A finitary criterion for selfless tracial C*-algebras

TL;DR

This paper establishes a finitary criterion for selfless tracial C*-algebras, showing equivalence to approximate selflessness and exploring implications for group C*-algebras and properties like nuclearity.

Contribution

It introduces a new finitary condition characterizing selfless tracial C*-algebras and proves their equivalence to the existing definition for separable cases.

Findings

Selflessness is equivalent to approximate selflessness in separable tracial C*-algebras.

A concise proof that certain group C*-algebras are C*-selfless.

Discussion of the relation to nuclearity and $\\mathcal{Z}$-stability.

Abstract

We study the class of selfless C*-probability spaces introduced by Robert. It is known that a selfless tracial algebra has strict comparison and a unique trace. We prove that for separable tracial C*-algebras, selflessness is equivalent to approximate selflessness, a finitary condition: for every finite set $F$, every $N \geq 1$ and $\varepsilon > 0$ there exists a unitary $u$ with $|\tau(u^k)| < \varepsilon$ ($1 \leq |k| \leq N$) and $|\tau(w)| < \varepsilon$ for all alternating words $w$ of length $\leq N$ built from centered elements of $F$ and powers $u^n$ ($|n| \leq N$). The equivalence is established using a diagonalisation argument in the tracial ultrapower. As an application, we give a concise proof that countable groups with a topologically-free extreme boundary are C*-selfless. We also discuss the relation to nuclearity and $\mathcal{Z}$-stability.