view of the dimension
function it looks as if it were the case — hence the name “dimension-
flatness”. The first part of the proof of Theorem 4.4 consists of a reduction
to the case when $X$ is finitely presented. This part is verbatim identical to
the corresponding proof for groups due to Lück (see [Lüc02, Theorem 6.37]),
but we include it here for the sake of completeness.

###### Proof.

Let an arbitrary left $\mathcal{A}$-module $X$ be given and choose a short
exact sequence $0\to Y\to F\to X\to 0$ of $\mathcal{A}$-modules in which $F$
is free. Then the corresponding long exact $\operatorname{Tor}$-sequence gives

$\operatorname{Tor}_{p+1}^{\mathcal{A}}(M,X)\simeq\operatorname{Tor}_{p}^{\mathcal{A}}(M,Y),$

and hence it suffices to prove that
$\dim_{M}\operatorname{Tor}_{1}^{\mathcal{A}}(M,X)=0$ for arbitrary $X$.
Recall that $\operatorname{Tor}$ commutes with direct limits and that the
dimension function $\dim_{M}(-)$ is also well behaved with respect to direct
limits [Lüc02, Theorem 6.13]); seeing that an arbitrary module is the directed
union of its finitely generated submodules we may therefore assume $X$ to be
finitely generated. Hence we can find a short exact sequence $0\to Y\to F\to
X\to 0$ with $F$ finitely generated and free. The module $Y$ is the directed
union of its system of finitely generated submodules $(Y_{j})_{j\in J}$ and
therefore $X$ can be realized as the direct limit $\varinjlim_{j}F/Y_{j}$.
Since each of the modules $F/Y_{j}$ is finitely presented by construction this
shows that it suffices to treat the case where $X$ is a finitely presented
module. In this case we may therefore choose a presentation of the form

$\mathcal{A}^{k}\overset{T_{0}}{\longrightarrow}\mathcal{A}^{l}\longrightarrow
X\longrightarrow 0.$

This presentation can be continued to a free resolution

$\cdots\overset{S_{0}}{\longrightarrow}\mathcal{A}^{k}\overset{T_{0}}{\longrightarrow}\mathcal{A}^{l}\longrightarrow
X\longrightarrow 0$

of $X$ that can be used to compute the $\operatorname{Tor}$ functor; in
particular we get

$\operatorname{Tor}_{1}^{\mathcal{A}}(M,X)\simeq\frac{\ker(\operatorname{id}_{M}\otimes_{\mathcal{A}}T_{0})}{{\operatorname{rg\hskip
1.13791pt}}(\operatorname{id}_{M}\otimes_{\mathcal{A}}S_{0})}.$

Denote by $T_{0}^{\operatorname{vN}}\colon M^{k}\to M^{l}$ the map induced by
$\operatorname{id}_{M}\otimes_{\mathcal{A}}T_{0}$ under the natural
identification $M\odot_{\mathcal{A}}\mathcal{A}^{i}=M^{i}$ ($i=k,l$) and by
$T\colon H^{k}\to H^{l}$ its continuous extension333Recall that “$\odot$”
denotes the algebraic tensor product and $H$ the GNS-space $L^{2}(M,\tau)$..
Since $M\odot_{\mathcal{A}}-$ is right exact and $S_{0}$ surjects onto
$\ker(T_{0})$, we see that ${\operatorname{rg\hskip
1.13791pt}}(\operatorname{id}_{M}\otimes_{\mathcal{A}}S_{0})\subseteq
M\odot_{\mathcal{A}}\mathcal{A}^{k}$ is identified with the $M$-submodule
$M\ker(T_{0})$ in $M^{k}$ generated by $\ker(T_{0})$; thus

(4.4)
$\displaystyle\dim_{M}\operatorname{Tor}_{1}^{\mathcal{A}}(M,X)=\dim_{M}\ker(T_{0}^{\operatorname{vN}})-\dim_{M}(M\ker(T_{0})).$

We now claim that

(4.5) $\displaystyle\dim_{M}\ker
T_{0}^{\operatorname{vN}}=\dim_{M}\ker(T)\quad\text{ and
}\quad\dim_{M}(M\ker(T_{0}))=\dim_{M}\overline{\ker(T_{0})},$

where the closure is taken in the Hilbert space $H^{k}$. The first equality
follows from the fact that the $L^{2}$-completion functor has an exact and
dimension preserving inverse [Lüc02, Theorem 6.24]. By Lemma 2.6, the
dimension of a submodule in $M^{k}$ coincides with the dimension of its
closure in $H^{k}$, so to prove the the second equality it suffices to prove
that $M\ker(T_{0})$ and $\ker(T_{0})$ have the same closure in $H^{k}$. But
this follows from the following simple approximation argument: Take $x\in M$
and $a\in\ker(T_{0})$ and consider $xa\in M\ker(T_{0})$. By picking a net
$x_{\alpha}\in\mathcal{A}$ converging in the strong operator topology to $x$
we obtain that $\|xa-x_{\alpha}a\|_{2}\to 0$ and $x_{\alpha}a\in\ker(T_{0})$;
hence $M\ker(T_{0})\subseteq\overline{\ker(T_{0})}$ and the proof of (4.5) is
complete. By (4.4) and (4.5) it is therefore sufficient to prove that
$\overline{\ker(T_{0})}=\ker(T)$. The adjoint operator $T^{*}\colon H^{l}\to
H^{k}$ is the extension of the operator
$T_{0}^{*}\colon\mathcal{A}^{l}\to\mathcal{A}^{k}$ which multiplies from the
right with the adjoint of the matrix defining $T_{0}$. From this we obtain
$\ker(T_{0}^{*}T_{0}^{\phantom{*}})=\ker(T_{0})$ and since
$\ker(T^{*}T)=\ker(T)$ it is equivalent to prove that

$\overline{\ker(T_{0}^{*}T_{0}^{\phantom{*}})}=\ker(T^{*}T),$

but this follows from Lemma 4.2 and Remark 4.3. ∎

######  Remark 4.5.

A careful examination of the results in this section reveals that the
dimension-flatness theorem can actually be obtained under slightly less
restrictive assumptions. Namely, the proof goes through as long as
$N\subseteq\mathcal{A}\subseteq M$ satisfies the requirements (i)-(iii) from
Proposition 3.3, but with the uniform convergence in (iii) replaced by weak∗
convergence. However, for the results in the following section the uniform
convergence will be of importance which is the reason for it being included in
the definition of the strong Følner property.

##  5\. Operator algebraic amenability

In this section we explore the connection between the strong Følner property
and the existing operator algebraic notions of amenability. Consider therefore
again a trace-preserving inclusion $N\subseteq M$ of finite von Neumann
algebras and an intermediate $*$-algebra $\mathcal{A}$ which is weakly dense
in $M$. The main goal is to prove the following result:

######  Theorem 5.1.

If $\mathcal{A}$ has the strong Følner property relative to $N$ then $M$ is
amenable relative to $N$. In particular, $M$ is injective if $\mathcal{A}$ has
the strong Følner property and $N$ is injective.

The notion of relative amenability for von Neumann algebras dates back to
Popa’s work in [Pop86]; we briefly recall the basics here following the
exposition in [OP10]. Consider again the trace-preserving inclusion
$N\subseteq M$ of finite von Neumann algebras and denote by $Q:=\langle
M,e_{N}\rangle=(JNJ)^{\prime}\cap B(L^{2}(M))$ the basic construction. Recall
that $Q$ has a unique normal, semifinite tracial weight $\Psi\colon
Q_{+}\to[0,\infty]$ with the property that

$\Psi(ae_{N}b)=\tau(ab)\text{ for all }a,b\in N.$

One way to construct the trace $\Psi$ is as follows: Since any normal
representation of $N^{\operatorname{{op}}}$ is an amplification followed by a
reduction, there exists a projection $q\in B(\ell^{2}({\mathbb{N}})\otimes
L^{2}(N))$ and a right $N$-equivariant unitary identification

$L^{2}(M)=q(\ell^{2}({\mathbb{N}})\otimes L^{2}(N)),$

where the right hand side is endowed with the diagonal right $N$-action. This
induces an identification $Q:=(JNJ)^{\prime}\cap
B(L^{2}(M))=q(B(\ell^{2}({\mathbb{N}})\otimes N))q$ and the trace $\Psi$ is
simply the pull back of the restriction of $\operatorname{Tr}\otimes\tau$
under this identification.

######  Theorem 5.2 ([OP10, Theorem 2.1]).

The following conditions are equivalent:

  * (i)

There exists a state $\varphi\colon Q\to{\mathbb{C}}$ such that
$\varphi|_{M}=\tau$ and $\varphi(xT)=\varphi(Tx)$ for all $x\in M$ and $T\in
Q$.

  * (ii)

There exists a conditional expectation $E\colon Q\to M$.

  * (iii)

There exists a net $\xi_{n}\in L^{2}(Q,\Psi)$ such that
$\lim_{n}\left\langle\xi_{n},x\xi_{n}\right\rangle_{L^{2}(Q,\Psi)}=\tau(x)$
and $\lim_{n}\|[x,\xi_{n}]\|_{2,\Psi}=0$ for every $x\in M$.

If $N\subseteq M$ satisfies one, and hence all, of these conditions then $M$
is said to be amenable relative to $N$ (or $N$ to be coamenable in $M$).

Note that (ii) in the above theorem gives that amenability of $M$ relative to
${\mathbb{C}}$ is equivalent to amenability (a.k.a. injectivity) of $M$. Note
also that if $\mathcal{F}\subseteq M$ is a complete, finitely generated,
projective right $N$-module then the projection $P_{F}$ onto its closure $F$
in $L^{2}(M)$ is an element in $Q$ and

$\dim_{N}(\mathcal{F})=\Psi(P_{F}).$

This follows from the construction of $\Psi$, the equality
$\dim_{N}(\mathcal{F})=\dim_{N}(F)$ and the definition of the von Neumann
dimension for normal $N$-modules; see [Lüc02, Definition 1.8] and the comments
following it. More generally, for any operator $T\in Q$ we have
$\Psi(P_{F}TP_{F})=\operatorname{Tr}_{N}(P_{F}TP_{F})$ where
$\operatorname{Tr}_{N}$ is the trace on $\operatorname{End}_{N}(F)$ considered
in Section 2 (see also Lemma 2.2). We are now ready to give the proof of
Theorem 5.1

###### Proof of Theorem 5.1..

Assume that $N\subseteq\mathcal{A}\subseteq M$ has the Følner property. Since
$M_{*}$ is assumed separable the unit ball $(M)_{1}$ is separable and
metrizable for the strong operator topology and by Kaplansky’s density theorem
the unit ball $(\mathcal{A})_{1}$ is strongly dense in $(M)_{1}$. We may
therefore choose a sequence $\\{T_{i}\\}_{i=1}^{\infty}$ in
$(\mathcal{A})_{1}$ which is strongly dense in $(M)_{1}$ and upon adding
further operators to this sequence we may assume that
$\\{T_{i}\\}_{i=1}^{\infty}$ is $*$-stable. Since $\mathcal{A}$ satisfies the
strong Følner condition we can, for each $T_{1},\dots,T_{n}$, find a complete,
finitely generated, projective $N$-submodule
$\mathcal{P}_{n}\subseteq\mathcal{A}$ and a submodule
$\mathcal{S}_{n}\subseteq\mathcal{P}_{n}$ such that

  * •

$T_{i}(\mathcal{S}_{n})\subseteq\mathcal{P}_{n}$ for all
$i\in\\{1,\dots,n\\}$,

  * •

$\frac{\dim_{N}\mathcal{S}_{n}}{\dim_{N}\mathcal{P}_{n}}\geqslant
1-\frac{1}{n}$,

  * •

$\|\varphi_{\mathcal{P}_{n}}-\tau\|_{M_{*}}\leqslant\frac{1}{n}$.

In what follows we denote by $P_{n}\in B(H)$ the orthogonal projection onto
the closure (in $H$) of $\mathcal{P}_{n}$, by $S_{n}$ the orthogonal
projection onto the closure of $\mathcal{S}_{n}$ and by $S_{n}^{\perp}$ the
difference $P_{n}-S_{n}$. Since $\mathcal{P}_{n}$ is complete and finitely
generated projective, the discussion preceding the proof implies that
$\Psi(P_{n})=\dim_{N}\mathcal{P}_{N}$; in particular $P_{n}\in L^{2}(Q,\Psi)$.
We aim at proving that the unit vectors

$\xi_{n}:=\frac{1}{\sqrt{\dim_{N}\mathcal{P}_{n}}}P_{n}\in L^{2}(Q,\Psi)$

satisfy condition (iii) of Theorem 5.2. The trace approximation is automatic
since

$\left\langle\xi_{n},x\xi_{n}\right\rangle_{L^{2}(Q,\Psi)}=\frac{\Psi(P_{n}xP_{n})}{\dim_{N}\mathcal{P}_{n}}=\frac{\operatorname{Tr}_{N}(P_{n}xP_{n})}{\dim_{N}\mathcal{P}_{n}}=\varphi_{\mathcal{P}_{n}}(x)\underset{n\to\infty}{\longrightarrow}\tau(x)$

for any $x\in M$. To prove the asymptotic commutation property, consider first
$x=T_{i_{1}}\in\\{T_{i}\\}_{i=1}^{\infty}$. Then $x^{*}=T_{i_{2}}$ for some
$i_{2}\in{\mathbb{N}}$ so for for
$n\geqslant{\operatorname{max}}\\{i_{1},i_{2}\\}$ we have $P_{n}xS_{n}=xS_{n}$
and $P_{n}x^{*}S_{n}=x^{*}S_{n}$. Hence

$\displaystyle\|xP_{n}-P_{n}x\|_{2,\Psi}^{2}$
$\displaystyle=\Psi((xP_{n}-P_{n}x)^{*}(xP_{n}-P_{n}x))$
$\displaystyle=\Psi(P_{n}x^{*}xP_{n}-P_{n}x^{*}P_{n}x-x^{*}P_{n}xP_{n}+x^{*}P_{n}x)$
$\displaystyle=\Psi(P_{n}x^{*}xP_{n}-P_{n}x^{*}P_{n}xP_{n}-P_{n}xP_{n}x^{*}P_{n}+P_{n}xx^{*}P_{n})$
$\displaystyle=\Psi\left((P_{n}x^{*}xP_{n}-P_{n}x^{*}P_{n}xP_{n}-P_{n}xP_{n}x^{*}P_{n}+P_{n}xx^{*}P_{n})S_{n}^{\perp}\right)$
$\displaystyle=\Psi\left(S_{n}^{\perp}(P_{n}x^{*}xP_{n}-P_{n}x^{*}P_{n}xP_{n}-P_{n}xP_{n}x^{*}P_{n}+P_{n}xx^{*}P_{n})S_{n}^{\perp}\right)$
$\displaystyle=\left|\left\langle
S_{n}^{\perp},(P_{n}x^{*}xP_{n}-P_{n}x^{*}P_{n}xP_{n}-P_{n}xP_{n}x^{*}P_{n}+P_{n}xx^{*}P_{n})S_{n}^{\perp}\right\rangle_{L^{2}(Q,\Psi)}\right|$
$\displaystyle\leqslant\|S_{n}^{\perp}\|_{2,\Psi}\|(P_{n}x^{*}xP_{n}-P_{n}x^{*}P_{n}xP_{n}-P_{n}xP_{n}x^{*}P_{n}+P_{n}xx^{*}P_{n})S_{n}^{\perp}\|_{2,\Psi}$
$\displaystyle\leqslant\|S_{n}^{\perp}\|_{2,\Psi}^{2}\|P_{n}x^{*}xP_{n}-P_{n}x^{*}P_{n}xP_{n}-P_{n}xP_{n}x^{*}P_{n}+P_{n}xx^{*}P_{n}\|_{\infty}$
$\displaystyle\leqslant 4\Psi(S_{n}^{\perp})$
$\displaystyle=4(\dim_{N}\mathcal{P}_{n}-\dim_{N}\mathcal{S}_{n}).$

Hence

$\|[x,\xi_{n}]\|_{2,\Psi}=\frac{\|xP_{n}-P_{n}x\|_{2,\Psi}}{\sqrt{\dim_{N}\mathcal{P}_{n}}}\leqslant
2\sqrt{\frac{\dim_{N}\mathcal{P}_{n}-\dim_{N}\mathcal{S}_{n}}{\dim_{N}\mathcal{S}_{n}}}\underset{n\to\infty}{\longrightarrow}0.$

Now the general case follows from this and an approximation argument: let
$x\in M$ and ${\varepsilon}>0$ be given and assume without loss of generality
that $\|x\|_{\infty}\leqslant 1$. Choose a net
$x_{\alpha}\in\\{T_{i}\\}_{i=1}^{\infty}$ converging strongly to $x$. Then
$x_{\alpha}$ also converges to $x$ in $L^{2}(M,\tau)$ and we may therefore
choose an $\alpha$ such that

$\|x-x_{\alpha}\|_{2,\tau}<\varepsilon.$

We now have

$\|x\xi_{n}-\xi_{n}x\|_{2,\Psi}\leqslant\|(x-x_{\alpha})\xi_{n}\|_{2,\Psi}+\|x_{\alpha}\xi_{n}-\xi_{n}x_{\alpha}\|_{2,\Psi}+\|\xi_{n}(x_{\alpha}-x)\|_{2,\Psi}$

Considering the first term we get

$\displaystyle\|(x-x_{\alpha})\xi_{n}\|_{2,\Psi}^{2}$
$\displaystyle=\frac{\Psi(P_{n}(x-x_{\alpha})^{*}(x-x_{\alpha})P_{n})}{\dim_{N}\mathcal{P}_{n}}$
$\displaystyle=\varphi_{\mathcal{P}_{n}}((x-x_{\alpha})^{*}(x-x_{\alpha}))$
$\displaystyle\leqslant|(\varphi_{\mathcal{P}_{n}}-\tau)((x-x_{\alpha})^{*}(x-x_{\alpha}))|+\tau((x-x_{\alpha})^{*}(x-x_{\alpha}))$
$\displaystyle\leqslant\|\varphi_{\mathcal{P}_{n}}-\tau\|_{M_{*}}\|x-x_{\alpha}\|_{\infty}^{2}+\|x-x_{\alpha}\|_{2,\tau}^{2}$
$\displaystyle\leqslant\frac{2}{n}+\varepsilon^{2}.$

Thus, for a suitably chosen $n_{1}\in{\mathbb{N}}$ we have
$\|(x-x_{\alpha})\xi_{n}\|_{2,\Psi}^{2}\leqslant 2{\varepsilon}$ for all
$n\geqslant n_{1}$. Considering the third term we get, in a completely similar
manner, an $n_{3}\in{\mathbb{N}}$ such that
$\|\xi_{n}(x_{\alpha}-x)\|_{2,\Psi}\leqslant 2\varepsilon$ for $n\geqslant
n_{3}$. Since $x_{\alpha}\in\\{T_{i}\\}_{i=1}^{\infty}$, the second term
converges to zero by what was shown in the first part of the proof, and hence
there exists an $n_{2}\in{\mathbb{N}}$ such that
$\|x_{\alpha}\xi_{n}-\xi_{n}x_{\alpha}\|_{2,\Psi}\leqslant\varepsilon$ for
$n\geqslant n_{2}$. Thus, for
$n\geqslant{\operatorname{max}}\\{n_{1},n_{2},n_{3}\\}$ we have
$\|x\xi_{n}-\xi_{n}x\|_{2,\Psi}\leqslant 5\varepsilon$ and since
$\varepsilon>0$ was arbitrary this completes the proof of the amenability of
$M$ relative to $N$.

We now just have to prove the final claim regarding the injectivity of $M$. So
assume that $N$ is injective and, as before, that $\mathcal{A}\subseteq M$
satisfies the Følner condition relative to $N$. Injectivity of $N$ simply
means that $N$ is amenable relative to the subalgebra ${\mathbb{C}}\subseteq
N$ and by what was just proven $M$ is amenable relative to $N$ as well. By
[Pop86, Theorem 3.2.4] this means that $M$ is also amenable relative to
${\mathbb{C}}$; i.e $M$ is injective.

∎

##  6\. Examples and applications to $L^{2}$-Betti numbers

The first goal in this section is to show how our techniques can be used to
unify the dimension-flatness results known for amenable groups and quantum
groups and, more generally, to provide a vanishing result in the context of
operator algebraic $L^{2}$-Betti numbers. Secondly, in order to convince the
reader that the class of algebras with the strong Følner property is rich and
extends beyond the ones arising from amenable groups an quantum groups, we
give a number of such examples and show how they yield results about vanishing
of $L^{2}$-Betti numbers in various situations. As a first application we re-
obtain the original dimension flatness result of Lück [Lüc02, Theorem 6.37].

######  Corollary 6.1 ($L^{2}$-Betti numbers of amenable groups).

If $\Gamma$ is a countable, discrete, amenable group then ${\mathbb{C}}\Gamma$
has the strong Følner property with respect to the tower
${\mathbb{C}}\subseteq{\mathbb{C}}\Gamma\subseteq L\Gamma$. In particular, the
inclusion ${\mathbb{C}}\Gamma\subseteq L\Gamma$ is dimension flat and
$\beta_{p}^{(2)}(\Gamma)=0$ for $p\geqslant 1$.

###### Proof.

Let $T_{1},\dots,T_{r}\in{\mathbb{C}}\Gamma$ be given and put
$S=\bigcup_{i=1}^{r}\operatorname{supp}(T_{i})\subseteq\Gamma$. Since $\Gamma$
is amenable, we can find (see e.g. [BP92, F.6.8]) a sequence of finite subset
$F_{n}\subseteq\Gamma$ such that

$\frac{\operatorname{card}(\partial_{S}(F_{n}))}{\operatorname{card}(F_{n})}\underset{n\to\infty}{\longrightarrow}0,$

where $\partial_{S}(F):=\\{\gamma\in F\mid\exists s\in S:s\gamma\notin F\\}.$
Then putting $\mathcal{P}_{n}={\mathbb{C}}[F_{n}]\subseteq{\mathbb{C}}\Gamma$
and $\mathcal{S}_{n}={\mathbb{C}}[F_{n}\setminus\partial_{S}(F_{n})]$ we
clearly have $T_{i}(\mathcal{S}_{n})\subseteq\mathcal{P}_{n}$ for all
$i\in\\{1,\dots,r\\}$ and

$\frac{\dim_{{\mathbb{C}}}\mathcal{S}_{n}}{\dim_{\mathbb{C}}\mathcal{P}_{n}}=\frac{\operatorname{card}(F_{n})-\operatorname{card}(\partial_{S}(F))}{\operatorname{card}(F_{n})}\underset{n\to\infty}{\longrightarrow}1.$

Denoting by $\rho$ the right regular representation of $\Gamma$ we get for
$T\in L\Gamma$

$\displaystyle\varphi_{\mathcal{F}}(T)=\frac{\sum_{\gamma\in
F}\left\langle\delta_{\gamma},T\delta_{\gamma}\right\rangle}{\operatorname{card}(F)}=\frac{\sum_{\gamma\in
F}\left\langle\delta_{e},\rho_{\gamma}^{*}T\rho_{\gamma}\delta_{e}\right\rangle}{\operatorname{card}(F)}=\frac{\sum_{\gamma\in
F}\left\langle\delta_{e},T\delta_{e}\right\rangle}{\operatorname{card}(F)}=\tau(T).$

Hence ${\mathbb{C}}\Gamma$ has the strong Følner property and dimension
flatness therefore follows from Theorem 4.4. The fact that the $L^{2}$-Betti
numbers vanish in positive degrees follows from this since

$\beta_{p}^{(2)}(\Gamma)=\dim_{L\Gamma}\operatorname{Tor}_{p}^{{\mathbb{C}}\Gamma}(L\Gamma,{\mathbb{C}})=0.\qed$

Next we turn our attention to the class of compact/discrete quantum groups.
Since this is a bit of an aside we shall not elaborate on the basics of
quantum group theory; for back ground material the reader is referred to the
detailed survey articles [Wor98], [KT99] and [MVD98]. In what follows, we
denote by $\widehat{{\mathbb{G}}}$ a discrete quantum group of Kac type and by
${\mathbb{G}}=(C({\mathbb{G}}),\Delta_{\mathbb{G}})$ its compact dual.
Associated with ${\mathbb{G}}$ is a Hopf $*$-algebra
${\operatorname{Pol}}({\mathbb{G}})$ which is naturally included into a finite
von Neumann algebra $L^{\infty}({\mathbb{G}})$. Furthermore, the von Neumann
algebra $L^{\infty}({\mathbb{G}})$ carries a natural trace $\tau$, namely the
Haar state associated with ${\mathbb{G}}$. In [Kye08a] and [Kye08b]
$L^{2}$-Betti numbers were studied in the context of discrete quantum groups
of Kac type and we re-obtain [Kye08a, Theorem 6.1] by setting
$N={\mathbb{C}}$, $\mathcal{A}={\operatorname{Pol}}({\mathbb{G}})$ and
$M=L^{\infty}({\mathbb{G}})$:

######  Corollary 6.2 ($L^{2}$-Betti numbers of amenable quantum groups).

If $\widehat{{\mathbb{G}}}$ is a discrete amenable quantum group of Kac type
and ${\mathbb{G}}$ is its compact dual then
${\operatorname{Pol}}({\mathbb{G}})$ has the strong Følner property with
respect to the tower
${\mathbb{C}}\subseteq{\operatorname{Pol}}({\mathbb{G}})\subseteq
L^{\infty}({\mathbb{G}})$. In particular, the inclusion
${\operatorname{Pol}}({\mathbb{G}})\subseteq L^{\infty}({\mathbb{G}})$ is
dimension flat and $\beta_{p}^{(2)}(\widehat{{\mathbb{G}}})=0$ for $p\geqslant
1$.

###### Proof.

Choose a complete set $(u^{\alpha})_{\alpha\in I}$ of representatives for the
set of equivalence classes of irreducible unitary corepresentations of
${\mathbb{G}}$ and denote by $n_{\alpha}$ the matrix size of $u^{\alpha}$.
Recall that the corresponding matrix coefficients $u_{ij}^{\alpha}$ constitute
a linear basis for ${\operatorname{Pol}}({\mathbb{G}})$ and that the
normalized matrix coefficients $\sqrt{n_{\alpha}}u_{ij}^{\alpha}$ constitute
an orthonormal basis in
$L^{2}({\mathbb{G}}):=L^{2}({\operatorname{Pol}}({\mathbb{G}}),\tau)$. Let
$T_{1},\dots,T_{r}\in{\operatorname{Pol}}({\mathbb{G}})$ and $\varepsilon>0$
be given and denote by $S$ the joint support of $T_{1},\dots,T_{r}$; that is

$S=\\{u^{\gamma}\mid\exists l\in\\{1,\dots,r\\}\exists
p,q\in\\{1,\dots,n_{\gamma}\\}:\tau(u_{pq}^{\gamma*}T_{l})\neq 0\\}.$

In other words, $S$ consists of all the irreducible corepresentations that has
at least one of its matrix coefficients entering in the linear expansion of
one of the operators in question. According to the quantum Følner condition
[Kye08a, Corollary 4.10] we can find a sequence of finite subset
$F_{k}\subseteq(u^{\alpha})_{\alpha\in I}$ such that

$\sum_{u^{\alpha}\in\partial_{S}(F_{k})}n_{\alpha}^{2}<\frac{1}{k}\sum_{u^{\alpha}\in
F_{k}}n_{\alpha}^{2},$

where the boundary $\partial_{S}(F)$ is as in [Kye08a, Definition 3.2]. Define

$\displaystyle\mathcal{P}_{k}$
$\displaystyle:={\operatorname{span}}_{\mathbb{C}}\\{u_{ij}^{\alpha}\mid
u^{\alpha}\in F,1\leqslant i,j\leqslant n_{\alpha}\\};$
$\displaystyle\mathcal{S}_{k}$
$\displaystyle:={\operatorname{span}}_{\mathbb{C}}\\{u_{ij}^{\alpha}\mid
u^{\alpha}\in F\setminus\partial_{S}(F),1\leqslant i,j\leqslant
n_{\alpha}\\}.$

By construction we now have $T_{l}(\mathcal{S}_{k})\subseteq\mathcal{P}_{k}$
for every $l\in\\{1,\dots,r\\}$ and that

$\frac{\dim_{\mathbb{C}}\mathcal{S}_{k}}{\dim_{\mathbb{C}}\mathcal{P}_{k}}=\frac{\sum_{u^{\alpha}\in
F_{k}\setminus\partial_{S}(F_{k})}n_{\alpha}^{2}}{\sum_{\alpha\in
F_{k}}n_{\alpha}^{2}}\underset{n\to\infty}{\longrightarrow}1.$

Furthermore, denoting by $P_{\mathcal{P}_{n}}\in B(L^{2}({\mathbb{G}}))$ the
projection onto the closed subspace $\mathcal{P}_{n}\subseteq
L^{2}({\mathbb{G}})$ we get

$\displaystyle\operatorname{Tr}_{\mathbb{C}}(P_{\mathcal{P}_{n}}TP_{\mathcal{P}_{n}})$
$\displaystyle=\sum_{\alpha\in
F}\sum_{i,j=1}^{n_{\alpha}}\left\langle\sqrt{n_{\alpha}}u_{ij}^{\alpha},T\sqrt{n_{\alpha}}u_{ij}^{\alpha}\right\rangle$
$\displaystyle=\sum_{\alpha\in
F}\sum_{i,j=1}^{n_{\alpha}}n_{\alpha}\tau(u_{ij}^{\alpha*}Tu_{ij}^{\alpha})$
$\displaystyle=\sum_{\alpha\in
F}n_{\alpha}\tau\left(\left(\sum_{i,j=1}^{n_{\alpha}}u_{ij}^{\alpha}u_{ij}^{\alpha*}\right)T\right)$
$\displaystyle=\tau(T)\sum_{\alpha\in F}n_{\alpha}^{2}$
$\displaystyle=\tau(T)\dim_{{\mathbb{C}}}\mathcal{P}_{n},$

for every $T\in L^{\infty}({\mathbb{G}})$; hence
$\varphi_{\mathcal{P}_{n}}(T)=\tau(T)$. Thus
${\operatorname{Pol}}({\mathbb{G}})$ has the strong Følner property and
dimension flatness follows from Theorem 4.4. The vanishing of the
$L^{2}$-Betti numbers follows from dimension flatness since

$\beta_{p}^{(2)}(\widehat{{\mathbb{G}}}):=\dim_{L^{\infty}({\mathbb{G}})}\operatorname{Tor}_{p}^{{\operatorname{Pol}}({\mathbb{G}})}(L^{\infty}({\mathbb{G}}),{\mathbb{C}})=0.\qed$

Note that Corollary 6.1 is actually just a special case of Corollary 6.2, but
since we expect most readers to be more familiar with groups than quantum
groups we singled out this case in form of Corollary 6.1 and its proof.

######  Remark 6.3.

From Theorem 5.1 and Corollary 6.1 we re-obtain the classical fact that the
group von Neumann algebra $L\Gamma$ is hyperfinite when $\Gamma$ is an
amenable discrete group. Furthermore, we see that the group algebra
${\mathbb{C}}\Gamma$ associated with a discrete group $\Gamma$ has the strong
Følner property (relative to the chain
${\mathbb{C}}\subseteq{\mathbb{C}}\Gamma\subseteq L\Gamma$) if and only if
$\Gamma$ is amenable: We already saw in Corollary 6.1 that amenability implies
the strong Følner property. Conversely, if ${\mathbb{C}}\Gamma$ has the strong
Følner property then $L\Gamma$ is hyperfinite by Theorem 5.1 and thus $\Gamma$
is amenable (see e.g. [BO08, Theorem 2.6.8 and 9.3.3]). By a similar argument,
using [Rua96, Theorem 4.5] and Corollary 6.2, we obtain that a discrete
quantum group $\widehat{{\mathbb{G}}}$ of Kac type is amenable if and only if
the dual Hopf $*$-algebra ${\operatorname{Pol}}({\mathbb{G}})$ has the strong
Følner property relative to the chain
${\mathbb{C}}\subseteq{\operatorname{Pol}}({\mathbb{G}})\subseteq
L^{\infty}({\mathbb{G}})$. Thus, for algebras arising from groups and quantum
groups the strong Følner condition coincides exactly with the notion of
amenability of the underlying object.

Next we take a step up in generality by considering $L^{2}$-Betti numbers in a
purely operator algebraic context. In [CS05], Connes and Shlyakhtenko
introduce a notion of $L^{2}$-homology and $L^{2}$-Betti numbers for a dense
$*$-subalgebra in a tracial von Neumann algebra $(M,\tau)$. More precisely, if
$\mathcal{A}\subseteq M$ is a weakly dense unital $*$-subalgebra its
$L^{2}$-homology and $L^{2}$-Betti numbers are defined, respectively, as

$H_{p}^{(2)}(\mathcal{A})=\operatorname{Tor}_{p}^{\mathcal{A}\odot\mathcal{A}^{\operatorname{{op}}}}(M\bar{\otimes}M^{{\operatorname{{op}}}},\mathcal{A})\
\text{ and }\
\beta_{p}^{(2)}(\mathcal{A},\tau)=\dim_{M\bar{\otimes}M^{{\operatorname{{op}}}}}H^{(2)}_{p}(\mathcal{A}).$

This definition extends the definition for groups [CS05, Proposition 2.3] by
means of the formula
$\beta_{p}^{(2)}({\mathbb{C}}\Gamma,\tau)=\beta_{p}^{(2)}(\Gamma)$ and it also
fits with the notion for quantum groups studied above [Kye08b, Theorem 4.1].
In order to apply our techniques in the Connes-Shlyakhtenko setting we first
need to prove that the strong Følner property is preserved under forming
algebraic tensor products.

######  Proposition 6.4.

Let $\mathcal{A}$ and $\mathcal{B}$ be dense $*$-subalgebras in the tracial
von Neumann algebras $(M,\tau)$ and $(N,\rho)$, respectively. If $\mathcal{A}$
and $\mathcal{B}$ have the strong Følner property with respect to the towers
${\mathbb{C}}\subseteq\mathcal{A}\subseteq M$ and
${\mathbb{C}}\subseteq\mathcal{B}\subseteq N$ then so does
$\mathcal{A}\odot\mathcal{B}$ with respect to the tower
${\mathbb{C}}\subseteq\mathcal{A}\odot\mathcal{B}\subseteq M\bar{\otimes}N$.

###### Proof.

Let $T_{1},\dots,T_{r}\in\mathcal{A}\odot\mathcal{B}$ be given and write each
$T_{i}$ as $\sum_{k=1}^{N_{i}}a_{k}^{(i)}\otimes b_{k}^{(i)}$. Choose a strong
Følner sequence $\mathcal{P}_{n}^{\prime}\subseteq\mathcal{A}$, with
associated subspaces $\mathcal{S}_{n}^{\prime}$, for the $a_{k}^{(i)}$’s and a
strong Følner sequence $\mathcal{P}_{n}^{\prime\prime}\subseteq\mathcal{B}$,
with associated subspaces $\mathcal{S}_{n}^{\prime\prime}$, for the
$b_{k}^{(i)}$’s . Putting
$\mathcal{P}_{n}:=\mathcal{P}_{n}^{\prime}\odot\mathcal{P}_{n}^{\prime\prime}$
and
$\mathcal{S}_{n}:=\mathcal{S}_{n}^{\prime}\otimes\mathcal{S}_{n}^{\prime\prime}$
we clearly have that the sequence $(\mathcal{P}_{n},\mathcal{S}_{n})$
satisfies the first to requirements in Proposition 3.3, and since
$\varphi_{\mathcal{P}_{n}}=\varphi_{\mathcal{P}_{n}^{\prime}}\otimes\varphi_{\mathcal{P}_{n}^{\prime\prime}}$
we get

$\displaystyle\|\varphi_{\mathcal{P}_{n}}-\tau\otimes\rho\|_{(M\bar{\otimes}N)_{*}}$
$\displaystyle=\|\varphi_{\mathcal{P}_{n}^{\prime}}\otimes\varphi_{\mathcal{P}_{n}^{\prime\prime}}-\tau\otimes\varphi_{\mathcal{P}_{n}^{\prime\prime}}+\tau\otimes\varphi_{\mathcal{P}_{n}^{\prime\prime}}-\tau\otimes\rho\|_{(M\bar{\otimes}N)_{*}}$
$\displaystyle\leqslant\|\varphi_{\mathcal{P}_{n}^{\prime}}-\tau\|_{M_{*}}+\|\rho-\varphi_{\mathcal{P}_{n}^{\prime\prime}}\|_{N_{*}}\underset{n\to\infty}{\longrightarrow}0.$

Thus, $(\mathcal{P}_{n},\mathcal{S}_{n})$ is a strong Følner sequence for
$T_{1},\dots,T_{r}$. ∎

######  Corollary 6.5.

If $\mathcal{A}$ has the strong Følner property with respect to the tower
${\mathbb{C}}\subseteq\mathcal{A}\subseteq M$ then the Connes-Shlyakhtenko
$L^{2}$-Betti numbers of $(\mathcal{A},\tau)$ vanish in positive degrees.

###### Proof.

Since $\mathcal{A}$ is strongly Følner so is the opposite algebra
$\mathcal{A}^{\operatorname{{op}}}$ (see e.g. Remark 3.2) and by Proposition
6.4 the same is true for $\mathcal{A}\odot\mathcal{A}^{\operatorname{{op}}}$.
Hence by Theorem 4.4 the inclusion
$\mathcal{A}\odot\mathcal{A}^{\operatorname{{op}}}\subseteq
M\bar{\otimes}M^{\operatorname{{op}}}$ is dimension flat so in particular

$\beta^{(2)}_{p}(\mathcal{A},\tau)=\dim_{M\bar{\otimes}M^{\operatorname{{op}}}}\operatorname{Tor}_{p}^{\mathcal{A}\otimes\mathcal{A}^{\operatorname{{op}}}}(M\bar{\otimes}M^{\operatorname{{op}}},\mathcal{A})=0$

for all $p\geqslant 1$. ∎

######  Corollary 6.6 (Actions of amenable groups on probability spaces).

Let $\Gamma$ be a discrete, countable amenable group acting on a probability
space $(X,\mu)$ in a probability measure-preserving way and denote by $\alpha$
the induced action $\Gamma\to\operatorname{Aut}(L^{\infty}(X))$. Consider the
crossed product von Neumann algebra $M=L^{\infty}(X)\rtimes\Gamma$ and its
subalgebras $N=L^{\infty}(X)$ and
$\mathcal{A}=L^{\infty}(X)\rtimes_{\mathrm{alg}}\Gamma$. Then $\mathcal{A}$
has the strong Følner property relative to $N$ and hence the inclusion
$L^{\infty}(X)\rtimes_{\mathrm{alg}}\Gamma\subseteq
L^{\infty}(X)\rtimes\Gamma$ is dimension-flat. Moreover, if the action of
$\Gamma$ is essentially free, then the $L^{2}$-Betti numbers (in the sense of
Sauer [Sau05]) of the groupoid defined by the action vanish in positive
degrees.

We denote by $(u_{g})_{g\in\Gamma}$ the natural unitaries on $L^{2}(X)$
implementing the action; i.e. $\alpha_{g}(a)=u_{g}^{\phantom{*}}au_{g}^{*}$
for all $g\in\Gamma$ and all $a\in L^{\infty}(X)$.

###### Proof.

For any finite set $F\subseteq\Gamma$, the right $L^{\infty}(X)$-module

$\mathcal{F}=\left.\left\\{\sum_{g\in F}a_{g}u_{g}\,\right|\,a_{g}\in
L^{\infty}(X)\right\\}$

is free of dimension $\operatorname{card}(F)$ and the inner product on
$\mathcal{F}$ arising from the conditional expectation $E\colon M\to N$ is
given by

$\left\langle\sum_{g\in F}a_{g}u_{g},\sum_{g\in
F}b_{g}u_{g}\right\rangle_{N}=\sum_{g\in
F}\alpha_{g^{-1}}(a_{g}^{*}b_{g}^{\phantom{*}}).$

From this it follows that the map
$(L^{\infty}(X)^{\operatorname{card}(F)},\left\langle\cdot,\cdot\right\rangle_{\operatorname{st}})\to(\mathcal{F},\left\langle\cdot,\cdot\right\rangle_{N})$
given by $(a_{g})_{g\in F}\mapsto\sum_{g\in F}a_{g}u_{g}$ is a unitary
isomorphism and that $u_{1},\dots,u_{n}$ is a basis for $\mathcal{F}$. In
particular, $\mathcal{F}$ is complete and finitely generated projective. The
trace-approximation condition is automatic, because for $T\in M$ we have

$\varphi_{\mathcal{F}}(T)=\frac{\sum_{g\in F}\left\langle
u_{g},Tu_{g}\right\rangle}{\dim_{N}\mathcal{F}}=\frac{\sum_{g\in
F}\tau(u_{g}^{*}Tu_{g}^{\phantom{*}})}{\operatorname{card}(F)}=\tau(T).$

Hence the strong Følner condition for $\mathcal{A}$ relative to $N$ follows
from the classical Følner condition for $\Gamma$ and the inclusion
$L^{\infty}(X)\rtimes_{\mathrm{alg}}\Gamma\subseteq
L^{\infty}(X)\rtimes\Gamma$ is therefore dimension-flat. In the case of an
essentially free action we combine this with [Sau05, Lemma 5.4] and get that
the $L^{2}$-Betti numbers of the groupoid defined by an action of an amenable
group vanish. ∎

######  Remark 6.7.

Note that the last part of the statement in Corollary 6.6 is a well known
result which, for instance, follows from combining [Sau05, Theorem 5.5] and
[CG86, Theorem 0.2].

Next we generalize the above result to the setting of amenable discrete
measured groupoids; $L^{2}$-Betti numbers for discrete measured groupoids were
introduced by Sauer [Sau05] and generalize the notion for equivalence
relations [Gab02]. We recall only the basic definitions and notation here
referring to [Sau05, ADR01] for the general theory of discrete measured
groupoids. Recall that the object set $\mathscr{G}^{0}$ of a discrete measured
groupoid $\mathscr{G}$ comes equipped with a $\mathscr{G}$-invariant measure
$\mu$ which gives rise to a measure $\mu_{\mathscr{G}}$ on $\mathscr{G}$
defined on a Borel subset $A\subseteq\mathscr{G}$ as

$\mu_{\mathscr{G}}(A):=\int_{\mathscr{G}^{0}}\operatorname{card}(s^{-1}(x)\cap
A)\,d\mu(x).$

Here $s\colon\mathscr{G}\to\mathscr{G}^{0}$ denotes the source map of the
groupoid. A measurable subset $K\subseteq\mathscr{G}$ is called _bounded_ if
there exists an $M>0$ such that for almost every $x\in\mathscr{G}^{0}$ both
$\operatorname{card}(s^{-1}(x)\cap K)<M$ and
$\operatorname{card}(s^{-1}(x)\cap K^{-1})<M$. Let $\mathbb{C}\mathscr{G}$ be
the groupoid ring of $\mathscr{G}$, defined as

$\mathbb{C}\mathscr{G}:=\\{f\in
L^{\infty}(\mathscr{G},\mu_{\mathscr{G}})\,|\,\operatorname{supp}(f)\text{ is
a bounded subset of }\mathscr{G}\\}.$

(this is well-defined independently of the choice of the representative of
$f$). The convolution and involution on $\mathbb{C}\mathscr{G}$ are defined as

$(f\ast
g)(\gamma):=\sum_{\begin{subarray}{c}\gamma^{\prime},\gamma^{\prime\prime}\in\mathscr{G},\\\
\gamma^{\prime}\gamma^{\prime\prime}=\gamma\end{subarray}}f(\gamma^{\prime})g(\gamma^{\prime\prime}),$
$f^{*}(\gamma):=\overline{f(\gamma^{-1})}.$

This gives $\mathbb{C}\mathscr{G}$ the structure of a $\ast$-algebra which
forms a strongly dense $\ast$-subalgebra in the groupoid von Neumann algebra
$L\mathscr{G}$ [ADR00]. For arbitrary subsets $K\subseteq\mathscr{G}$,
$A\subseteq\mathscr{G}^{0}$, $B\subseteq\mathscr{G}^{0}$ we denote by
$K_{A}^{B}$ the set $K\cap s^{-1}(A)\cap r^{-1}(B)$. Notice in particular that
for every $x\in\mathscr{G}^{0}$ the set $\mathscr{G}^{x}_{x}$ is a group with
respect to the composition. It is called the stabilizer group of $x$. A
discrete measured groupoid is called ergodic if for every set
$A\subseteq\mathscr{G}^{0}$ of positive measure
$r(s^{-1}(A))=\mathscr{G}^{0}$, and every discrete measured groupoid admits an
ergodic decomposition into a direct integral of ergodic groupoids. If
$K\subseteq\mathscr{G}$ is a bounded measurable subset, then

$\mathcal{F}_{K}:=\\{f\in\mathbb{C}\mathscr{G}\,|\,\operatorname{supp}f\subseteq
K\\}$

is a finitely generated projective right Hilbert
$L^{\infty}(\mathscr{G}^{0})$-module when equipped with the
$L^{\infty}(\mathscr{G}^{0})$-valued inner product arising from the
conditional expectation $E\colon L\mathscr{G}\to L^{\infty}(\mathscr{G}^{0})$.
This can be easily seen as follows: take an $n\in\mathbb{N}$ such that almost
every $x\in\mathscr{G}^{0}$ has at most $n$ preimages from $K$ with respect to
$s$ and decompose $K$ as $K=K_{1}\sqcup K_{2}\sqcup\dots\sqcup K_{n}$ in such
a way that the map $s|_{K_{i}}$ is a measurable isomorphism onto its image
$X_{i}\subseteq\mathscr{G}^{0}$ [Sau05, Lemma 3.1]. Then $\mathcal{F}_{K}$ is
isomorphic as a (pre-)Hilbert $C^{*}$-module over
$L^{\infty}(\mathscr{G}^{0})$ to the direct sum
$\oplus_{i=1}^{n}\mathbf{1}_{X_{i}}L^{\infty}(\mathscr{G}^{0})$ endowed with
the standard inner product. This follows from the fact that for
$f\in\mathcal{F}_{K_{i}}$ and $x\in X_{i}$ we have

$E(f^{*}f)(x)=(f^{*}\ast
f)({\operatorname{id}}_{x})=\sum_{\begin{subarray}{c}\gamma^{\prime},\gamma\in
K_{i},\\\
\gamma^{\prime}\circ\gamma={\operatorname{id}}_{x}\end{subarray}}\overline{f\left({\gamma^{\prime}}^{-1}\right)}f(\gamma)=|f(\gamma(x))|^{2},$

where $\gamma(x)\in K_{i}$ is uniquely determined by the property
$s(\gamma(x))=x$. Note in particular that
$\dim_{L^{\infty}(\mathscr{G}^{0})}\mathcal{F}_{K}=\mu_{\mathscr{G}}(K)$.
Moreover, a direct computation gives the following formula for
$\varphi_{\mathcal{F}_{K}}$:

(6.1)
$\varphi_{\mathcal{F}_{K}}(T)=\frac{\sum_{i=1}^{n}\int_{X_{i}}E(T)\,d\mu}{\mu_{\mathscr{G}}(K)}=\frac{\sum_{i=1}^{n}\int_{X_{i}}E(T)\,d\mu}{\sum_{i=1}^{n}\mu(X_{i})},\quad
T\in L\mathscr{G}.$

The result now is as follows.

######  Corollary 6.8 (Discrete measured amenable groupoids).

Let $\mathscr{G}$ be an amenable discrete measured groupoid for which the
stabilizers $\mathscr{G}_{x}^{x}$ are finite for almost every
$x\in\mathscr{G}^{0}$. Then the $L^{2}$-Betti numbers of $\mathscr{G}$ vanish
in positive degrees:

$\beta_{p}^{(2)}(\mathscr{G}):=\dim_{L\mathscr{G}}\operatorname{Tor}_{p}^{\mathbb{C}\mathscr{G}}(L\mathscr{G},L^{\infty}(\mathscr{G}^{0}))=0.$

In particular, the $L^{2}$-Betti numbers of an amenable equivalence relation
vanish.

Note that Corollary 6.8 is just a special case of [ST10, Theorem 1.1]. In
particular, the assumption that the stabilizers are finite is not important
for the validity of the statement in Corollary 6.8, but only needed in order
for our methods to apply.

###### Proof.

First of all, we observe that by [ST10, Remark 1.7] it is sufficient to prove
the statement for ergodic groupoids, and we will restrict ourselves to this
case for the rest of the proof. Let $\mathscr{R}$ be the equivalence relation
on $\mathscr{G}^{0}$ induced by $\mathscr{G}$; it can be considered as a
quotient groupoid of $\mathscr{G}$ after dividing out all stabilizers and is
known as the derived groupoid of $\mathscr{G}$. Since $\mathscr{G}$ is ergodic
the same is true for $\mathscr{R}$. By amenability of $\mathscr{G}$, the
quotient $\mathscr{R}$ is an amenable equivalence relation [ADR01, Theorem
2.11], which is therefore hyperfinite [Tak03, Theorem XIII.4.10]. That is,
there exists an increasing sequence of subrelations $\mathscr{R}_{n}$ each of
which has finite orbits of cardinality $k_{n}$ and such that
$\mathscr{R}\setminus\bigcup_{n}\mathscr{R}_{n}$ is a zero set. Let
$\mathscr{G}_{n}$ be the subgroupoid of $\mathscr{G}$ generated by the
stabilizers $\mathscr{G}_{x}^{x}$ and the lifts of the morphisms from
$\mathscr{R}_{n}$ to $\mathscr{G}$. This subgroupoid is well-defined because
two lifts of a morphism from $\mathscr{R}_{n}$ differ by an element in the
stabilizer. Let $\mathcal{A}:=\varinjlim\mathbb{C}\mathscr{G}_{n}$ be the
algebraic direct limit of the corresponding groupoid algebras. By construction
of $\mathscr{G}_{n}$, $\mathscr{G}\setminus\bigcup_{n}\mathscr{G}_{n}$ is a
measure zero subset of $\mathscr{G}$ and therefore $\mathcal{A}$ is an
_$L^{\infty}(\mathscr{G}^{0})$ -rank dense_ subalgebra of
$\mathbb{C}\mathscr{G}$; i.e. for each $f\in\mathbb{C}\mathscr{G}$ and for
each $\varepsilon>0$ there exists a projection $p\in
L^{\infty}(\mathscr{G}^{(0)})$ with $\tau(p)>1-\varepsilon$ such that
$pf\in\mathcal{A}$. (for instance, one can take the characteristic function of
a subset $X_{f}$ in $\mathscr{G}^{0}$ of measure bigger than $1-\varepsilon$
with the property that $f$ is non-zero only on those morphisms from
$\mathscr{G}\setminus\mathscr{G}_{n}$ whose target is in the complement of
$X_{f}$). Furthermore, as $\mathbb{C}\mathscr{G}$ is dimension-compatible as
an $L^{\infty}(\mathscr{G}^{0})$-bimodule [Sau05, Lemma 4.8] the rank-density
together with [Sau05, Theorem 4.11] implies that

(6.2)
$\beta_{p}^{(2)}(\mathscr{G}):=\dim_{L\mathscr{G}}\operatorname{Tor}^{\mathbb{C}\mathscr{G}}_{p}(L\mathscr{G},L^{\infty}(\mathscr{G}^{0}))=\dim_{L\mathscr{G}}\operatorname{Tor}^{\mathcal{A}}_{p}(L\mathscr{G},L^{\infty}(\mathscr{G}^{0})).$

Let us now consider $\mathbb{C}\mathscr{G}_{n}$ as a right
$L^{\infty}(\mathscr{G}^{0})$-module. By construction, it is equal to
$\mathcal{F}_{\mathscr{G}_{n}}$, where $\mathscr{G}_{n}$ is considered as a
bounded subset of $\mathscr{G}$. Its dimension can be easily computed as

$\dim_{L^{\infty}(\mathscr{G}^{0})}\mathbb{C}\mathscr{G}_{n}=k_{n}\cdot\operatorname{card}\mathscr{G}_{x}^{x},$

where $\operatorname{card}\mathscr{G}_{x}^{x}$ is the cardinality of the
stabilizer at $x\in\mathscr{G}^{0}$; it is an essentially constant function on
$\mathscr{G}^{0}$ because of the ergodicity hypothesis. Moreover, (6.1) gives
that

$\varphi_{\mathscr{G}_{n}}(T)=\tau(T),\quad T\in L\mathscr{G}.$

Now, putting $M:=L(\mathscr{G})$ and considering
$N:=L^{\infty}(\mathscr{G}^{0})$ as a subalgebra of $\mathcal{A}$ via the
inclusion $\mathscr{G}^{0}\subseteq\mathscr{G}_{n}$, we infer that the
subalgebra $\mathcal{A}$ has the strong Følner property. Therefore the
inclusion $\mathcal{A}\subseteq M$ is dimension-flat and formula (6.2) gives
us the vanishing result for the positive degree $L^{2}$-Betti numbers of
$\mathscr{G}$. ∎

######  Example 6.9 (The irrational rotation algebra).

Let $\theta\in]0,1[$ be an irrational number and recall that the irrational
rotation algebra $A_{\theta}$ is the universal $C^{*}$-algebra generated by
two unitaries $u$ and $v$ subject to the relation $uv=e^{2\pi i\theta}vu$. The
set

$\mathcal{A}_{\theta}:=\left.\left\\{\sum_{p,q\in{\mathbb{Z}}}z_{p,q}u^{p}v^{q}\,\right|\,z_{p,q}\in{\mathbb{C}},z_{p,q}=0\text{
for all but finitely many }(p,q)\in{\mathbb{Z}}^{2}\right\\}$

forms a dense subalgebra in $A_{\theta}$ and carries a natural filtering
sequence of subspaces

$\mathcal{F}_{n}:=\left.\left\\{\sum_{-n\leqslant p+q\leqslant
n}z_{p,q}u^{p}v^{q}\,\right|\,z_{p,q}\in{\mathbb{C}}\right\\}.$

Recall that $A_{\theta}$ has a unique trace which on $\mathcal{A}_{\theta}$ is
given by
$\tau\left(\sum_{p,q\in{\mathbb{Z}}}z_{p,q}u^{p}v^{q}\right)=z_{0,0}$. We aim
at proving that $\mathcal{A}_{\theta}$ has the strong Følner property relative
to $N:={\mathbb{C}}$ and the enveloping von Neumann algebra
$M_{\theta}:=A_{\theta}^{\prime\prime}\subseteq B(L^{2}(A_{\theta},\tau))$.
Let ${\varepsilon}>0$ and $T_{1},\dots,T_{r}\in\mathcal{A}_{\theta}$ be given.
Then there exists an $m_{0}\in{\mathbb{N}}$ such that
$T_{i}(\mathcal{F}_{n})\subseteq\mathcal{F}_{n+m_{0}}$ for each
$i\in\\{1,\dots,r\\}$ and $n\in{\mathbb{N}}$, and we now define
$\mathcal{P}_{n}:=\mathcal{F}_{nm_{0}}$ and
$\mathcal{S}_{n}:=\mathcal{F}_{(n-1)m_{0}}$. Using the fact that
$H=L^{2}(A_{\theta},\tau)$ has an orthonormal basis consisting of the
unitaries

$\\{u^{p}v^{q}\mid p,q\in{\mathbb{Z}}\\}$

it is not difficult to see that $\varphi_{\mathcal{P}_{n}}(T)=\tau(T)$ for
every $T\in M_{\theta}$ and furthermore it implies that
$\dim_{\mathbb{C}}{\mathcal{P}_{n}}=2(nm_{0})^{2}+2nm_{0}+1$ from which it
follows that
$\frac{\dim_{\mathbb{C}}\mathcal{S}_{n}}{\dim_{\mathbb{C}}\mathcal{P}_{n}}\longrightarrow
1$. Thus $(\mathcal{P}_{n},\mathcal{S}_{n})$ is a strong Følner sequence for
$T_{1},\dots,T_{r}$. From our results we therefore obtain that the Connes-
Shlyakhtenko $L^{2}$-Betti numbers
$\beta^{(2)}_{p}(\mathcal{A}_{\theta},\tau)$ vanish for $p\geqslant 1$ as well
as the well-known fact that the enveloping von Neumann algebra $M_{\theta}$ is
hyperfinite.

######  Example 6.10 (UHF-algebras).

Let $A$ be a UHF-algebra; i.e. $A$ is a $C^{*}$-algebraic direct limit of a
sequence $({\mathbb{M}}_{k(n)}({\mathbb{C}}),\alpha_{n})$ of full matrix
algebras. Denoting by $\mathcal{A}$ the algebraic direct limit we get a
natural dense $\ast$-subalgebra in $A$ and the tracial states on the matrix
algebras ${\mathbb{M}}_{k(n)}({\mathbb{C}})$ give rise to a unique tracial
state $\tau$ on $A$, for which the enveloping von Neumann algebra
$M:=A^{\prime\prime}\subseteq B(L^{2}(A,\tau))$ is the hyperfinite II1-factor.
Denote by $\mathcal{F}_{n}$ the image of ${\mathbb{M}}_{k(n)}({\mathbb{C}})$
in $\mathcal{A}$. Then for given $T_{1},\dots,T_{r}\in\mathcal{A}$ there
exists an $n_{0}$ such that they are all in $\mathcal{F}_{n_{0}}$ and thus
$T_{i}(\mathcal{F}_{n})\subseteq\mathcal{F}_{n}$ for all $i\in\\{1,\dots,r\\}$
and $n\geqslant n_{0}$. Putting
$\mathcal{P}_{n}=\mathcal{S}_{n}:=\mathcal{F}_{n_{0}+n}$ we get a strong
Følner sequence (relative to ${\mathbb{C}}\subseteq\mathcal{A}\subseteq M)$
for $T_{1},\dots,T_{r}$: the two first conditions in Proposition 3.3 are
obviously fulfilled and the a direct computation444This can be seen either
using matrix units or a collection of unitaries matrices forming an
orthonormal basis for the Hilbert-Schmidt inner product. shows that
$\varphi_{\mathcal{P}_{n}}(T)=\tau(T)$. Thus the Connes-Shlyakhtenko
$L^{2}$-Betti numbers of $(\mathcal{A},\tau)$ vanish in positive degrees.

######  Remark 6.11.

Note that by [Soł10, Corollary 3.2 & Theorem 3.3] neither the UHF-algebra nor
the irrational rotation algebra is the algebra associated with a discrete
group or quantum group; hence Example 6.10 and Example 6.9 provide honest new
examples of dimension flat inclusions.

######  Remark 6.12.

Generalizing Example 6.10, we may consider an arbitrary finite factor $N$ and
an inductive system of the form $({\mathbb{M}}_{k(n)}(N),\alpha_{n})$ and the
inclusion of the algebraic direct limit $\mathcal{A}$ in the von Neumann
algebraic direct limit $M$. Then, by exactly the same reasoning as in Example
6.10, $\mathcal{A}$ has the strong Følner property relative to the chain
$N\subseteq\mathcal{A}\subseteq M$. Hence the inclusion $\mathcal{A}\subseteq
M$ is dimension flat even though $\mathcal{A}$ can be “far from amenable” ($N$
might for instance be a II1-factor with property (T)). Compare e.g. with
[Lüc02, Conjecture 6.48] where it is conjectured that for $N={\mathbb{C}}$ and
$\Gamma$ a discrete group, dimension flatness of the inclusion
${\mathbb{C}}\Gamma\subseteq L\Gamma$ is equivalent to amenability of
$\Gamma$.

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