Finite approximation properties of $C^{\ast }$-modules III

Abstract

We introduce and study a notion of module nuclear dimension for a $C^{\ast }$-algebra $A$ which is $C^{\ast }$-module over another $C^{\ast }$-algebra $\mathfrak{A}$ with compatible actions. We show that the module nuclear dimension of $A$ is zero if $A$ is $\mathfrak{A}$-NF. The converse is shown to hold when $\mathfrak{A}$ is a $C(X)$-algebra with simple fibers, with $X$ compact and totally disconnected. We also introduce a notion of module decomposition rank, and show that when $\mathfrak{A}$ is unital and simple, if the module decomposition rank of $A$ is finite then $A$ is $\mathfrak{A}$-QD. We study the set ${\mathcal{T}}_{\mathfrak{A}}(A)$ of $\mathfrak{A}$-valued module traces on $A$ and relate the Cuntz semigroup of $A$ with lower semicontinuous affine functions on the set ${\mathcal{T}}_{\mathfrak{A}}(A)$. Along the way, we also prove a module Choi-Effros lifting theorem. We give estimates of the module nuclear dimension for a class of examples.