The generator problem for Z-stable C*-algebras

TL;DR

This paper proves that all unital, separable, Z-stable C*-algebras are singly generated, solving a longstanding problem and demonstrating applications to tensor products involving reduced group C*-algebras.

Contribution

It establishes that Z-stable C*-algebras are singly generated, providing a solution to the generator problem for this class and exploring related applications.

Findings

Z-stable C*-algebras are singly generated

Z embeds into reduced group C*-algebras with non-cyclic free subgroups

Tensor products of reduced free group C*-algebras are singly generated

Abstract

The generator problem was posed by Kadison in 1967, and it remains open until today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z tensorially. More precisely, we show that every unital, separable, Z-stable C*-algebras A is singly generated, which means that there exists an element x in A that is not contained in any proper sub-C*-algebra of A. To give applications of our result, we observe that Z can be embedded into the reduced group C*-algebra of a discrete group that contains a non-cyclic, free subgroup. It follows that certain tensor products with reduced group C*-algebras are singly generated. In particular, the tensor product of two reduced free group C*-algebras is singly generated.