1 Introduction
Ultraproducts play an important role in the study of operator algebras and their automorphisms (cf. [Reference Ando and HaagerupAH14]). In this context, Reiji Tomatsu initiated the study of Rohlin flows on von Neumann algebras using ultraproduct techniques [Reference Masuda and TomatsuMT16] and recognized a close relationship between ultraproducts and crossed products, leading to the following theorem.
Theorem 1.1 [Reference TomatsuTom17, Theorem 3.6]
Let
$\alpha $
be an action of a second countable locally compact Hausdorff group G on a von Neumann algebra M with separable predual. Then the following statements hold:
-
(1) There exists a canonical embedding from $M^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega }}G$
into
$(M\rtimes _{\alpha }G)^{\omega }$
. -
(2) If G is abelian, then the embedding induces an isomorphism from $M^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega }}G$
onto
$(M\rtimes _{\alpha }G)^{\omega }_{\hat {\alpha }}$
.
Theorem 1.1 provides a powerful tool for analyzing the structure of ultraproduct von Neumann algebras through continuous or discrete crossed product decompositions. As an application, let
$M=N\rtimes _{\theta } \mathbb {R}$
be the continuous crossed product decomposition of a properly infinite von Neumann algebra M, where N is a semifinite von Neumann algebra and
$\theta $
is a trace scaling one-parameter automorphism group. Then Theorem 1.1 implies that the ultraproduct von Neumann algebra
$M^{\omega }$
also admits a continuous crossed product decomposition,
$M^{\omega }=N^{\omega }\rtimes _{\theta ^{\omega }}\mathbb {R}$
. Hence, the continuous decomposition structure is preserved under ultraproducts, allowing one to lift the analysis of the flow of weights to ultraproduct von Neumann algebras.
The main difficulty is that continuous group actions do not continuously extend to ultraproduct C*algebras. This leads us to the notion of continuous part introduced by Kishimoto [Reference KishimotoKis96]. For a C*algebra
$\mathcal {A}$
, we simply write
$\ell ^{\infty }(\mathcal {A})=\ell ^{\infty }(\mathbb {N},\ \mathcal {A})$
. Define
$C^{\omega }(\mathcal {A}) = \{x \in \ell ^{\infty }(\mathcal {A}) : \lim _{n \to \omega } \|x(n)\| = 0 \}\triangleleft \ell ^{\infty }(\mathcal {A})$
and the ultraproduct of
$\mathcal {A}$
is given by
$\mathcal {A}^{\omega } = \ell ^{\infty }(\mathcal {A}) / C^{\omega }(\mathcal {A})$
. Given a continuous action
$\alpha : G\curvearrowright \mathcal {A}$
, it naturally induces an algebraic ultra-action
$\alpha ^{\omega }: G \to \mathrm {Aut}(\mathcal {A}^{\omega })$
on the ultrapower (note that
$\mathcal {A}^{\omega }$
is non-separable in general and
$\mathrm {Aut}(\mathcal {A}^{\omega })$
may not be a Polish group). By restricting to the continuous part
$\mathcal {A}^{\omega }_{\alpha }:=\ell ^{\infty }_{\alpha }(\mathcal {A})/C^{\omega }(\mathcal {A})$
, we then obtain a C* dynamical system. Here,
$\ell ^{\infty }_{\alpha }(\mathcal {A}):=\{x\in \ell ^{\infty }(\mathcal {A}):~ s\in G,~s\mapsto \bar {\alpha }_{s}(x) ~\mathrm {is~continuous}\}$
, where
$\bar {\alpha }$
acts on
$\ell ^{\infty }(\mathcal {A})$
in the natural way. In this article, we first consider the case of a discrete group, which we always denote by
$\Gamma $
. We conclude with the following theorems.
Theorem 1.2 [Theorems 3.2 and 4.6]
Let
$\alpha $
be an action of an exact discrete group on a C* algebra
$\mathcal {A}$
. Then the following statements hold:
-
(1) There exists a canonical embedding from $\mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega },r}\Gamma $
into
$(\mathcal {A}\rtimes _{\alpha ,r}\Gamma )^{\omega }$
. -
(2) If $\Gamma $
is abelian, then the embedding induces an isomorphism from
$\mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega }}\Gamma $
onto
$(\mathcal {A}\rtimes _{\alpha }\Gamma )^{\omega }_{\hat {\alpha }}$
.
2 Preliminaries
Definition 2.1 Let
$\Gamma $
be a discrete group and let
$\lambda $
be the left regular representation of
$\Gamma $
on
$l^{2}(\Gamma )$
. For a
$\Gamma $
-C*algebra
$\mathcal {A}\stackrel {\pi }{\subset }B(\mathcal {H})$
, define the representation
$\tilde {\pi }$
of
$\mathcal {A}$
on
$\mathcal {H}\otimes \ell ^{2}(\Gamma )$
by
$\tilde {\pi }(a)(\xi \otimes \delta _{g})=(\alpha _{g^{-1}}(a)\xi )\otimes \delta _{g}$
. We can see that
$(\tilde {\pi },1\otimes \lambda ,\mathcal {H}\otimes \ell ^{2}(\Gamma ))$
is a covariant representation. We define the reduced crossed product
$\mathcal {A}\rtimes _{\alpha ,r}\Gamma $
by
$\overline {\mathrm {span}}\{\tilde {\pi }(a)(1\otimes \lambda _{s}):a\in \mathcal {A}, s\in \Gamma \}\subset B(\mathcal {H}\otimes \ell ^{2}(\Gamma ))$
.
Definition 2.2 A C*algebra
$\mathcal {A}$
is said to be exact if for every short exact sequence of
$C^*$
-algebras
$0\to I\to E\to Q\to 0$
, the sequence
$0\to I\otimes _{\mathrm {min}}\mathcal {A}\to E\otimes _{\mathrm {min}}\mathcal {A}\to Q\otimes _{\mathrm {min}}\mathcal {A} \to 0$
is exact. Moreover, a discrete group is exact if its reduced group C*algebra is exact.
For an exact group, the reduced crossed product preserves short exact sequences (see also [Reference Brown and OzawaBO08, Theorem 5.1.10]). Consequently, for the C*short exact sequence, the sequence
is exact and hence induces a well-defined
$*$
isomorphism
2.1 Crossed product for locally compact group
(See [Reference WilliamsDW07, Chapter 2]) Let G be a locally compact Hausdorff group, which is always assumed to be second countable. Suppose that
$(\mathcal {A},G,\alpha )$
is a C* dynamical system and that, for each
$f\in C_{c}(G,\mathcal {A}),$
we define
where
$(\pi ,U)$
is a covariant representation of
$(\mathcal {A},G)$
and
$\pi \rtimes U$
is the integrated form of
$(\pi ,U)$
. The crossed product
$\mathcal {A}\rtimes _{\alpha }G$
is the completion of
$C_{c}(G,\mathcal {A})$
with respect to
$\|\cdot \|_{u}$
. As is well known, there is a one-to-one correspondence between nondegenerate covariant representations of
$(\mathcal {A},G,\alpha )$
and nondegenerate representations of
$\mathcal {A}\rtimes _{\alpha }G$
. Another well-known fact is that, if G is amenable, then
$\mathcal {A}\rtimes _{\alpha }G$
coincides with the reduced crossed product
$\mathcal {A}\rtimes _{\alpha ,r}G$
.
2.2 Duality
(See [Reference RaeburnRae88, Theorem 6]) For a locally compact abelian group G,
extends to a continuous action
$\hat {\alpha }:\hat {G}\curvearrowright \mathcal {A}\rtimes _{\alpha }G$
. By the Takai duality, we have an isomorphism from
$(\mathcal {A}\rtimes _{\alpha }G)\rtimes _{\hat {\alpha }}\hat {G}$
onto
$A\otimes \mathbb {K}(L^{2}(G))$
such that
$\hat {\hat {\alpha }}$
is carried into
$\alpha \otimes \mathrm {Ad}\rho $
, where
$\rho $
denotes right regular representation.
For C*algebra-valued functions, one can avoid issues with approximation by simple functions by using a more direct approach than the Bochner integral. The following well-known lemma provides a convenient definition of integration and its basic properties.
Lemma 2.3 (See [Reference RaeburnRae88, Lemma 7])
For
$f\in C_{c}(G,\mathcal {A})$
, there exists a unique element
$\int f(s)ds\in \mathcal {A}$
, such that for any nondegenerate representation
$\pi $
of
$\mathcal {A}$
,
where
$ds$
denotes the Haar measure on G. Further, the following properties hold:
-
• $(\int f(s)\,ds)\,a = \int f(s)a\,ds \quad \text {for } a\in \mathcal {A}$
; -
• $(\int f(s)\,ds)^{*} = \int f(s)^{*}\,ds$
; -
• $\varphi (\int f(s)\,ds) = \int \varphi (f(s))\,ds$
for any homomorphism
$\varphi :\mathcal {A}\to \mathcal {B}$
.
3 Main result
3.1 On embedding relations
Lemma 3.1 Let
$\Gamma $
be a discrete group, then
$\ell ^{\infty }(\mathcal {A})\rtimes _{\bar {\alpha },r}\Gamma $
is a C*subalgebra of
$\ell ^{\infty }(\mathcal {A}\rtimes _{\alpha ,r}\Gamma )$
. (An action
$\bar {\alpha }$
on
$\ell ^{\infty }(\mathcal {A})$
is naturally considered as
$\bar {\alpha }_{g}((x_{n})_{n})=(\alpha _{g}(x_{n}))_{n}$
.)
Proof
$C_{c}(\Gamma ,\ell ^{\infty }(\mathcal {A}))\ni \sum _{s\in F;F\Subset \Gamma }(a_{s}(n))_{n}u_{s}\mapsto (\sum _{s\in F}a_{s}(n)u_{s})_{n}\in \ell ^{\infty }(C_{c}(\Gamma ,\ \mathcal {A}))$
is a well-defined
$*$
homomorphism. Let
$\iota :\mathcal {A}\hookrightarrow B(\mathcal {H})$
be a faithful representation. Then the
$\ell ^{\infty }$
-direct sum of
$\iota $
denoted by
$\bigoplus _{n}\iota $
defines a faithful representation of
$\ell ^{\infty }(\mathcal {A})$
. Consequently, the induced representation
is also faithful. Since
$\mathcal {H}^{\oplus _{n}}\otimes \ell ^{2}(\Gamma )\simeq \bigoplus _{n}(\mathcal {H}\otimes \ell ^{2}(\Gamma ))$
via a unitary isomorphism, it is straightforward to verify that
$\widetilde {\bigoplus _{n}\iota }\rtimes \oplus (1\otimes \lambda )$
and
$\bigoplus _{n}(\tilde {\iota }\rtimes (1\otimes \lambda ))$
are unitarily equivalent. Thus, we compute
Therefore, the
$*$
homomorphism defined above extends to an injective homomorphism from
$\ell ^{\infty }(\mathcal {A})\rtimes _{\bar {\alpha },r}\Gamma $
into
$\ell ^{\infty }(\mathcal {A}\rtimes _{\alpha ,r}\Gamma )$
.
Let us denote
$C^{\omega }(\mathcal {A}\rtimes _{\alpha ,r}\Gamma )$
by
$\bar {C}^{\omega }$
. For
$a\in C^{\omega }$
,
$\lim _{n}\|\bar {\alpha }_{g}(a)(n)\|\leq \lim _{\omega }\|a(n)\|(^{\forall }g\in \Gamma )$
implies
$\bar {\alpha }_{g}(C^{\omega })=C^{\omega }~(^{\forall }g\in \Gamma )$
and thus
$\bar {\alpha }$
is also an action on
$C^{\omega }$
. Therefore, the following relationship exists:
Theorem 3.2 Let
$\Gamma $
be an exact discrete group, for any actions
$\alpha :\Gamma \curvearrowright \mathcal {A}$
, we have
$\mathcal {A}^{\omega }\rtimes _{\alpha ^{\omega },r}\Gamma \subset (\mathcal {A}\rtimes _{\alpha ,r}\Gamma )^{\omega }$
.
Proof Let
$F\Subset \Gamma $
, and by Lemma 3.1, a map
is a
$*$
isometry. It is well known that, for an exact group, the reduced crossed product preserves short exact sequences [Reference Brown and OzawaBO08, Chapter 5]. Therefore, we obtain the following double exact sequences:
Here, q and
$\bar {q}$
are the canonical quotient maps. As mentioned after Definition 2.2, we note that
$\ell ^{\infty }(\mathcal {A})/C^{\omega }\rtimes _{\alpha ^{\omega },r}\Gamma $
is nothing but
$\ell ^{\infty }(\mathcal {A})\rtimes _{\bar {\alpha },r}\Gamma /C^{\omega }\rtimes _{\bar {\alpha },r}\Gamma $
. We define a
$*$
homomorphism
$\Theta $
so that the right-hand side of the above diagram commutes, that is,
Furthermore, by Lemma 3.1,
$C^{\omega }\rtimes _{\bar {\alpha },r}\Gamma \subset \bar {C}^{\omega }$
as a C*subalgebra. Therefore,
$\Theta $
is well-defined. The following argument is inspired by a technique appearing in [Reference PisierPis09]. It remains to show that
$\Theta $
is injective. Assume
$x\in \ell ^{\infty }(\mathcal {A})/C^{\omega }\rtimes _{\alpha ^{\omega },r}\Gamma $
,
$\Theta (x)=0$
. For
$y\in \ell ^{\infty }(\mathcal {A})\rtimes _{\bar {\alpha },r}\Gamma $
such that
$q\rtimes \mathrm {id}(y)=x$
, then
$\mathcal {J}(y)\in \bar {C}^{\omega }$
, that is,
We define
$p_{S}:\ell ^{\infty }(\mathcal {A})\to \ell ^{\infty }(\mathcal {A})$
as follows: for
$a\in \ell ^{\infty }(\mathcal {A})$
,
$p_{S}(a)_{n}=0$
if
$n\notin S$
and the n-th component remains the same if
$n\in S$
. Clearly,
$p_{S}$
is a projection. For any
$(a_{n})_{n}\in \ell ^{\infty }$
, we have
Indeed, for a fixed
$\varepsilon $
as above, by the definition of S, we have
$S\subset \{i:\|p_{S}(a_{i})-a_{i}\|<\varepsilon \}$
, which implies that
$\{i:\|p_{S}(a_{i})-a_{i}\|<\varepsilon \} \in \omega $
. Because
$p_{S}$
is
$\Gamma $
-equivariant, we can consider
$p_{S}\rtimes \mathrm {id}_{\Gamma }$
, which acts as the identity on
$\mathcal {A}^{\omega }\rtimes _{\alpha ^{\omega },r}\Gamma $
.
Thus, estimating the norm of x, we have
Since
$\varepsilon $
is arbitrary, we conclude
$x=0$
.
Remark If
$\mathcal {B}$
is an exact C*algebra, then
$\mathcal {A}^{\omega }\otimes _{\mathrm {{min}}}\mathcal {B}$
is just the quotient C*algebra of
$\ell ^{\infty }(\mathcal {A})\otimes _{\mathrm {{min}}}\mathcal {B}$
as mentioned after Definition 2.2. Similarly, by the proof of Theorem 3.2, we obtain the following corollary, which is well known to experts and appears, at least implicitly, in [Reference Brown and OzawaBO08].
Corollary 3.3 If
$\mathcal {B}$
is an exact C*algebra, then
$\mathcal {A}^{\omega }\otimes _{\mathrm {min}}\mathcal {B}\subset (\mathcal {A}\otimes _{\mathrm {min}}\mathcal {B})^{\omega }$
.
Example The nuclearity of
$\mathbb {K}(L^{2}(G))$
implies exactness and so
$\mathcal {A}^{\omega }\otimes \mathbb {K}(L^{2}(G))\subset (\mathcal {A}\otimes \mathbb {K}(L^{2}(G)))^{\omega }$
holds.
Proposition 3.4 Let G be a finite group. Then, for any actions
$\alpha :G\curvearrowright \mathcal {A}$
,
$\mathcal {A}^{\omega }\rtimes _{\alpha ^{\omega },r}G=(\mathcal {A}\rtimes _{\alpha ,r}G)^{\omega }$
holds.
Proof By Lemma 3.1, it suffices to show that
$\bar {C}^{\omega }\subset C^{\omega }\rtimes _{\bar {\alpha },r}G$
. Since G is finite, every element of
$\mathcal {A}\rtimes _{r}G$
can be written as a finite sum. Hence, every element of
$(\mathcal {A}\rtimes _{r} G)^{\omega }$
is represented by a bounded sequence of such finite sums.
We can take conditional expectation on reduced crossed product,
$E:C^{\omega }\rtimes _{\bar {\alpha },r}G\to C^{\omega };\sum _{s}[(a_{s}(n))]_{\omega }u_{s}\mapsto [(a_{e}(n))]_{\omega }$
and
This completes the proof.
In the case of a locally compact group, Lemma 3.1 still holds. Indeed, there is a well-defined G-equivariant homomorphism
However, ultralimits do not satisfy the dominated convergence theorem, and hence
$\Theta $
cannot be constructed as in Theorem 3.2. This is due to the fact that the norm estimate required for the well-definedness of
$\Theta $
involves an integral, and in general one cannot interchange the ultralimit and the integral.
Proposition 3.5 Let G be a locally compact group, there exists a homomorphism from
$\mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega }}G$
into
$(\mathcal {A}\rtimes _{\alpha }G)^{\omega }$
.
Proof Let
$(\iota ,u)$
be the canonical covariant representation of
$(\mathcal {A},G,\alpha )$
into the multiplier algebra
$M(\mathcal {A}\rtimes _{\alpha }G)$
. Thanks to the natural inclusion
$M(\mathcal {A}\rtimes _{\alpha }G)^{\omega }\subset M((A\rtimes _{\alpha }G)^{\omega })$
. The (restriction of) ultraproduct representation
where
$u^{\omega }(g)=[u_{g}]_{\omega }$
, are also covariant. However, the strict continuity of
$u^{\omega }$
may fail in general (see Remark). Define the strictly continuous part by
which is a hereditary C*subalgebra of
$(\mathcal {A}\rtimes G)^{\omega }$
.
We now show that
$\iota ^{\omega }(\mathcal {A}^{\omega }_{\alpha })\subset M(\mathcal {U})$
. For
$x\in \mathcal {A}^{\omega }_{\alpha }$
, by covariance, we have
$u^{\omega }_{s}\iota ^{\omega }(x)=\iota ^{\omega }(\alpha ^{\omega }_{s}(x))u^{\omega }_{s}$
and thus for
$y\in \mathcal {U}$
,
Taking adjoints, we conclude that
$\iota ^{\omega }(\mathcal {A}^{\omega }_{\alpha })\subset M(\mathcal {U})$
. Similarly, by definition of
$\mathcal {U}$
, we have
$u^{\omega }_{G}\subset M(\mathcal {U})$
. Hence, by universality, the integrated form
$\iota ^{\omega }\rtimes u^{\omega }$
yields a homomorphism from
$\mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha }G$
into
$M(\mathcal {U})$
. Actually, the image of
$\iota ^{\omega }\rtimes u^{\omega }$
lies in
$\mathcal {U}$
, that is, for
$a\in \mathcal {A}^{\omega }_{\alpha },~f\in C_{c}(G)$
, and
$\iota ^{\omega }\rtimes u^{\omega }(af)=\iota ^{\omega }(a)u^{\omega }(f)\in \mathcal {U}$
. This follows from the observation that the maps
are norm continuous, where
$s.f$
and
$f.s$
denote the left and right translations, respectively. Thus, we obtain the desired map from
$\mathcal {A}^{\omega }_{\alpha }\rtimes G$
to
$(\mathcal {A}\rtimes G)^{\omega }$
.
Remark Let
$\lambda :\mathbb {R}\curvearrowright L^{2}(\mathbb {R})$
be the left regular representation and let
$\lambda ^{\omega }:\mathbb {R}\to M(\mathbb {K}(L^{2}(\mathbb {R}))^{\omega })$
denote its ultraproduct representation.
Choose
$\xi _{n}\in L^{2}(\mathbb {R})$
such that
$\mathrm {supp}\,\xi _{n}\subset [-\frac {1}{n},\frac {1}{n}]~,~\|\xi _{n}\|_{2}=1$
for each
$n\in \mathbb {N}$
. Then, for every fixed
$t\in \mathbb {R}\backslash \{0\}$
, we have
$\|\lambda _{t}\xi _{n}-\xi _{n}\|^{2}_{2}=2$
for all sufficiently large n. Define
$a:=[(\xi _{n}\otimes \xi _{n})]_{\omega }\in \mathbb {K}(L^{2}(\mathbb {R})^{\omega })$
, we obtain
$\|\lambda _{t}^{\omega }a-a\|=\lim _{n\to \omega }\|\lambda _{t}\xi _{n}-\xi _{n}\|_{2}=\sqrt {2}$
, which implies
$\lim _{t\to 0}\lambda ^{\omega }_{t}a\neq a$
.
For the reduced crossed product, we can faithfully represent
$\mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega },r}G\stackrel {\widetilde {\pi ^{\omega }}\rtimes (1\otimes \lambda )}{{\hookrightarrow}
}B(\mathcal {H}^{\omega }\otimes L^{2}(G))$
and
$(\mathcal {A}\rtimes _{\alpha ,r}G)^{\omega }\stackrel {(\tilde {\pi }\rtimes (1\otimes \lambda ))^{\omega }}{{\hookrightarrow}
}B((\mathcal {H}\otimes L^{2}(G))^{\omega })$
. Hence, it is crucial to consider the transformation between the two Hilbert spaces,
$\mathcal {H}^{\omega }\otimes L^{2}(G)$
and
$(\mathcal {H}\otimes L^{}(G))^{\omega }$
.
Lemma 3.6 The map
$V:\mathcal {H}^{\omega }\otimes _{\mathrm {alg}}L^{2}(G)\longrightarrow (\mathcal {H}\otimes L^{2}(G))^{\omega }$
defined by
$V((\xi _{n})_{\omega }\otimes \eta )=(\xi _{n}\otimes \eta )_{\omega }$
extends to an isometry on
$\mathcal {H}^{\omega }\otimes L^{2}(G)$
. Moreover, for each
$a\in \mathcal {A}_{\alpha }^{\omega }$
and
$g\in G,$
the following relations hold:
Proof This follows from a straightforward calculation.
Theorem 3.7 If G is amenable, then
$\mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega }}G\subset (\mathcal {A}\rtimes _{\alpha }G)^{\omega }$
.
Proof Because G is amenable, the above discussion allows us to faithfully represent
$\mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega }}G$
on
$\mathcal {H}^{\omega }\otimes L^{2}(G)$
by
$\widetilde {\pi ^{\omega }}\rtimes (1\otimes \lambda )$
. By Lemma 3.6, the injective homomorphism
maps
$af$
to
Therefore, for each
$af\in \mathcal {A}^{\omega }_{\alpha }\otimes _{\mathrm {alg}}C_{c}(G)$
,
$\iota ^{\omega }\rtimes u^{\omega }(af)VV^{*}=\mathrm {Ad}V\circ \widetilde {\pi ^{\omega }}\rtimes (1\otimes \lambda )(af)$
. Thus, this equality holds for every
$x\in \mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega }}G$
. This proves that
$\iota ^{\omega }\rtimes u^{\omega }$
is injective.
For abelian G, by a simple calculation, we obtain
and hence, by density,
$\iota ^{\omega }\rtimes u^{\omega }$
is
$\hat {G}$
-equivariant. Since the dual action is automatically continuous, in particular, we have
$\mathcal {A}^{\omega }_{\alpha }\rtimes _{\alpha ^{\omega }}G\subset (\mathcal {A}\rtimes _{\alpha }G)^{\omega }_{\hat {\alpha }}$
. The question remains whether the reverse inclusion holds.
4 Reverse inclusion
The following diagram summarizes the relationships between ultraproduct of crossed products by a discrete abelian group and the corresponding dual actions:
A key step in the proof of Theorem 1.1(2) is the equality
$(M \otimes B(L^{2}(G)))_{\alpha \otimes \operatorname {Ad} \rho }^{\omega } = M_{\alpha }^{\omega } \otimes B(L^{2}(G))$
, which follows from the tensor product decomposition for a type I factor
$Q\subset M$
, namely,
$M=Q\vee (Q^\prime \cap M)$
(cf. [Reference Masuda and TomatsuMT16, Lemma 2.8]). However, in our situation, we do not have sufficient information about the inverse of
$\sigma $
. Therefore, the von Neumann algebra argument cannot be directly applied.
For a free ultrafilter
$\omega $
, the non-separability of
$c_{0}(\mathbb {Z})^{\omega }$
can be proved in the same way as for
$\ell ^{\infty }$
.
Lemma 4.1 The ultraproduct
$c_{0}(\mathbb {Z})^{\omega }$
is not separable.
Proof Let
$I \subset \mathbb {N}$
, and define
$e_I \in \ell ^{\infty }(c_0(\mathbb {Z}))$
by
For
$I,J\subset \mathbb {N}$
, we have
$\|e_{I}(n)-e_{J}(n)\|=\chi _{I\triangle J}(n)$
. Let
$q:\ell ^{\infty }(c_{0})\twoheadrightarrow c_{0}^{\omega }$
be the natural quotient map. Define an equivalence relation on
$\mathcal {P}(\mathbb {N})$
by
$I \sim J :\Longleftrightarrow I \triangle J \notin \omega $
. Since
$\omega $
is free, we have
$\vert \mathcal {P}(\mathbb {N})/\sim \vert ~= 2^{\aleph _{0}}$
.
For
$I\nsim J$
, we have
$\lim _{\omega }\chi _{I\triangle J}(n)=1$
and thus
$\|q(e_{I})-q(e_{J})\|=1$
.
Now let
$S \subset c_{0}^\omega $
be dense. For each
$I \in \mathcal {P}(\mathbb {N})/\sim $
, choose
$s_{I}\in S$
such that
${\|q(e_{I})-s_{I}\|<\frac {1}{2}}$
.
If
$I\nsim J$
, then by the triangle inequality,
Hence, the map
$I\mapsto s_I$
is injective, which implies that
$c_{0}^{\omega }$
is not separable.
It is natural to ask whether the reverse inclusion always holds. The main point is that for a compact group, its dual is discrete, so the ultra-action is automatically continuous, which enlarges the right-hand side.
Proposition 4.2 There exists a locally compact abelian group G and a C*algebra
$\mathcal {A}$
such that
Proof Let us assume that
$\mathcal {A}=\mathbb {C}$
,
$G=\mathbb {T}$
, and let
$\tau $
denote the trivial action.
By the Pontryagin duality theorem, we have
$C^{*}\mathbb {T}\simeq C_{0}(\mathbb {Z})=c_{0}(\mathbb {Z}),$
which is separable. Moreover,
$C^{*}\mathbb {T}^{\omega }_{\tau }=C^{*}\mathbb {T}^{\omega }=c_{0}(\mathbb {Z})^{\omega }$
.
Hence, by Lemma 4.1,
$\mathbb {C}\rtimes _{\tau }\mathbb {T}\subsetneq (\mathbb {C}\rtimes _{\tau }\mathbb {T})^{\omega }_{\hat {\tau }}$
.
To avoid the situation in Theorem 4.2, let us set that
$G=\Gamma $
is a discrete abelian group. In this case, does the reverse inclusion hold? For this purpose, we introduce some notation and a few auxiliary lemmas.
Notation 4.3 (cf. [Reference PedersenPed79])
For
$s\in \Gamma $
, set the closed subspace
For
$x\in \mathcal {A}\rtimes \Gamma \backslash \{0\}$
,
$F_{x}:\hat {\Gamma }\ni \gamma \mapsto \overline {\langle {s,\gamma }\rangle}\hat {\alpha }_{\gamma }(x)$
is continuous. Since
$\mathrm {supp}(F_{x})=\hat {\Gamma }$
is compact, the Haar integral of
$F_{x}$
exists. More precisely, by Lemma 2.3, we can define
Moreover, it is straightforward to check that
$E_{s}$
is a map onto
$\mathcal {A}\rtimes _{\alpha }\Gamma (s)$
and satisfies
$\|E_{s}(x)\|\leq \|x\|,~E_{s}^{2}=E_{s}$
.
Lemma 4.4
$\mathcal {A}\lambda _{s}=\mathcal {A}\rtimes \Gamma (s),~s\in \Gamma $
.
Proof Take
$x\in \mathcal {A}\rtimes \Gamma (s)$
. Then
$\hat {\alpha }_{\gamma }(x\lambda ^{*}_{s})=\langle{s,\gamma }\rangle x\overline {\langle{s,\gamma }\rangle}\lambda ^{*}_{s}=x\lambda ^{*}_{s}$
Thus,
$x\lambda ^{*}_{s}\in \mathcal {A}\rtimes _{\alpha }\Gamma (e)=(\mathcal {A}\rtimes _{\alpha }\Gamma )^{\hat {\alpha }}=\mathcal {A}$
. Therefore,
$x\in \mathcal {A}\lambda _{s}$
.
By the preceding lemma, we know that the C*subalgebra
$\overline {\mathrm {span}}_{s}\mathcal {A}\rtimes _{\alpha }\Gamma (s)\subset \mathcal {A}\rtimes _{\alpha }\Gamma $
actually coincides with the entire crossed product
$\mathcal {A}\rtimes _{\alpha }\Gamma $
.
Lemma 4.5 For the compact group action
$\hat {\alpha }^{\omega }:\hat {\Gamma }\curvearrowright (\mathcal {A}\rtimes _{\alpha }\Gamma )^{\omega }_{\hat {\alpha }}$
, we have
$(\mathcal {A}\rtimes _{\alpha }\Gamma )^{\omega }_{\hat {\alpha }}=\overline {\mathrm {span}}_{s}(\mathcal {A}\rtimes \Gamma )^{\omega }(s)$
.
Proof The continuity of
$\hat {\alpha }^{\omega }$
allows us to consider
$E_{s,\omega }:(\mathcal {A}\rtimes _{\alpha }\Gamma )^{\omega }_{\hat {\alpha }}\twoheadrightarrow (\mathcal {A}\rtimes \Gamma )^{\omega }(s)$
;
$E_{s,\omega }(x)=\int _{\hat {\Gamma }}\overline {\langle{s,\gamma }\rangle }\hat {\alpha }^{\omega }_{\gamma }(x)d\gamma $
, which is well defined by Lemma 2.3.
Let us assume that
$(\mathcal {A}\rtimes _{\alpha }\Gamma )^{\omega }_{\hat {\alpha }}\backslash \overline {\mathrm {span}}_{s}(\mathcal {A}\rtimes \Gamma )^{\omega }(s)$
is nonempty, and that x belongs to it. By the Hahn–Banach theorem, there exists a linear functional
$\varphi \in (\mathcal {A}\rtimes _{\alpha }\Gamma )^{\omega ~*}_{\hat {\alpha }}$
such that
$\varphi \neq 0$
and
$\varphi (y)=0$
for all
$y\in \mathrm {span}_{s}(\mathcal {A}\rtimes \Gamma )^{\omega }(s)$
.
Hence, for all
$s\in \Gamma $
,
$\varphi ({E_{s,\omega }}(x))=\int _{\hat {\Gamma }}\overline {\langle{s,\gamma }\rangle}\varphi (\hat {\alpha }^{\omega }_{\gamma }(x))d\gamma =0$
. By the uniqueness of Fourier coefficients,
$\varphi (\hat {\alpha }^{\omega }_{\gamma }(x))=0$
for all
$\gamma $
. In particular, taking
$\gamma =e$
gives a contradiction.
Note that
$(x_{n})_{n}\in \ell ^{\infty }(\mathcal {A}\rtimes \Gamma )$
is uniformly bounded. Since
$E_{s}$
is bounded, we can define
$E^{\omega }_{s}:(\mathcal {A}\rtimes \Gamma )^{\omega }_{\hat {\alpha }}\to \mathcal {A}\rtimes \Gamma (s)^{\omega }$
;
$ E^{\omega }_{s}([(x_{n})_{n}]_{\omega }):=[(E_{s}(x_{n}))_{n}]_{\omega }$
. Furthermore, by Lemma 4.4, the image of
$E_{s}^{\omega }$
is contained in
$(\mathcal {A}\lambda _{s})^{\omega }=\mathcal {A}^{\omega }\lambda _{s}$
.
Theorem 4.6 For a discrete abelian group
$\Gamma $
, we have
$\mathcal {A}^{\omega }\rtimes _{\alpha ^{\omega }}\Gamma =(\mathcal {A}\rtimes _{\alpha }\Gamma )^{\omega }_{\hat {\alpha }}$
.
Proof For
$x\in (\mathcal {A}\rtimes \Gamma )^{\omega }(s)$
, there exists
$[(y_{n})_{n}]_{\omega }\in (\mathcal {A}\rtimes \Gamma )^{\omega }_{\hat {\alpha }}$
, where the lift
$(y_{n})$
of x lies in the continuous part of
$\ell ^{\infty }(\mathcal {A}\rtimes \Gamma )$
, such that
$E_{s,\omega }([(y_{n})_{n}]_{\omega })=x$
. Then
Since the restriction of the quotient map
$[\cdot ]_{\omega }:\ell ^{\infty }(\mathcal {A}\rtimes \Gamma )\twoheadrightarrow (\mathcal {A}\rtimes \Gamma )^{\omega }$
to the continuous part is also bounded (or equivalently, by using Lemma 2.3 on the properties of integrals of C*algebra-valued functions), the above formula equals
This shows that
$x\in \overline {\mathrm {span}}_{s}\mathcal {A}^{\omega }\lambda _{s}=\mathcal {A}^{\omega }\rtimes \Gamma $
. Hence, by Lemma 4.5, we obtain the desired conclusion.
Acknowledgements
The author is grateful to Toshihiko Masuda for his support and valuable comments on this article as my supervisor. The author would also like to thank Miho Mukohara for helpful discussions regarding this work.
