Equivariant (co)module nuclearity of $C^{\ast }$-crossed products

Abstract

We define an equivariant and equicovariant versions of the notion of module nuclearity. More precisely, for a discrete group $\Gamma$ and operator $\mathcal{A}$-$\Gamma$-(co)module $\mathcal{B}$, $\mathcal{E}$ over a $\Gamma$-C${}^{\ast }$-algebra $\mathcal{A}$, we define $\mathcal{E}$-$\Gamma$-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 $\Gamma$-action on $\mathcal{E}$. For trivial actions of $\Gamma$, 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 $\Gamma$ acting amenably on a unital C${}^{\ast }$-algebra $\mathcal{A}$, we show that the reduced crossed product $\mathcal{A}{⋊}_r\Gamma$ is $\mathcal{A}$-$\Gamma$-nuclear. Conversely, if $\mathcal{A}$ is a nuclear C${}^{\ast }$-algebra with a $\Gamma$-invariant state $\rho$ and $\mathcal{A}{⋊}_r\Gamma$ is $\mathcal{A}$-$\Gamma$-nuclear, then we deduce that $\Gamma$ is amenable. We show that when $\mathcal{A}{⋊}_r\Gamma$ is $\mathcal{A}$-$\Gamma$-nuclear and $\mathcal{A}$ has the completely bounded approximation property (resp., is exact), then so is $\mathcal{A}{⋊}_r\Gamma$. We prove similar results for $\mathcal{A}{⋊}_r\Gamma$, regarded as an $\mathcal{A}$-$\Gamma$-comodule.