A finitary criterion for selfless tracial C*-algebras

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 $N\ge 1$ and $\epsilon >0$ there exists a unitary $u$ with $\mathrm{\mid }\tau (u^k)\mathrm{\mid }<\epsilon$ ($1\le \mathrm{\mid }k\mathrm{\mid }\le N$) and $\mathrm{\mid }\tau (w)\mathrm{\mid }<\epsilon$ for all alternating words $w$ of length $\le N$ built from centered elements of $F$ and powers $u^n$ ($\mathrm{\mid }n\mathrm{\mid }\le N$). 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.