Selfless Inclusions of C*-Algebras

TL;DR

This work extends the notion of selflessness from C*-probability spaces to inclusions of C*-probability spaces, defining selfless inclusions via existential first-factor embeddings and ultrapower considerations. It develops a PHP property for inclusions and shows that PHP implies selflessness, adapting Ozawa’s group-theoretic techniques to the operator-algebraic setting and deriving Avitzour-type results for reduced free products. A central theme is the use of free complexification and Z-stability as sources of selfless inclusions, enabling new selfless examples beyond classical free-products and group C*-algebras. As key applications, the authors prove selflessness for the reduced free unitary quantum groups A_u(n) and the half-liberated unitary quantum groups A_u^*(n) for n\\ge 2, with further consequences for regularity properties like stable rank and comparison, broadening the landscape of selfless C*-probability spaces.

Abstract

We introduce and study a natural notion of selflessness for inclusions of C*-probability spaces, which in particular implies that all intermediate C*-algebras are selfless in the sense of Robert. We identify natural sources of selfless inclusions in the realms of Z-stable and free product C*-algebras. As an application of this, we prove selflessness for a new family of C*-probability spaces outside the regime of free products and group C*-algebras. These include the reduced free unitary compact quantum groups.

Theorems & Definitions (33)

• Theorem 1.1: Theorem \ref{['thm:cpct-qntm-unitry-slflss']}
• Corollary 1.2
• Lemma 2.1
• Theorem 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Lemma 2.6