Selfless reduced free products and graph products of $\mathrm{C}^\ast$-algebras

TL;DR

This paper demonstrates that under mild conditions, reduced free and graph products of C*-algebras are selfless, leading to new examples of simple, monotracial C*-algebras with desirable properties.

Contribution

It establishes selflessness of reduced free and graph products of C*-algebras without the rapid decay assumption, expanding the class of known simple C*-algebras.

Findings

New examples of simple, monotracial C*-algebras with strict comparison

Existence of stable rank one C*-algebras with unique Jiang-Su embedding

Selflessness of reduced free and graph products under mild assumptions

Abstract

Under mild assumptions, we show that reduced free products and reduced graph products of $\mathrm{C}^\ast$-algebras are selfless in the sense of L. Robert, without assuming the rapid decay property. In particular, our main theorems yield numerous new examples of simple, monotracial $\mathrm{C}^\ast$-algebras with strict comparison, stable rank one, and admitting a unique unital embedding of the Jiang-Su algebra $\mathcal{Z}$ up to approximate unitary equivalence.