Finite approximation properties of C*-modules III

We introduce and study a notion of module nuclear dimension for a C*-algebra A which is a C*-module over another C*-algebra (sic) with compatible actions. We show that the module nuclear dimension of A is zero if A is (sic)-NF. The converse is shown to hold when (sic) 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 (sic) is unital and simple, if the module decomposition rank of A is finite then A is (sic)-QD. We study the set T-(sic)(A) of (sic)-valued module traces on A and relate the Cuntz semigroup of A with lower semicontinuous affine functions on the set T-(sic)(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.