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Equivariant (co)module nuclearity of $C^*$-crossed products

Massoud Amini, Qing Meng

Abstract

We define an equivariant and equicovariant versions of the notion of module nuclearity. More precisely, for a discrete group $Γ$ and operator $\mathcal A$-$Γ$-(co)module $\mathcal B$, $\mathcal E$ over a $Γ$-C$^*$-algebra $\mathcal A$, we define $\mathcal E$-$Γ$-nuclearity of $\mathcal B$, as an equivariant version of the notion of $\mathcal E$-nuclearity, in which the identity map on $\mathcal B$ is required to be approximately factored through matrix algebras on $\mathcal E$ with module structures coming both from the original module structure of $\mathcal E$ and the $Γ$-action on $\mathcal E$. For trivial actions of $Γ$, this is shown to reduce to the notion of module nuclearity, introduced and studied by the first author. As a concrete example, for a discrete group $Γ$ acting amenably on a unital C$^*$-algebra $\mathcal A$, we show that the reduced crossed product $\mathcal A\rtimes_{r} Γ$ is $\mathcal A$-$Γ$-nuclear. Conversely, if $\mathcal A$ is a nuclear C$^*$-algebra with a $Γ$-invariant state $ρ$ and $\mathcal A\rtimes_{r} Γ$ is $\mathcal A$-$Γ$-nuclear, then we deduce that $Γ$ is amenable. We show that when $\mathcal A\rtimes_{r} Γ$ is $\mathcal A$-$Γ$-nuclear and $\mathcal A$ has the completely bounded approximation property (resp., is exact), then so is $\mathcal A\rtimes_{r} Γ$. We prove similar results for $\mathcal A\rtimes_{r} Γ$, regarded as an $\mathcal A$-$Γ$-comodule.

Equivariant (co)module nuclearity of $C^*$-crossed products

Abstract

We define an equivariant and equicovariant versions of the notion of module nuclearity. More precisely, for a discrete group and operator --(co)module , over a -C-algebra , we define --nuclearity of , as an equivariant version of the notion of -nuclearity, in which the identity map on is required to be approximately factored through matrix algebras on with module structures coming both from the original module structure of and the -action on . For trivial actions of , this is shown to reduce to the notion of module nuclearity, introduced and studied by the first author. As a concrete example, for a discrete group acting amenably on a unital C-algebra , we show that the reduced crossed product is --nuclear. Conversely, if is a nuclear C-algebra with a -invariant state and is --nuclear, then we deduce that is amenable. We show that when is --nuclear and has the completely bounded approximation property (resp., is exact), then so is . We prove similar results for , regarded as an --comodule.
Paper Structure (3 sections, 13 theorems, 52 equations)

This paper contains 3 sections, 13 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Equivariant (Co)module Nuclearity
  3. Main Results

Key Result

Lemma 2.10

Let $\mathcal{B}$ and $\mathcal{D}$ be operator $\mathcal{A}$-$\Gamma$-(co)modules and $\mathcal{E}$ be an operator system such that $\mathcal{E}\otimes\mathbb B(\ell^2(\Gamma^\infty))$ has an $\mathcal{A}$-$\Gamma$-(co)module structure. Let $\theta: \mathcal{B} \rightarrow \mathcal{D}$ be a c.c.p.

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 1
  • ...and 26 more