The generator problem for Z-stable C*-algebras

Thiel Hannes, Winter Wilhelm

Abstract

The generator problem was posed by Kadison in 1967, and it remains open today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z{stroke} tensorially. More precisely, we show that every unital, separable, Z{stroke}-stable C*-algebra A is singly generated, which means that there exists an element x ∈ A that is not contained in any proper sub-C*-algebra of A. To give applications of our result, we observe that Z{stroke} 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, C*r(F∞) ⊗ C*r(F∞) is singly generated. © 2014 American Mathematical Society.