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

TL;DR

The paper proves that reduced free and reduced graph products of C*-algebras are selfless in the sense of L. Robert without assuming rapid decay, yielding a broad class of new simple, monotracial C*-algebras with strict comparison, stable rank one, and unique embeddings of the Jiang–Su algebra $\mathcal{Z}$. It develops a universal framework using graph-product constructions, ultrapower techniques, and Avitzour-type unitary hypotheses to obtain complete selflessness for both graph products (under connected complements) and two-vertex free products. The results provide new instances of $\mathcal{Z}$-stability and strict comparison in non-nuclear settings, extend known simplicity and trace properties, and supply explicit mechanisms for embeddings into ultrapowers that preserve regularity. Overall, the work broadens the landscape of selfless C*-algebras and contributes tools for classification-program regularity phenomena in reduced products.

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.

Theorems & Definitions (25)

• Definition : see Robert25
• Definition : see Ozawa25
• Theorem 1: see Theorem \ref{['Selflessness2']}
• Corollary 2
• Theorem 3: see Theorem \ref{['Selflessness1']}
• Corollary 4
• Corollary 5
• Proposition 1.1: see CaspersFima17
• Proposition 2.1: see Klisse25
• Theorem 2.2: Theorem \ref{['MainTheorem1']}