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

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.