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

TL;DR

This paper introduces and analyzes new notions of nuclear dimension and decomposition rank for $C^*$-modules, establishing their properties and relationships with other invariants, and providing estimates for specific examples.

Contribution

It defines module nuclear dimension and decomposition rank, explores their properties, and relates them to module traces and the Cuntz semigroup, advancing the theory of $C^*$-modules.

Findings

Module nuclear dimension is zero for $rak A$-NF $A$.

Finite module decomposition rank implies $A$ is $rak A$-QD.

Provides estimates of module nuclear dimension for certain examples.

Abstract

We introduce and study a notion of module nuclear dimension for a $C^{*}$-algebra $A$ which is $C^*$-module over another $C^*$-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.