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

Abstract

Under mild assumptions, we show that reduced free products and reduced graph products of ${\mathrm{C}}^{\ast }$-algebras are completely selfless, 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, and of purely infinite ${\mathrm{C}}^{\ast }$-algebras in the traceless case.

v2: Updated results and exposition; v3: Minor changes and updates, now Theorem C and Corollary D cover infinite graphs. This is the final version and will appear in Advances in Mathematics