1 Introduction
In the setting of group actions on operator algebras, there are a number of questions which have satisfactory answers for actions of discrete groups, but which are still open for the case of actions of locally compact groups.
-
(1)
-
(a) How is injectivity of the crossed product corresponding to a
$w^*$
-dynamical system
$(M,G,\alpha)$
related to amenability of the action
$\alpha $
and injectivity of M? -
(b) How is nuclearity of the (full or reduced) crossed product of a
$C^*$
-dynamical system
$(A,G,\alpha)$
related to amenability of the action
$\alpha $
and nuclearity of A?
-
-
(2) How can one carry out Haagerup’s averaging arguments, in order to pass from approximation properties of a crossed product to approximation properties of the components of a (
$C^*$
or
$w^*$
)-dynamical system? (See, e.g., [Reference McKee, Skalski, Todorov and Turowska27, Proposition 3.4] for the answer in the discrete case.)
In this paper, we show that these problems can be solved by accounting for the module structure over the Fourier algebra of the acting group, which crossed products naturally carry.
Restricting for a moment to group
$C^*$
-algebras and group von Neumann algebras (the case of trivial actions), question (1) goes back to Lance [Reference Lance24, Theorem 4.3], who showed that a discrete group G is amenable if and only if
$C^*_\lambda (G)$
is a nuclear
$C^*$
-algebra (and a similar technique shows
$\mathrm {vN}(G)$
is an injective von Neumann algebra). Connes [Reference Connes12] showed that this does not generalize directly to locally compact groups; indeed, although amenable locally compact groups have injective group von Neumann algebras, there exist nonamenable locally compact groups G for which
$\mathrm {vN}(G)$
is injective (any nonamenable connected Lie group is an example). Lau and Paterson [Reference Paterson30, p. 85] recognized that it is only in the class of inner amenable groups that injectivity of
$\mathrm {vN}(G)$
implies amenability of G. Because every amenable group is inner amenable, this provides a complete answer to the problem of understanding amenability of locally compact groups in terms of their group von Neumann algebras. Recently, Crann and Tanko [Reference Crann and Tanko17, Theorem 3.4] gave a reinterpretation of this result in terms of module maps: it is injectivity of
$\mathrm {vN}(G)$
as an
$A(G)$
-module, which implies that G is amenable, and inner amenable groups are the locally compact groups for which injectivity of
$\mathrm {vN}(G)$
implies injectivity of
$\mathrm {vN}(G)$
as an
$A(G)$
-module (see Theorem 2.6). All discrete groups are inner amenable, so in the discrete case, one does not need to account for the
$A(G)$
-module structure.
For trivial actions, the averaging arguments mentioned in (2) were introduced by Haagerup [Reference Haagerup21] (written in 1986 and circulated as an unpublished manuscript), where such arguments were used to study discrete groups. Latter work of Lau and Paterson [Reference Paterson30, p. 85] indicated that one can use inner amenability, rather than discreteness, to give such averaging arguments, extending these techniques beyond discrete groups. Crann [Reference Crann14] (see also [Reference Crann and Tanko17, Section 3]) rephrases inner amenability in terms of relative module injectivity, clarifying the exact relationship between amenability of a locally compact group and injectivity of its group von Neumann algebra.
Moving on to group actions, Anantharaman-Delaroche [Reference Anantharaman-Delaroche1] introduced a definition of amenable action of a locally compact group on a von Neumann algebra, building on Zimmer’s work [Reference Zimmer35–Reference Zimmer37] on amenable actions on commutative von Neumann algebras. For actions of discrete groups, this definition is used to solve question (1a) [Reference Anantharaman-Delaroche1, Corollaire 4.2], and was subsequently adapted to solve question (1b) [Reference Anantharaman-Delaroche3, Théorème 4.5]. We caution the reader that there is another definition of amenable action on a set in the literature which is different from Zimmer’s; in this paper, we consider only Zimmer’s notion and its generalization by Anantharaman-Delaroche. The question of how to define amenable actions of locally compact groups on
$C^*$
-algebras remained open (see [Reference Anantharaman-Delaroche5, Section 9]). This question has attracted significant recent attention—we note work of Bearden and Crann [Reference Bearden and Crann6], Buss, Echterhoff, and Willett [Reference Buss, Echterhoff, Willett, Cortiñas and Weibel10, Reference Buss, Echterhoff and Willett11], Ozawa and Suzuki [Reference Ozawa and Suzuki29], and Suzuki [Reference Suzuki32], all of which are concerned with amenable actions of locally compact groups on
$C^*$
-algebras.
Since we began this work, there has been a flurry of papers on the topic of amenable actions of locally compact groups. As well as the papers [Reference Bearden and Crann6, Reference Buss, Echterhoff, Willett, Cortiñas and Weibel10, Reference Buss, Echterhoff and Willett11, Reference Ozawa and Suzuki29, Reference Suzuki32] mentioned above, which focus on amenable actions on
$C^*$
-algebras, we have recent work of Crann [Reference Crann15] and Bearden and Crann [Reference Bearden and Crann7]. We have rewritten this paper several times in attempts to account for these works. In particular, Crann [Reference Crann15] and Bearden and Crann [Reference Bearden and Crann7] also discovered the relationship between amenable actions and module injectivity, and gave another related module property. Our results were obtained independently, but we note that we have been heavily influenced by Crann’s earlier work on module injectivity for quantum groups [Reference Crann13, Reference Crann14].
The organization of this paper is as follows. In Section 2, we review the definitions and results surrounding the notion of group operator algebras and crossed products. Then, we review the module versions of injectivity, and their link to amenability, which we aim to generalize. In Section 3, we generalize amenable actions on von Neumann algebras by using the natural module structure of a crossed product over the Fourier algebra. In Section 4, we introduce inner amenable actions on von Neumann algebras. We give some basic properties of this notion and extend Lau and Paterson’s result [Reference Lau and Paterson25, Theorem 3.1] to inner amenable actions on noncommutative injective von Neumann algebras. In Section 5, we consider the weak* completely bounded approximation property for crossed product von Neumann algebras, giving an example of how inner amenable actions can be used for averaging arguments in Proposition 5.2. Section 6 is devoted to amenable actions on
$C^*$
-algebras. We use the natural extension of Anantharaman-Delaroche’s definition of amenable action of a discrete group on a
$C^*$
-algebra to locally compact groups, and relate this to nuclearity of the corresponding crossed products. We also give a survey of known results, mainly from work of Buss et al. [Reference Buss, Echterhoff and Willett11], Bearden and Crann [Reference Bearden and Crann6], and Suzuki [Reference Suzuki32], to demonstrate how close our notion of amenable action comes to satisfying the requirements for an amenable action on a
$C^*$
-algebra suggested by Anantharaman-Delaroche [Reference Anantharaman-Delaroche5, Section 9.2].
2 Preliminaries
For von Neumann algebras M and N, we will write their normal spatial tensor product as
$M \mathbin {\overline {\otimes }} N$
. We will use throughout the theory of operator spaces and completely bounded maps; the reference [Reference Effros and Ruan19] is suggested for this background material.
2.1 Group operator algebras and crossed products
Let G be a locally compact group. We write
$\lambda $
for the left regular representation
$$ \begin{align*} \lambda: G \to {\mathcal{B}(L^2(G))};\ \lambda_r \xi(s):= \xi(r^{-1} s), \quad r,s \in G, \ \xi \in L^2(G), \end{align*} $$
which is a unitary representation of G, so extends to a
$*$
-representation of
$C_c(G)$
on
$L^2(G)$
by
$$ \begin{align*} \lambda: C_c(G) \to {\mathcal{B}(L^2(G))};\ \lambda(f):= \int_G f(r) \lambda_r dr, \quad f \in C_c(G). \end{align*} $$
The reduced group
$C^*$
-algebra
$C^*_\lambda (G)$
is the
$C^*$
-algebra obtained by completing
$C_c(G)$
in the norm induced by
$\lambda $
, while the group von Neumann algebra is the double commutant
$\mathrm {vN}(G):= C^*_\lambda (G)'' = \{ \lambda _r: r \in G \}''$
.
The Fourier algebra of G, denoted by
$A(G)$
, is the set
$$ \begin{align*} A(G) = \{ u: G \to \mathbb{C}: u(r) = \left\langle \lambda_r \xi, \eta \right\rangle, \ \xi,\eta \in L^2(G) \}, \end{align*} $$
with pointwise operations and the norm
$\Vert u \Vert = \inf \Vert \xi \Vert \Vert \eta \Vert $
, where the infimum is taken over all possible representations
$u(\cdot) = \left \langle \lambda _{\cdot } \xi, \eta \right \rangle $
. The Fourier algebra is a completely contractive Banach algebra, and is naturally identified with the predual of
$\mathrm {vN}(G)$
by the pairing
$\left \langle \lambda _r, u \right \rangle = u(r)$
(
$u \in A(G)$
). We refer to [Reference Eymard20] for further background.
Now, let A be a
$C^*$
-algebra, and
$\alpha: G \to \mathop {\mathrm {Aut}}(A)$
a homomorphism which is continuous in the point-norm topology, i.e., let
$(A, G, \alpha)$
be a
$C^*$
-dynamical system. A pair
$(\phi, \rho)$
, where
$\phi: A \to {\mathcal {B}(\mathcal {H})}$
is a
$*$
-representation and
$\rho: G \to {\mathcal {B}(\mathcal {H})}$
is a unitary representation, is called a covariant pair for
$(A,G,\alpha)$
if
$\rho _r \phi (a) \rho _r^* = \phi (\alpha _r(a))$
, for all
$r \in G,\ a \in A$
. In particular, a regular covariant pair is constructed as follows: suppose that
$\pi: A \to {\mathcal {B}(\mathcal {H})}$
is a faithful
$*$
-representation and define
$$ \begin{align} \begin{gathered} \pi_{\alpha}: A \to {\mathcal{B}(L^2(G) \otimes \mathcal{H})}; \ \pi_{\alpha}(a) \xi(s):= \alpha_{s^{-1}}(a) \xi(s), \\ \lambda: G \to {\mathcal{B}(L^2(G) \otimes \mathcal{H})}; \ \lambda_r \xi(s):= \xi(r^{-1} s). \end{gathered} \end{align} $$
A covariant pair
$(\phi, \rho)$
as above defines a
$*$
-representation of
$C_c(G, A)$
on
$\mathcal {H}$
by
$$ \begin{align*} \phi \rtimes \rho (f):= \int_G \phi \big( f(r) \big) \rho_r dr, \quad f \in C_c(G, A). \end{align*} $$
The reduced crossed product
$A \rtimes _{{\alpha },r} G$
is the
$C^*$
-algebra obtained by completing
$C_c(G,A)$
in the norm induced by
$\pi _{\alpha } \rtimes \lambda $
; it can be shown that this definition is independent of the original faithful
$*$
-representation
$\pi $
. The full crossed product
$A \rtimes _{\alpha } \! G$
is the completion of
$C_c(G,A)$
in the universal norm
$$ \begin{align*} \Vert f \Vert:= \mathrm{sup}\{ \Vert \phi \rtimes \rho(f) \Vert: (\phi,\rho) \text{ is a covariant pair for } (A,G,\alpha) \}. \end{align*} $$
We refer to Williams [Reference Williams34, Chapter 2] for the details of this construction, and the fact that the full crossed product is universal in the following sense: every
$*$
-representation of the full crossed product
$A \rtimes _{\alpha } \! G$
is of the form
$\phi \rtimes \rho $
for some covariant pair
$(\phi, \rho)$
for
$(A,G,\alpha)$
. In particular, we will use the covariant pair
$(i_A, i_G)$
for which
$i_A \rtimes i_G$
is the universal
$*$
-representation of
$A \rtimes _{\alpha } \! G$
in Section 6.
If
$M \subset {\mathcal {B}(\mathcal {H})}$
is a von Neumann algebra and
$\alpha $
is a point-weak* continuous action of G on M, then we say that
$(M,G,\alpha)$
is a
$w^*$
-dynamical system. In this case, the definition of
$\pi _{\alpha }$
in (1) is an injective, weak*-continuous homomorphism
$\pi _{\alpha }: M \to L^\infty (G) \mathbin {\overline {\otimes }} M$
satisfying
$$ \begin{align*} (\mathrm{id} \otimes \pi_{\alpha}) \circ \pi_{\alpha} = (\Delta_{L^\infty(G)} \otimes \mathrm{id}) \circ \pi_{\alpha}. \end{align*} $$
Here,
$\Delta _{L^\infty (G)}$
is the natural coproduct on
$L^\infty (G)$
, given by
$\Delta _{L^\infty (G)}(\phi)(s,t):= \phi (st)$
(
$\phi \in L^\infty (G),\ s,t \in G$
). The crossed product associated to
$(M,G,\alpha)$
is the von Neumann algebra
$M \rtimes _{\alpha } \! G:= \{ \pi _\alpha (M), \mathrm {vN}(G) \otimes \mathbb {C} \}'' \subset {\mathcal {B}(L^2(G) \otimes \mathcal {H})}$
. We write
$\pi _{\hat {\alpha }}: M \rtimes _{\alpha } \! G \to \mathrm {vN}(G) \mathbin {\overline {\otimes }} M \rtimes _{\alpha } \! G $
for the natural coaction of G on
$M \rtimes _{\alpha } \! G$
(see [Reference Nakagami and Takesaki28, Proposition 2.4]), given by
$$ \begin{gather*} \pi_{\hat{\alpha}} \big( \pi_\alpha(a) \big):= 1 \otimes \pi_\alpha(a), \quad \pi_{\hat{\alpha}} (\lambda_r \otimes 1):= \lambda_r \otimes \lambda_r \otimes 1, \end{gather*} $$
for all
$r \in G$
and all
$a \in M$
. This coaction induces a module action of
$A(G)$
on
$M \rtimes _{\alpha } \! G$
, with the module structure given by
$$ \begin{align*} u * x:= (u \otimes \mathrm{id}) \pi_{\hat{\alpha}} (x), \quad u \in A(G),\ x \in M \rtimes_{\alpha} \! G. \end{align*} $$
In particular,
$u * (\pi _{\alpha }(a) \lambda _r) = u(r) \pi _{\alpha }(a) \lambda _r$
. In fact, with this definition,
$M \rtimes _{\alpha } \! G$
is a faithful
$A(G)$
-module. This module structure induces an inclusion
$$ \begin{align*} \Delta: M \rtimes_{\alpha} \! G \hookrightarrow \mathcal{C}\mathcal{B}(A(G), M \rtimes_{\alpha} \! G);\ \Delta(x)(u):= u * x, \quad u \in A(G),\ x \in M \rtimes_{\alpha} \! G. \end{align*} $$
There is a natural
$A(G)$
-module structure on
$\mathcal {C}\mathcal {B}(A(G), M \rtimes _{\alpha } \! G)$
given by
$$ \begin{align*} (u \cdot \Psi)(v):= \Psi(v u), \quad u,v \in A(G), \end{align*} $$
and
$\Delta $
is an
$A(G)$
-module map under this definition.
Remark 2.1 By, e.g., [Reference Effros and Ruan19, Corollary 7.1.5], there is a natural isomorphism
$$ \begin{align*} \mathcal{C}\mathcal{B}(A(G), M \rtimes_{\alpha} \! G) \cong \mathrm{vN}(G) \mathbin{\overline{\otimes}} (M \rtimes_{\alpha} \! G), \end{align*} $$
and under this isomorphism, the inclusion
$\Delta $
is induced by the coaction
$\pi _{\hat {\alpha }}$
. Indeed, the image of
$\Delta $
is identified by this isomorphism with the image of
$\pi _{\hat {\alpha }}$
in
$\mathrm {vN}(G) \mathbin {\overline {\otimes }} (M \rtimes _{\alpha } \! G)$
.
The coproduct on
$\mathrm {vN}(G)$
is implemented by the multiplicative unitary
$U \in \mathrm {vN}(G) \mathbin {\overline {\otimes }} L^\infty (G)$
, given by
$$ \begin{align*} U \xi (s,t):= \xi(t s, t), \quad \xi \in L^2(G) \otimes L^2(G),\ s,t \in G, \end{align*} $$
so that
$\Delta _{\mathrm {vN}(G)}(x) = U^* (1 \otimes x) U$
(
$x \in \mathrm {vN}(G)$
). This extends to a coaction of G on
${\mathcal {B}(L^2(G))}$
by the same formula. Thus, the module action
$*$
of
$A(G)$
on
$\mathrm {vN}(G)$
extends to a module action on
${\mathcal {B}(L^2(G))}$
by
$$ \begin{align*} u * x:= (u \otimes \mathrm{id}) U^* (1 \otimes x) U, \quad u \in A(G),\ x \in {\mathcal{B}(L^2(G))}. \end{align*} $$
We include the following easy lemma, because it is used several times in our arguments.
Lemma 2.2 Let
$(M,G,\alpha)$
and
$(N,G,\beta)$
be
$w^*$
-dynamical systems and suppose that
$\Phi: M \to N$
is a unital, completely bounded, G-equivariant map, i.e.,
$\Phi \circ \alpha _r = \beta _r \circ \Phi $
, for all
$r \in G$
. Then, there is a completely bounded
$A(G)$
-module map
$\tilde {\Phi }: {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M \to {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} N$
, which restricts to an
$A(G)$
-module map
$\tilde {\Phi } |_{M \rtimes _{\alpha } \! G}: M \rtimes _{\alpha } \! G \to N \rtimes _{\beta } \! G$
. Moreover,
$\Vert \tilde {\Phi } \Vert _{\mathrm {cb}} \leq \Vert \Phi \Vert _{\mathrm {cb}}$
.
Proof Define
$\tilde {\Phi }:= \mathrm {id} \otimes \Phi $
; it is clear that
$\tilde {\Phi }$
is an
$A(G)$
-module map, and well known that
$\Vert \tilde {\Phi } \Vert _{\mathrm {cb}} \leq \Vert \Phi \Vert _{\mathrm {cb}}$
. Suppose
$N \subset {\mathcal {B}(\mathcal {H}_N)}$
; because
$\Phi $
is G-equivariant, we have, for
$a \in M,\ r \in G,\ \xi \in L^2(G,\mathcal {H}_N)$
,
$$ \begin{align*} \tilde{\Phi}\big( \pi_{\alpha}(a) \big) \xi(r) = \Phi \big( \alpha_{r^{-1}}(a) \big) \xi(r) = \beta_{r^{-1}}\big( \Phi(a) \big) \xi(r) = \pi_{\beta}\big( \Phi(a) \big) \xi(r). \end{align*} $$
It follows that
$\tilde {\Phi }(\pi _{\alpha }(M)) \subset \pi _{\beta }(N)$
, so the claim about the restriction of
$\tilde {\Phi }$
follows, because it is unital and therefore acts as the identity on
$\mathrm {vN}(G) \otimes \mathbb {C} \subset M \rtimes _{\alpha } \! G$
.▪
2.2 Approximation properties for operator modules
The following definitions are given by Crann [Reference Crann15, Section 7].
Let X be a completely contractive Banach algebra. An operator space A will be called a left operator module over X if A is a left X-module and the module action extends to a complete contraction
$X \mathbin {\hat {\otimes }} A \to A$
(here,
$\mathbin {\hat {\otimes }}$
denotes the operator space projective tensor product). If
${X}_{(1)}:= X \oplus \mathbb {C}$
denotes the unitization of X, then the module action extends to a complete contraction
${X}_{(1)} \mathbin {\hat {\otimes }} A \to A$
by
$(x,c) \cdot a:= x \cdot a + ca$
(
$x \in X,\ a \in A,\ c \in \mathbb {C}$
). We will often omit the adjective “operator” in the rest of the paper, for example, writing left module over X in place of left operator module over X. Write
$X - \mathbf {mod}$
for the category of left X-modules with completely bounded module maps as morphisms.
Definition 2.3 Let X be a completely contractive Banach algebra and
$A,B$
be left modules over X (respectively, left modules over X which are dual spaces). A map
$\theta: A \to B$
is called nuclear in
$X - \mathbf {mod}$
(respectively, weakly nuclear in
$X - \mathbf {mod}$
) if there are morphisms:
$$ \begin{align*} \varphi_k: A \to M_{n_k}({X}_{(1)}^*), \quad \psi_k: M_{n_k}({X}_{(1)}^*) \to B, \end{align*} $$
such that
$\psi _k \circ \varphi _k$
converges to
$\theta $
in the point-norm (respectively, the point-weak*) topology.
The following definitions are natural generalizations of the usual ones for operator spaces.
Definition 2.4 Let X be a completely contractive Banach algebra and A be a left module over X. We say A is injective in
$X - \mathbf {mod}$
if, for any two X-modules B and C, any morphism
$\phi: B \to A$
, and any completely isometric morphism
$\kappa: B \to C$
, there is a morphism
$\tilde {\phi }: C \to A$
such that
$\phi = \tilde {\phi } \circ \kappa $
.
We will make repeated use of the obvious fact that injectivity in
$X - \mathbf {mod}$
is preserved under taking conditional expectations which are X-module maps.
The following results are known; we record them here for latter use.
Lemma 2.5
-
(i) For any locally compact group G, the algebra
${\mathcal {B}(L^2(G))}$
is injective in
$A(G) - \mathbf {mod}$
. -
(ii) Suppose that M is an injective von Neumann algebra. Then,
${\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M$
, endowed with the left
$A(G)$
-module structure induced by that on
${\mathcal {B}(L^2(G))}$
, is injective in
$A(G) - \mathbf {mod}$
.
Proof (i) This is essentially proved in [Reference Crann and Neufang16, Theorem 5.5]. Indeed, [Reference Renaud31] shows, for any locally compact group G, that
$\mathrm {vN}(G)$
has an
$A(G)$
-invariant state. Then, the same proof as [Reference Crann and Neufang16, Theorem 5.5] implies the claim.
(ii) If
$M = {\mathcal {B}(\mathcal {H})}$
, this follows routinely from (i) (see, e.g., [Reference Takesaki33, Proposition XV.3.2]). The general case follows.▪
2.3 Amenability and injectivity
Recall that a von Neumann algebra M is called injective if, for all unital
$C^*$
-algebras A with a unital inclusion
$M \subset A$
, there is a projection of norm 1 from A to M. Equivalently, if
$M \subset {\mathcal {B}(\mathcal {H})}$
, there is a projection of norm 1 from
${\mathcal {B}(\mathcal {H})}$
to M. We say that a crossed product
$M \rtimes _{\alpha } \! G$
is relatively injective in
$A(G) - \mathbf {mod}$
if there is a morphism
$\Phi: \mathcal {C}\mathcal {B}(A(G),M \rtimes _{\alpha } \! G) \to M \rtimes _{\alpha } \! G$
which is an
$A(G)$
-module map and satisfies
$\Phi \circ \Delta = \mathrm {id}$
. When
$M = \mathbb {C}$
and
$\alpha $
is trivial, this definition reduces to relative injectivity of
$\mathrm {vN}(G)$
as used by Crann and Tanko [Reference Crann and Tanko17].
Recall that a locally compact group is called amenable if there is a state on
$L^\infty (G)$
which is invariant under the left translation action
$\tau $
of G on
$L^\infty (G)$
. A locally compact group is called inner amenable if there is a state on
$L^\infty (G)$
which is invariant under the conjugation action
$\beta $
of G on
$L^\infty (G)$
. Crann and Tanko [Reference Crann and Tanko17, Proposition 3.2] showed that inner amenability of G is equivalent to the existence of a state on
$\mathrm {vN}(G)$
which is invariant under the conjugation action of G on
$\mathrm {vN}(G)$
; it is this condition which we generalize in Definition 4.1. Note that there is another notion of inner amenable locally compact group, introduced by Effros [Reference Effros18], which is different to the one just introduced, because it excludes the inner invariant state
$\delta _e$
.
We will mostly use amenability in the form of injectivity properties of operator algebras. Here, we summarize what is known about amenable groups in the language of injectivity. A discrete group is amenable if and only if its group von Neumann algebra is injective.
For locally compact groups, inner amenability is equivalent to relative injectivity of the group von Neumann algebra in
$A(G) - \mathbf {mod}$
.
Crann and Tanko [Reference Crann and Tanko17, Theorem 3.4] observe that Lau and Paterson’s result [Reference Lau and Paterson25, Corollary 3.2] can be phrased in this way, generalizing the above result.
Theorem 2.6 Let G be a locally compact group. The following are equivalent:
-
(i) G is amenable;
-
(ii)
$\mathrm {vN}(G)$
is injective in
$A(G) - \mathbf {mod}$
; -
(iii)
$\mathrm {vN}(G)$
is relatively injective in
$A(G) - \mathbf {mod}$
and injective in
$\mathbb {C} - \mathbf {mod}$
.
Our aim is to generalize the latter result to crossed products. Crann and Tanko [Reference Crann and Tanko17, Theorem 3.4] show how relative module injectivity is precisely what is needed for averaging arguments to work. This paper began with the idea to look for similar averaging techniques for crossed products.
3 Amenable actions on von Neumann algebras
Recall Anantharaman-Delaroche’s definition of an amenable action [Reference Anantharaman-Delaroche1].
Definition 3.1 Let
$\alpha $
be an action of a locally compact group G on a von Neumann algebra M. We say that
$\alpha $
is amenable if there is a projection of norm
$1$
$$ \begin{align*} P: L^\infty(G) \mathbin{\overline{\otimes}} M \to M \ \text{such that } P \circ (\tau_r \otimes \alpha_r) = \alpha_r \circ P, \quad r \in G. \end{align*} $$
That is, there is a G-equivariant conditional expectation from
$L^\infty (G) \mathbin {\overline {\otimes }} M$
to M. Here,
$\tau $
is the natural action of G on
$L^\infty (G)$
by left translation.
When
$M = \mathbb {C}$
, so
$\alpha $
is trivial, the projection P is a left-invariant mean, so G is amenable.
We now aim to improve [Reference Anantharaman-Delaroche1, Proposition 3.11] by accounting for the
$A(G)$
-module structure. Recall that the coproduct on
$\mathrm {vN}(G)$
is unitarily implemented, so extends to
${\mathcal {B}(L^2(G))}$
where it induces a module action of
$A(G)$
.
Proposition 3.2 Let
$\alpha $
be an action of G on M. The following are equivalent:
-
(i)
$\alpha $
is an amenable action; -
(ii) there is a norm-1 projection
${\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M \to M \rtimes _{\alpha } \! G$
which is an
$A(G)$
-module map.
Proof (i)
$\implies $
(ii) Let
$P: L^\infty (G) \mathbin {\overline {\otimes }} M \to M$
be the norm-1 equivariant projection from the definition of an amenable action. By Lemma 2.2, P extends to an
$A(G)$
-module map
$\tilde {P}: {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} L^\infty (G) \mathbin {\overline {\otimes }} M \to {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M$
, which restricts to a norm-1
$A(G)$
-module projection
$$ \begin{align*} P_\alpha: \big( L^\infty(G) \mathbin{\overline{\otimes}} M \big) \rtimes_{\tau \otimes \alpha} \! G \to M \rtimes_{\alpha} \! G. \end{align*} $$
By [Reference Anantharaman-Delaroche1, Lemme 3.10],
$\big ( L^\infty (G) \mathbin {\overline {\otimes }} M \big) \rtimes _{\tau \otimes \alpha } \! G$
is isomorphic to
${\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M$
, so
$P_\alpha $
is identified with a norm-1 projection
${\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M \to M \rtimes _{\alpha } \! G$
. Moreover, for each
$r \in G$
, the isomorphism in [Reference Anantharaman-Delaroche1, Lemme 3.10] identifies
$\lambda _r \otimes 1_{L^\infty (G) \mathbin {\overline {\otimes }} M} \in \big ( L^\infty (G) \mathbin {\overline {\otimes }} M \big) \rtimes _{\tau \otimes \alpha } \! G$
with
$\lambda _r \otimes 1_M \in {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M$
, so it follows that the natural
$A(G)$
-module structure on the domain of
$P_\alpha $
is transformed under this isomorphism to the natural
$A(G)$
-module structure on
${\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M$
. It follows that the map which corresponds to
$P_\alpha $
under this identification is an
$A(G)$
-module map.
(ii)
$\implies $
(i) Let
$P_\alpha $
be the projection in (ii) and take
$x \in L^\infty (G) \mathbin {\overline {\otimes }} M$
and
$u \in A(G)$
; we have
$$ \begin{align*} u * P_\alpha(x) = P_\alpha(u * x) = u(e) P_\alpha(x). \end{align*} $$
As this holds for all
$u \in A(G)$
, we conclude
$\pi _{\hat {\alpha }} (P_\alpha (x)) = 1_{\mathrm {vN}(G)} \otimes P_\alpha (x)$
; it is standard (see, e.g., [Reference Nakagami and Takesaki28, p. 21]) that this implies
$P_\alpha (x) \in \pi _{\alpha }(M) \cong M$
. To see that the restriction of
$P_\alpha $
to
$L^\infty (G) \mathbin {\overline {\otimes }} M$
is equivariant identify
$P_\alpha $
with the corresponding map
$$ \begin{align*} (L^\infty(G) \mathbin{\overline{\otimes}} M) \rtimes_{\tau \otimes \alpha} \! G \to (\mathbb{C} \mathbin{\overline{\otimes}} M) \rtimes_{\mathrm{id} \otimes \alpha} \! G \end{align*} $$
using [Reference Anantharaman-Delaroche1, Lemme 3.10]. Then, for
$x \in L^\infty (G) \mathbin {\overline {\otimes }} M$
,
$$ \begin{align*} P_\alpha \big( \pi_{\tau \otimes \alpha} \big( (\tau_r \otimes \alpha_r)x \big) \big) = P_\alpha \big( \lambda_r \pi_{\tau \otimes \alpha}(x) \lambda_r^* \big) = \lambda_r P_\alpha \big( \pi_{\tau \otimes \alpha}(x) \big) \lambda_r^* \end{align*} $$
by the bimodule property of
$P_\alpha $
. Because
$P_\alpha ( \pi _{\tau \otimes \alpha }(x)) \in \pi _{\mathrm {id} \otimes \alpha }(M)$
, this shows that
$P_\alpha $
is equivariant.▪
The
$A(G)$
-module structure allows us to recover some results for locally compact groups which are only proved for discrete groups in [Reference Anantharaman-Delaroche1].
Remark 3.3 Anantharaman-Delaroche [Reference Anantharaman-Delaroche1, Proposition 3.6] has shown that every action of an amenable group is amenable. There are also many examples of nonamenable groups which have amenable actions; this follows from, e.g., [Reference Brodzki, Cave and Li8, Theorem 5.8], because there exist nonamenable exact groups.
The following result is given by Anantharaman-Delaroche [Reference Anantharaman-Delaroche1, Proposition 3.6]. We will use the implication (ii)
$\implies $
(i) later in Theorem 4.6, so we give a proof of this implication using our techniques. For a von Neumann algebra M, we write
$\mathcal {Z}({M})$
for the center of M.
Proposition 3.4 Let
$\alpha $
be an action of G on M. The following are equivalent:
-
(i) G is amenable;
-
(ii)
$\alpha $
is amenable and
$\mathcal {Z}({M})$
has a G-invariant state.
Proof (i)
$\implies $
(ii) See Remark 3.3 and [Reference Anantharaman-Delaroche1, Proposition 3.6].
(ii)
$\implies $
(i) Let
$P_\alpha: {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M \to M \rtimes _{\alpha } \! G$
be a norm-1 projection. For each
$x \in {\mathcal {B}(L^2(G))}$
, we claim
$P_\alpha (x \otimes 1_M) \in \mathcal {Z}({M}) \rtimes _{\alpha } \! G$
. Because
$\mathrm {vN}(G) \otimes \mathbb {C} \subset M \rtimes _{\alpha } \! G$
and
$P_\alpha $
is an
$M \rtimes _{\alpha } \! G$
-bimodule map, we have
$P_\alpha (\mathrm {vN}(G) \otimes \mathbb {C}) = \mathrm {vN}(G) \otimes \mathbb {C} \subset \mathcal {Z}({M}) \rtimes _{\alpha } \! G$
. As
${\mathcal {B}(L^2(G))}$
is generated by
$\mathrm {vN}(G)$
and
$L^\infty (G)$
, it remains to verify the claim for
$x \in L^\infty (G)$
; in this case, the bimodule property of
$P_\alpha $
means that, for
$y \in \pi _{\alpha }(M)$
,
$$ \begin{align*} y P_\alpha(x \otimes 1_M) = P_\alpha \big( y (x \otimes 1_M) \big) = P_\alpha \big( (x \otimes 1_M) y \big) = P_\alpha(x \otimes 1_M) y, \end{align*} $$
so
$P_\alpha (x) \in M \rtimes _{\alpha } \! G \cap \pi _{\alpha }(M)' \subset \mathcal {Z}({M}) \rtimes _{\alpha } \! G$
. Now, let
$\phi: \mathcal {Z}({M}) \to \mathbb {C}$
be a G-invariant state, and
$\tilde {\phi }: \mathcal {Z}({M}) \rtimes _{\alpha } \! G \to \mathrm {vN}(G)$
the corresponding map given by Lemma 2.2. The composition
$\tilde {\phi } \circ P_\alpha $
is a norm-1
$A(G)$
-module projection from
${\mathcal {B}(L^2(G))} \otimes \mathbb {C}$
to
$\mathrm {vN}(G)$
, so G is amenable by Theorem 2.6.▪
Now, we show how accounting for the
$A(G)$
-module structure helps the study of injectivity of crossed products; this result generalizes [Reference Anantharaman-Delaroche1, Proposition 3.12 and Corollaire 4.2]. Part of this result was also obtained recently by Crann [Reference Crann15, Theorem 8.2] and Bearden and Crann [Reference Bearden and Crann7, Theorem 5.2].
Theorem 3.5 Let
$\alpha $
be an action of G on M. The following are equivalent:
-
(i)
$\alpha $
is amenable and M is injective; -
(ii)
$M \rtimes _{\alpha } \! G$
is injective in
$A(G) - \mathbf {mod}$
.
Proof (i)
$\implies $
(ii) Because M is injective, we have
${\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M$
is injective in
$A(G) - \mathbf {mod}$
by Lemma 2.5. By Proposition 3.2, there is a norm-1
$A(G)$
-module projection
$P_\alpha: {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M \to M \rtimes _{\alpha } \! G$
. Hence, (ii) holds.
(ii)
$\implies $
(i) It is routine to obtain a norm-1
$A(G)$
-module projection
$E: {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} {\mathcal {B}(\mathcal {H})} \to M \rtimes _{\alpha } \! G$
. Clearly, E restricts to a norm-1 projection
${\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M \to M \rtimes _{\alpha } \! G$
giving amenability of
$\alpha $
. Take
$x \in L^\infty (G) \mathbin {\overline {\otimes }} {\mathcal {B}(\mathcal {H})}$
and
$u \in A(G)$
, so that
$u * x = u(e) x$
. Because E is an
$A(G)$
-module map,
$u * E(x) = E(u * x) = u(e) E(x)$
; as this holds for all
$u \in A(G)$
, we conclude that
$E(x) \in \pi _{\alpha }(M)$
as in the proof of Proposition 3.2. Thus, E restricts to a norm-1 projection
$L^\infty (G) \mathbin {\overline {\otimes }} {\mathcal {B}(\mathcal {H})} \to \pi _{\alpha }(M) \cong M$
, so M is injective.▪
4 Inner amenable actions on von Neumann algebras
Recall that a locally compact group G is called inner amenable if there is an inner-invariant state on
$L^\infty (G)$
, and that Crann and Tanko [Reference Crann and Tanko17, Proposition 3.2] showed that inner amenability of G is equivalent to the existence of an inner-invariant state on
$\mathrm {vN}(G)$
, i.e., a state
$\phi $
on
$\mathrm {vN}(G)$
satisfying
$\phi (\beta _r(x)) = \phi (x)$
(
$r \in G, x \in \mathrm {vN}(G)$
), where
$$ \begin{align*} \beta_r(x):= \lambda_r x \lambda_r^*, \quad x \in \mathrm{vN}(G),\ r \in G. \end{align*} $$
It is this latter condition which we generalize to define inner amenable actions.
Definition 4.1 Let
$(M,G,\alpha)$
be a
$w^*$
-dynamical system. We say
$\alpha $
is inner amenable if there is a projection of norm 1
$$ \begin{align*} Q: \mathrm{vN}(G) \mathbin{\overline{\otimes}} M \to M \text{ such that } Q \circ (\beta_r \otimes \alpha_r) = \alpha_r \circ Q, \quad r \in G. \end{align*} $$
That is, there is a G-equivariant conditional expectation from
$\mathrm {vN}(G) \mathbin {\overline {\otimes }} M$
to M.
If
$M = \mathbb {C}$
and
$\alpha $
is trivial, then the equivariant projection Q in Definition 4.1 is an inner-invariant state on
$\mathrm {vN}(G)$
, so the above definition reduces to inner amenability of G by [Reference Crann and Tanko17, Proposition 3.2].
Remark 4.2 To see that every action of an inner amenable group is inner amenable, suppose that
$(M,G,\alpha)$
is a
$w^*$
-dynamical system, and that G is inner amenable. Let
$\phi: \mathrm {vN}(G) \to \mathbb {C}$
be a state which is invariant under the action
$\beta $
and define
$$ \begin{align*} Q: \mathrm{vN}(G) \mathbin{\overline{\otimes}} M \to M;\ Q(x):= (\phi \otimes \mathrm{id}) (x), \quad x \in \mathrm{vN}(G) \mathbin{\overline{\otimes}} M. \end{align*} $$
It is easily seen that Q satisfies Definition 4.1; hence,
$\alpha $
is inner amenable.
We would like to know if there is a way to formulate our definition of inner amenable actions in terms of an equivariant projection on
$L^\infty (G) \mathbin {\overline {\otimes }} M$
, generalizing the original definition of inner amenable groups as the existence of an inner-invariant mean on
$L^\infty (G)$
.
In order to use inner amenable actions for averaging arguments, we now show that Definition 4.1 is equivalent to a crossed product version of relative injectivity, which is exactly the property required for the arguments in Theorem 4.6 and Section 5.
Proposition 4.3 Let
$(M,G,\alpha)$
be a
$w^*$
-dynamical system. The following are equivalent:
-
(i)
$\alpha $
is inner amenable; -
(ii)
$M \rtimes _{\alpha } \! G$
is relatively injective in
$A(G) - \mathbf {mod}$
(i.e., there is a norm-1
$A(G)$
-module projection from
$\mathrm {vN}(G) \mathbin {\overline {\otimes }} (M \rtimes _{\alpha } \! G)$
onto
$\pi _{\hat {\alpha }}(M \rtimes _{\alpha } \! G)$
).
Proof (i)
$\implies $
(ii) Let
$U \in \mathrm {vN}(G) \mathbin {\overline {\otimes }} L^\infty (G)$
be the unitary on
$L^2(G) \otimes L^2(G)$
given by
$U \xi (s,t):= \xi (t s,t)$
, and let
$V:= \sigma U$
, where
$\sigma $
is the flip operator. Routine calculations show that, for
$x \in \mathrm {vN}(G)$
,
$$ \begin{align*} V (x \otimes 1_{\mathrm{vN}(G)}) V^* = \pi_{\beta}(x) \quad \text{and} \quad V^* (x \otimes 1_{\mathrm{vN}(G)}) V = \Delta_{\mathrm{vN}(G)}(x). \end{align*} $$
Thus, we identify the
$A(G)$
-modules
$\mathrm {vN}(G) \rtimes _{\beta } \! G$
with
$\mathrm {vN}(G) \mathbin {\overline {\otimes }} \mathrm {vN}(G)$
by conjugating with
$\sigma V^* = \sigma U^* \sigma $
(the
$A(G)$
-module structure on
$\mathrm {vN}(G) \mathbin {\overline {\otimes }} \mathrm {vN}(G)$
comes from applying the coproduct to the left component). Under this identification,
$\mathrm {vN}(G) \otimes \mathbb {C} \subset \mathrm {vN}(G) \rtimes _{\beta } \! G$
is identified with
$\Delta _{{\mathrm {vN}(G)}}(\mathrm {vN}(G))$
.
Similarly, if M acts on the Hilbert space
$\mathcal {H}$
, conjugating by the unitary
$\sigma V^* \otimes \mathrm {id}_{\mathcal {H}}$
, we identify
$(\mathrm {vN}(G) \mathbin {\overline {\otimes }} M) \rtimes _{{\beta \otimes \alpha }} \! G$
with
$(\sigma \otimes \mathrm {id})(\mathrm {vN}(G) \mathbin {\overline {\otimes }} (M \rtimes _{\alpha } \! G))(\sigma \otimes \mathrm {id})$
. Under this identification,
$(\mathbb {C} \otimes M) \rtimes _{\beta \otimes \alpha } \! G$
is identified with
$\pi _{\hat {\alpha }}(M \rtimes _{\alpha } \! G)$
.
Now, let
$Q: \mathrm {vN}(G) \mathbin {\overline {\otimes }} M \to M$
be the norm-1 equivariant projection from Definition 4.1, so the
$A(G)$
-module map
$\tilde {Q}$
given by Lemma 2.2 gives a norm-1
$A(G)$
-module projection
$$ \begin{align*} (V \otimes \mathrm{id}_{\mathcal{H}})^* \tilde{Q} (V \otimes \mathrm{id}_{\mathcal{H}}): \mathrm{vN}(G) \mathbin{\overline{\otimes}} (M \rtimes_{\alpha} \! G) \to \pi_{\hat{\alpha}}(M \rtimes_{\alpha} \! G), \end{align*} $$
as required.
(ii)
$\implies $
(i) Let
$Q_\alpha $
denote the projection in (ii), and as in the first part of the proof above conjugate with the unitary
$(V \otimes \mathrm {id}_{\mathcal {H}})$
to identify
$Q_\alpha $
with a norm-1
$A(G)$
-module projection from
$(\mathrm {vN}(G) \mathbin {\overline {\otimes }} M) \rtimes _{\beta \otimes \alpha } \! G$
onto
$(\mathbb {C} \otimes M) \rtimes _{\mathrm {id} \otimes \alpha } \! G \cong M \rtimes _{\alpha } \! G$
. The rest of the proof proceeds as in Proposition 3.2.▪
Because a left-invariant mean on
$L^\infty (G)$
is automatically two-sided invariant, all amenable groups are automatically inner amenable. We can generalize this fact to actions on injective von Neumann algebras.
Proposition 4.4 Suppose that
$(M,G,\alpha)$
is a
$w^*$
-dynamical system, with M injective and
$\alpha $
amenable. Then,
$\alpha $
is inner amenable.
Proof Because
$\alpha $
is amenable and M is injective, it follows from Theorem 3.5 that
$M \rtimes _{\alpha } \! G$
is injective in
$A(G) - \mathbf {mod}$
. It follows easily that
$M \rtimes _{\alpha } \! G$
is relatively injective in
$A(G) - \mathbf {mod}$
, so
$\alpha $
is inner amenable.▪
Examples 4.5
-
(i) The action
$\tau $
of G on
$L^\infty (G)$
is always amenable [Reference Anantharaman-Delaroche2, Section 1.4], hence always inner amenable. Because
${\mathcal {B}(L^2(G))}$
is injective, hence relatively injective, in
$A(G) - \mathbf {mod}$
, and
${\mathcal {B}(L^2(G))}$
is isomorphic to
$L^\infty (G) \rtimes _{\tau } \! G$
, inner amenability of
$\tau $
also follows Proposition 4.3. -
(ii) Let G denote a second-countable, connected, nonamenable locally compact group, for example,
$G = \mathrm {SL}(2, \mathbb {R})$
. Such groups cannot be inner amenable, but they are exact [Reference Kirchberg and Wassermann23, Theorem 6.8]. Brodzki, Cave, and Li [Reference Brodzki, Cave and Li8, Theorem 5.8] have shown that such G admit an amenable action on a compact space X; by Proposition 4.4, this action is inner amenable.
By analogy with the class of exact groups, it may be interesting to study the class of locally compact groups which admit an inner amenable action on
$L^\infty (X)$
for some compact space X. By Remark 4.2, this class contains all inner amenable groups, and the same argument as in Example 4.5(ii) shows this class contains all second-countable exact groups.
To close this section, we generalize a result of Lau and Paterson [Reference Lau and Paterson25, Theorem 3.1] to actions on noncommutative von Neumann algebras.
Theorem 4.6 Let
$(M,G,\alpha)$
be a
$w^*$
-dynamical system. The following are equivalent:
-
(i) G is amenable and M is injective;
-
(ii)
$M \rtimes _{\alpha } \! G$
is injective,
$\alpha $
is inner amenable and
$\mathcal {Z}({M})$
has a G-invariant state.
Proof (i)
$\implies $
(ii) By Remark 4.2,
$\alpha $
is inner amenable, and Proposition 3.4 shows
$\alpha $
is amenable and
$\mathcal {Z}({M})$
has a G-invariant state. By Proposition 3.2, there is a norm-1 projection
${\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} M \to M \rtimes _{\alpha } \! G$
, so because M is injective, so is
$M \rtimes _{\alpha } \! G$
.
(ii)
$\implies $
(i) Because
$\alpha $
is inner amenable, by Proposition 4.3, we can upgrade injectivity of
$M \rtimes _{\alpha } \! G$
to injectivity of
$M \rtimes _{\alpha } \! G$
in
$A(G) - \mathbf {mod}$
as in [Reference Crann13, Proposition 2.3]. It follows that M is injective as in the proof of Theorem 3.5. To show G is amenable, we follow a similar idea to Proposition 3.4. Let
$E: {\mathcal {B}(L^2(G))} \mathbin {\overline {\otimes }} {\mathcal {B}(\mathcal {H}_M)} \to M \rtimes _{\alpha } \! G$
be a norm-1
$A(G)$
-module projection, and observe that, for
$x \in {\mathcal {B}(L^2(G))}$
, we have
$E(x \otimes 1_M) \in \mathcal {Z}({M}) \rtimes _{\alpha } \! G$
, as in the proof of Proposition 3.4. Let
$\phi $
be a G-invariant state on
$\mathcal {Z}({M})$
, and
$\tilde {\phi }: \mathcal {Z}({M}) \rtimes _{\alpha } \! G \to \mathrm {vN}(G)$
the associated
$A(G)$
-module map given by Lemma 2.2. The composition
$\tilde {\phi } \circ E$
is a norm-1
$A(G)$
-module projection from
${\mathcal {B}(L^2(G))} \otimes \mathbb {C}$
to
$\mathrm {vN}(G)$
, so G is amenable by Theorem 2.6.▪
5 A sample averaging argument on crossed products
Our definition of inner amenable actions is designed to enable averaging arguments. One example of such an argument has already occurred in the proof that (ii) implies (i) in Theorem 4.6, where inner amenability of the action allows us to obtain injectivity of the crossed product in
$A(G) - \mathbf {mod}$
from the assumption that the crossed product is injective in
$\mathbb {C} - \mathbf {mod}$
. In the setting of
$C^*$
-algebra crossed products, discreteness of the acting group is used in the same way in [Reference McKee, Skalski, Todorov and Turowska27, Proposition 3.4]. In this section, we briefly give a further example of this technique.
Lemma 5.1 Let
$(M,G,\alpha)$
be a
$w^*$
-dynamical system. Suppose that
$\alpha $
is inner amenable, with associated norm-1
$A(G)$
-module projection
$Q_\alpha: \mathrm {vN}(G) \mathbin {\overline {\otimes }} M \rtimes _{\alpha } \! G \to \pi _{\hat {\alpha }}(M \rtimes _{\alpha } \! G)$
, and let
$\Phi: M \rtimes _{\alpha } \! G \to M \rtimes _{\alpha } \! G$
be a completely bounded map. Then,
$$ \begin{align*} S_\Phi: M \rtimes_{\alpha} \! G \to M \rtimes_{\alpha} \! G;\ S_\Phi:= \pi_{\hat{\alpha}}^{-1} \circ Q_\alpha \circ (\mathrm{id} \otimes \Phi) \circ \pi_{\hat{\alpha}} \end{align*} $$
is a completely bounded
$A(G)$
-module map, with
$\Vert S_\Phi \Vert _{\mathrm {cb}} \leq \Vert \Phi \Vert _{\mathrm {cb}}$
.
Proof Let
$Q_\alpha: \mathrm {vN}(G) \mathbin {\overline {\otimes }} M \rtimes _{\alpha } \! G \to \pi _{\hat {\alpha }}(M \rtimes _{\alpha } \! G)$
be the
$A(G)$
-module norm-1 projection given by inner amenability of
$\alpha $
in Proposition 4.3. It is easily checked that
$\pi _{\hat {\alpha }}(u * x) = (u \otimes \mathrm {id}) * \pi _{\hat {\alpha }}(x)$
, for all
$x \in M \rtimes _{\alpha } \! G$
. Then, for any
$u \in A(G)$
and
$x \in M \rtimes _{\alpha } \! G$
, we have
$$ \begin{align*} \begin{aligned} S_\Phi(u*x) &= \pi_{\hat{\alpha}}^{-1} \circ Q_\alpha \circ (\mathrm{id} \otimes \Phi) \big( (u \otimes \mathrm{id}) * \pi_{\hat{\alpha}}(x) \big) \\ &= \pi_{\hat{\alpha}}^{-1} \circ Q_\alpha \circ \Big( (u \otimes \mathrm{id}) * \big( \mathrm{id} \otimes \Phi) \circ \pi_{\hat{\alpha}}(x) \big) \Big) \\ &= u* \big(\pi_{\hat{\alpha}}^{-1} \circ Q_\alpha \circ (\mathrm{id} \otimes \Phi) \circ \pi_{\hat{\alpha}}(x)\big) = u*S_\Phi(x); \end{aligned} \end{align*} $$
hence,
$S_\Phi $
is an
$A(G)$
-module map. The norm inequality is obvious.▪
A von Neumann algebra N is said to have the weak* completely bounded approximation property (
$w^*$
CBAP) if there exists a net of ultraweakly continuous, finite-rank, completely bounded maps
$(\Phi _i: N \to N)_i$
such that
$\Phi _i \to \mathrm {id}_N$
in the point-ultraweak topology and a constant C for which
$\Vert \Phi _i \Vert _{\mathrm {cb}} \leq C$
, for each i. The Haagerup constant
$\Lambda _{\mathrm {cb}}(N)$
is the infimum of those C for which such a net
$(\Phi _i)_i$
exists, and
$\Lambda _{\mathrm {cb}} (N)= \infty $
if N does not have the
$w^*$
CBAP.
Proposition 5.2 Let
$(M,G,\alpha)$
be a
$w^*$
-dynamical system. Consider the conditions:
-
(i) M has the weak* completely bounded approximation property;
-
(ii)
$M \rtimes _{\alpha } \! G$
has the weak* completely bounded approximation property.
If
$\alpha $
is amenable, then (i) implies (ii). If
$\alpha $
is inner amenable, and the corresponding projection
$Q_\alpha $
can be chosen weak* continuous, then (ii) implies (i).
Proof (i)
$\implies $
(ii) This was shown by Anantharaman-Delaroche [Reference Anantharaman-Delaroche4, 4.10].
(ii)
$\implies $
(i) Let
$Q_\alpha: \mathrm {vN}(G) \mathbin {\overline {\otimes }} M \rtimes _{\alpha } \! G \to \pi _{\hat {\alpha }}(M \rtimes _{\alpha } \! G)$
be the norm-1
$A(G)$
-module projection arising from inner amenability of
$\alpha $
in Proposition 4.3. Let
$(\Phi _i: M \rtimes _{\alpha } \! G \to M \rtimes _{\alpha } \! G)_i$
be a net of maps which implement the
$w^*$
CBAP of
$M \rtimes _{\alpha } \! G$
, and define
$$ \begin{align*} S_{\Phi_i}: M \rtimes_{\alpha} \! G \to M \rtimes_{\alpha} \! G;\ S_{\Phi_i}:= \pi_{\hat{\alpha}}^{-1} \circ Q_\alpha \circ (\Phi_i \otimes \mathrm{id}) \circ \pi_{\hat{\alpha}} \end{align*} $$
as in Lemma 5.1; now, each
$S_{\Phi _i}$
is weak* continuous as a composition of weak* continuous maps. The maps
$(S_{\Phi _i})_i$
satisfy
$\Vert S_{\Phi _i} \Vert _{\mathrm {cb}} \leq \Vert \Phi _i \Vert _{\mathrm {cb}}$
, also implement the
$w^*$
CBAP of
$M \rtimes _{\alpha } \! G$
, and they are
$A(G)$
-module maps. Because
$u * (S_{\Phi _i}(x)) = S_{\Phi _i}(u * x)$
, for all
$x \in M \rtimes _{\alpha } \! G$
and all
$u \in A(G)$
, we have
$S_{\Phi _i}(\pi _{\alpha }(M)) \subset \pi _{\alpha }(M)$
. It follows that
$\pi _{\alpha }(M) \cong M$
has the
$w^*$
CBAP and
$\Lambda _{\mathrm {cb}}(M) \leq \Lambda _{\mathrm {cb}}(M \rtimes _{\alpha } \! G)$
.▪
The above proof illustrates one of our motivations for defining inner amenable actions: inner amenability of the action allows us to “average” maps which implement an approximation property of
$A \rtimes _{{\alpha },r} G$
into
$A(G)$
-module maps which implement the approximation property. The resulting
$A(G)$
-module maps may be viewed as Herz–Schur multipliers, as in [Reference McKee26]; this is the perspective adopted in [Reference McKee, Skalski, Todorov and Turowska27], where averaging arguments are explicitly used to produce Herz–Schur multipliers.
6 Amenable actions on
$C^*$
-algebras
In this section,
$(A,G,\alpha)$
will be a
$C^*$
-dynamical system. Anantharaman-Delaroche [Reference Anantharaman-Delaroche3, Définition 4.1] gave a definition of amenability for actions of discrete groups on
$C^*$
-algebras: if
$(A,G,\alpha)$
is a
$C^*$
-dynamical system with G discrete, then
$\alpha $
is called amenable if the corresponding double dual action
$\alpha ^{**}$
of G on
$A^{**}$
is amenable in the sense of Definition 3.1. In [Reference Anantharaman-Delaroche3, Section 4], it is shown that this definition has several nice properties.
The work of Anantharaman-Delaroche leaves open the question of defining amenable actions of locally compact groups on
$C^*$
-algebras. It is proposed in [Reference Anantharaman-Delaroche5, Section 9.2] that one might define an action of G on A to be amenable if
$A \rtimes _{\alpha } \! G = A \rtimes _{{\alpha },r} G$
, that is, if the canonical quotient map
$A \rtimes _{\alpha } \! G \to A \rtimes _{{\alpha },r} G$
is an isomorphism, and several questions about the behavior of this proposed definition are raised. Suzuki [Reference Suzuki32] constructed examples showing that
$A \rtimes _{\alpha } \! G = A \rtimes _{{\alpha },r} G$
does not satisfy the functoriality properties in [Reference Anantharaman-Delaroche5, Section 9.2]. We interpret the questions asked by Anantharaman-Delaroche as requirements for a sensible definition of an amenable action.
Problem 6.1 Let
$(A,G,\alpha)$
be a
$C^*$
-dynamical system, with G locally compact. Give a definition of amenability of
$\alpha $
with the following properties:
-
(1) if A is nuclear and
$\alpha $
is amenable, then the crossed product
$A \rtimes _{\alpha } \! G$
and/or the reduced crossed product
$A \rtimes _{{\alpha },r} G$
is also nuclear; -
(2) if
$\alpha $
is amenable and H is a closed subgroup of G, then the restriction of
$\alpha $
is an amenable action of H on A; -
(3) if
$(B,G,\beta)$
is a
$C^*$
-dynamical system such that there is an equivariant
$*$
-homomorphism
$\Phi: A \to \mathrm {M}(B)$
with
$\Phi (A) B$
dense in B and
$\Phi (\mathcal {Z}({\mathrm {M}(A)})) \subset \mathcal {Z}({\mathrm {M}(B)})$
, then amenability of
$\alpha $
implies amenability of
$\beta $
; -
(4) if A is a simple
$C^*$
-algebra, then amenability of
$\alpha $
implies amenability of G; -
(5) if
$\alpha $
is amenable, then the canonical quotient map
$A \rtimes _{\alpha } \! G \to A \rtimes _{{\alpha },r} G$
is an isomorphism.
Anantharaman-Delaroche [Reference Anantharaman-Delaroche5, Section 9.2] suggests taking condition (5) above as the definition of an amenable action; because we will use a different definition of amenable action, we have added condition (5) to the list of requirements for an amenable action. Versions of this problem have been considered by a number of authors; we note the recent work of Bearden and Crann [Reference Bearden and Crann6] and Buss et al. [Reference Buss, Echterhoff and Willett11], which targets (5) above and along the way addresses a number of the other properties. Our main contribution is to the nuclearity problem (1): we use
$A(G)$
-modules to relate amenable actions and nuclearity of crossed products. We also survey known results, showing that the notion of amenable action we use satisfies several of the points in Problem 6.1.
We follow Anantharaman-Delaroche [Reference Anantharaman-Delaroche3, Définition 4.1] in defining an action of G on a
$C^*$
-algebra A to be amenable if a suitable double dual action is amenable in the sense of Definition 3.1. The double dual action
$\alpha ^{**}$
on
$A^{**}$
can fail to be weak*-continuous [Reference Ikunishi22], but fortunately there is a suitable replacement, which was introduced by Ikunishi [Reference Ikunishi22].
Definition 6.2 Let
$(A,G,\alpha)$
be a
$C^*$
-dynamical system. Denote by
$(i_A,i_G)$
the covariant representation underlying the universal representation
$i_A \rtimes i_G: A \rtimes _{\alpha } \! G \to {\mathcal {B}(\mathcal {H}_u)}$
. We define
${A}^{**}_{\alpha }:= i_A(A)'' \subset {\mathcal {B}(\mathcal {H}_u)}$
.
This definition is not the original one given by Ikunishi; we refer to Buss et al. [Reference Buss, Echterhoff and Willett11, Section 2], where it is shown that Definition 6.2 is equivalent to Ikunishi’s definition, and several useful properties of this object are shown. The von Neumann algebra
${A}^{**}_{\alpha }$
carries an action of G given by
$\mathrm {Ad} i_G$
; we will abuse notation and write
$\alpha ^{**}$
for this action, which gives rise to a
$w^*$
-dynamical system
$({A}^{**}_{\alpha }, G, \alpha ^{**})$
. To define an amenable action, we follow [Reference Buss, Echterhoff and Willett11, Definition 3.1], where such actions are called von Neumann amenable.
Definition 6.3 Let
$(A,G,\alpha)$
be a
$C^*$
-dynamical system. Say that
$\alpha $
is amenable if the corresponding universal action
$\alpha ^{**}$
of G on
${A}^{**}_{\alpha }$
is amenable in the sense of Definition 3.1.
For actions of discrete groups,
${A}^{**}_{\alpha } = A^{**}$
[Reference Buss, Echterhoff and Willett11, Section 2.2], so this definition extends the one given by Anantharaman-Delaroche [Reference Anantharaman-Delaroche3, Définition 4.1] for discrete groups. The remainder of this section is concerned with investigating if Definition 6.3 has the properties in Problem 6.1.
First, we consider nuclearity of crossed products, and aim to generalize [Reference Anantharaman-Delaroche3, Théorème 4.5]. The assumption that the action is inner amenable is discussed in Remark 6.5(iii) below.
Theorem 6.4 Let
$(A,G,\alpha)$
be a
$C^*$
-dynamical system, and suppose that
$\alpha ^{**}$
is an inner amenable action of G on
${A}^{**}_{\alpha }$
. The following are equivalent:
-
(i)
$A \rtimes _{\alpha } \! G$
is nuclear; -
(ii)
$A \rtimes _{{\alpha },r} G$
is nuclear; -
(iii)
${A}^{**}_{\alpha } \rtimes _{\alpha ^{**}} \! G$
is injective in
$A(G) - \mathbf {mod}$
; -
(iv)
$\alpha $
is amenable and
${A}^{**}_{\alpha }$
is injective.
Proof (i)
$\implies $
(ii) Nuclearity passes to quotients.
(ii)
$\implies $
(iii) By hypothesis,
$(A \rtimes _{{\alpha },r} G)^{**}$
is injective (in
$\mathbb {C} - \mathbf {mod}$
). Because
$i_A$
is a faithful representation of A, we may form the covariant pair
$((i_A)_\alpha, \lambda)$
as in equation (1). It is shown in [Reference Buss, Echterhoff and Willett11, Remark 2.6] that
$$ \begin{align} {A}^{**}_{\alpha} \rtimes_{\alpha^{**}} \! G = \big( (i_A)_\alpha \rtimes \lambda \big)^{**} \big( (A \rtimes_{{\alpha},r} G)^{**} \big), \end{align} $$
so it follows that
${A}^{**}_{\alpha } \rtimes _{\alpha ^{**}} \! G$
is injective (in
$\mathbb {C} - \mathbf {mod}$
). Because
$\alpha $
is assumed to be inner amenable,
${A}^{**}_{\alpha } \rtimes _{\alpha ^{**}} \! G$
is relatively injective in
$A(G) - \mathbf {mod}$
by Proposition 4.3, so by a standard argument (see [Reference Crann13, Proposition 2.3]),
${A}^{**}_{\alpha } \rtimes _{\alpha ^{**}} \! G$
is injective in
$A(G) - \mathbf {mod}$
.
(iii)
$\implies $
(iv) This is shown in Theorem 3.5.
(iv)
$\implies $
(i) Recent results allow us to adapt Anantharaman-Delaroche’s proof [Reference Anantharaman-Delaroche3, Théorème 4.5] to the locally compact case, as we now explain. We will show that for a (nondegenerate) covariant representation
$(\phi, \rho)$
of
$(A,G,\alpha)$
on the Hilbert space
$\mathcal {H}$
, the von Neumann algebra
$(\phi \rtimes \rho)(A \rtimes _{\alpha } \! G)''$
is injective. In fact, we will show the equivalent statement that the commutant
$(\phi \rtimes \rho)(A \rtimes _{\alpha } \! G)'$
is injective. Observe (as in [Reference Anantharaman-Delaroche3, Théorème 4.5]) that the latter is the space
$$ \begin{align*} \phi(A)^{\prime}_G:= \{ x \in \phi(A)': \rho_r x \rho_r^* = x \text{ for all }r \in G \}. \end{align*} $$
Applying [Reference Buss, Echterhoff and Willett11, Corollary 2.3] to the map
$\phi: A \to \phi (A)$
, we obtain a surjective, equivariant extension
$\phi ^{**}: {A}^{**}_{\alpha } \to \phi (A)''$
. Because
${A}^{**}_{\alpha }$
is injective, so is
$\phi (A)''$
, therefore also
$\phi (A)'$
. Now, because
$\alpha $
is amenable in our sense, it is also amenable in the sense of [Reference Buss, Echterhoff and Willett11, Definition 3.4], by [Reference Bearden and Crann6, Theorem 3.6] (see also [Reference Buss, Echterhoff and Willett11, Section 8]); therefore, by [Reference Buss, Echterhoff and Willett11, Proposition 5.9],
$\alpha $
is commutant amenable in the sense of [Reference Buss, Echterhoff and Willett11, Definition 5.7]. Therefore, by [Reference Bearden and Crann6, Theorem 3.6], the action
$\mathrm {Ad} \rho $
of G on
$\phi (A)'$
given by
$$ \begin{align*} \mathrm{Ad} \rho_r(x):= \rho_r x \rho_r^*, \quad r \in G,\ x \in \phi(A)', \end{align*} $$
is also amenable. Let
$R: L^\infty (G) \mathbin {\overline {\otimes }} \phi (A)' \to \phi (A)'$
be the corresponding norm-1 equivariant projection and define
$$ \begin{align*} R_G: \phi(A)' \to \phi(A)^{\prime}_G;\ R_G(x):= R(\hat{x}), \quad x \in \phi(A)', \end{align*} $$
where
$\hat {x} \in L^\infty (G) \mathbin {\overline {\otimes }} \phi (A)'$
is defined by
$\hat {x}(r):= \mathrm {Ad} \rho _r(x)$
. Because
$\phi (A)'$
is injective and
$R_G$
is a norm-1 projection, we have shown that
$\phi (A)^{\prime }_G$
is injective, as required.▪
Remarks 6.5
-
(i) The implication (iii)
$\implies $
(iv) fails if we do not account for the
$A(G)$
-module structure: there exist locally compact groups, for example,
$\operatorname {SL}(2,\mathbb {C})$
, for which the group von Neumann algebra is injective but the group is not amenable [Reference Connes12] (see also [Reference Brown and Ozawa9, Remark 2.6.10]). Anantharaman-Delaroche works only with discrete groups, where this difficulty does not arise. See also Theorem 2.6. -
(ii) Note that if A is nuclear, then
${A}^{**}_{\alpha }$
is injective, so condition (iv) above holds if
$\alpha $
is amenable and A is nuclear. If G is discrete, then
${A}^{**}_{\alpha } = A^{**}$
, so (iv) is equivalent to (iv)’
$\alpha $
is amenable and A is nuclear. In general, we do not see any reason for (iv) and (iv)’ to be equivalent. -
(iii) Our original goal was to prove a full generalization of [Reference Anantharaman-Delaroche3, Théorème 4.5], without the assumption of inner amenable actions but using nuclearity of
$A \rtimes _{\alpha } \! G$
as an
$A(G)$
-module in (i) and nuclearity of
$A \rtimes _{{\alpha },r} G$
as an
$A(G)$
-module in (ii) (see [Reference Crann15, Section 7] for the definition of these notions). Unfortunately, it seems that such a result would require module versions of the deep results linking injectivity, semidiscreteness, and nuclearity, which do not appear to be known in general.
The following shows that the class of inner amenable locally compact groups is an answer to [Reference Buss, Echterhoff and Willett11, Question 8.3].
Corollary 6.6 Let
$(A,G,\alpha)$
be a
$C^*$
-dynamical system. Suppose that
$A \rtimes _{{\alpha },r} G$
is nuclear and G is inner amenable. Then,
$\alpha $
is amenable.
Proof By hypothesis,
$(A \rtimes _{{\alpha },r} G)^{**}$
is injective, so it follows from [Reference Buss, Echterhoff and Willett11, Remark 2.6] that
${A}^{**}_{\alpha } \rtimes _{\alpha ^{**}} \! G$
is injective. Because G is inner amenable, it follows from Remark 4.2 that
$\alpha ^{**}$
is inner amenable, that is,
${A}^{**}_{\alpha } \rtimes _{\alpha ^{**}} \! G$
is relatively injective in
$A(G) - \mathbf {mod}$
. Now, [Reference Crann13, Proposition 2.3] shows that
${A}^{**}_{\alpha } \rtimes _{\alpha ^{**}} \! G$
is injective in
$A(G) - \mathbf {mod}$
. By Theorem 3.5,
$\alpha $
is amenable.▪
For question (2) on restriction to closed subgroups, we refer to Ozawa and Suzuki [Reference Ozawa and Suzuki29, Corollary 3.4], where the following result was recently shown.
Proposition 6.7 Let
$(A,G,\alpha)$
be a
$C^*$
-dynamical system with
$\alpha $
amenable. If H is a closed subgroup of G, then the restriction
$\alpha |_H$
is also amenable.
A solution to question (3) follows from work of Buss, Echterhoff, and Willett and Bearden and Crann.
Proposition 6.8 Let
$(A,G,\alpha)$
and
$(B,G,\beta)$
be
$C^*$
-dynamical systems and
$\Phi: A \to \mathrm {M}(B)$
a G-equivariant
$*$
-homomorphism such that
$\Phi (A) B$
is dense in B and
$\Phi (\mathcal {Z}({\mathrm {M}(A)})) \subset \mathcal {Z}({\mathrm {M}(B)})$
. If
$\alpha $
is amenable, then so is
$\beta $
.
Proof By [Reference Bearden and Crann6, Theorem 3.6],
$\alpha $
is amenable in our sense if and only if it is amenable in the sense of [Reference Buss, Echterhoff and Willett11, Definition 3.4] (see also [Reference Buss, Echterhoff and Willett11, Section 8]). Thus, the claim is equivalent to [Reference Buss, Echterhoff and Willett11, Lemma 3.17].▪
Examples given by Suzuki [Reference Suzuki32] show that our definition of amenable action does not satisfy the requirements of problem (4) (see [Reference Buss, Echterhoff, Willett, Cortiñas and Weibel10, Section 3] and [Reference Buss, Echterhoff and Willett11, Corollary 3.13]). However, we are able to solve problem (4) in the special case where A is the compact operators on some Hilbert space (see [Reference Buss, Echterhoff and Willett11, Observation 5.24]), as this is the only case for which we know that
${A}^{**}_{\alpha }$
is a factor. It would be interesting to have other examples where
${A}^{**}_{\alpha }$
is a factor.
Proposition 6.9 Let
$(\mathcal {K}(\mathcal {H}),G,\alpha)$
be a
$C^*$
-dynamical system with
$\alpha $
amenable and
$\mathcal {K}(\mathcal {H})$
the
$C^*$
-algebra of compact operators on the Hilbert space
$\mathcal {H}$
. Then, G is amenable.
Proof Ikunishi [Reference Ikunishi22, Example 1] shows that
$\mathcal {K}(\mathcal {H})^{**}_\alpha = {\mathcal {B}(\mathcal {H})}$
, so the claim follows from Proposition 3.4.▪
A solution to question (5) can also be deduced from the work of Bearden and Crann, and Buss, Echterhoff, and Willett.
Proposition 6.10 Let
$(A,G,\alpha)$
be a
$C^*$
-dynamical system with
$\alpha $
amenable. Then, the canonical quotient map
$A \rtimes _{\alpha } \! G \to A \rtimes _{{\alpha },r} G$
is an isomorphism.
Proof It follows from the result of Bearden and Crann [Reference Bearden and Crann6, Theorem 3.6] that if
$\alpha $
is amenable in our sense, then it is amenable in the sense of [Reference Buss, Echterhoff and Willett11, Definition 3.4]. The conclusion then follows from [Reference Buss, Echterhoff and Willett11, Theorem 5.9].▪
Remark 6.11 Buss et al. [Reference Buss, Echterhoff and Willett11, Proposition 5.28 and Example 5.29] have shown that the converse to Proposition 6.10 does not hold for general locally compact groups. We refer to [Reference Buss, Echterhoff and Willett11, Section 5] for an investigation of other conditions which are related to this weak containment property.
Acknowledgment
The second author would like to thank Alireza Medghalchi for his continuous encouragement and the Department of Mathematics of Kharazmi University for support. Parts of this work were completed when the second author was visiting the first author at the Chalmers University of Technology and the University of Gothenburg; she would also like to thank the Department of Mathematical Sciences at the Chalmers University of Technology and the University of Gothenburg for their warm hospitality. Further progress was made when the authors attended the 7th Workshop on Operator Algebras and Their Applications in Tehran, in January 2020; we are very grateful to the organizers for their hospitality. We thank Lyudmila Turowska and Massoud Amini for their valuable comments on early versions of our results and Fatemeh Khosravi for helpful discussions during this work. Finally, thanks to Siegfried Echterhoff for pointing out a mistake in a previous version of Section 6.
