The stable rank of some free product C*-algebras

TL;DR

This paper proves that certain reduced free product C*-algebras, including those from free products of discrete groups, have stable rank one, indicating a dense invertible element group under specific conditions.

Contribution

It establishes that reduced free product C*-algebras with the Avitzour condition have stable rank one, extending known results to a broader class of algebras.

Findings

Reduced group C*-algebras of free product groups have stable rank one.

Generalization to reduced free products of unital C*-algebras under the Avitzour condition.

Stable rank one implies the invertible elements are dense in these algebras.

Abstract

It is proved that the reduced group C*-algebra C*_{red}(G) has stable rank one (i.e. its group of invertible elements is a dense subset) if G is a discrete group arising as a free product G_1*G_2 where |G_1|>=2 and |G_2|>=3. This follows from a more general result where it is proved that if (A,tau) is the reduced free product of a family (A_i,tau_i), i\in I, of unital C*-algebras A_i with normalized faithful traces tau_i, and if the family satisfies the Avitzour condition (i.e. the traces, tau_i, are not too lumpy in a specific sense), then A has stable rank one.