The generator rank of subhomogeneous C*-algebras

TL;DR

This paper calculates the generator rank of subhomogeneous C*-algebras using the dimension of their primitive ideal space and shows that Z-stable ASH-algebras are generically generated by a single element, solving the generator problem for certain classifiable C*-algebras.

Contribution

It provides a formula for the generator rank of subhomogeneous C*-algebras and proves that Z-stable ASH-algebras have generator rank one, with implications for classifiable nuclear C*-algebras.

Findings

Generator rank expressed via primitive ideal space dimension.

Z-stable ASH-algebras have generator rank one.

Generic elements generate the algebra in certain classifiable cases.

Abstract

We compute the generator rank of a subhomgeneous C*-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every Z-stable ASH-algebra has generator rank one, which means that a generic element in such an algebra is a generator. This leads to a strong solution of the generator problem for classifiable, simple, nuclear C*-algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear C*-algebras.