The Selfless Dichotomy

Abstract

The purpose of this note is to address the gap in the stably finite/purely infinite dichotomy of selfless $C^*$-probability spaces. In particular, we show that nonfaithful selfless $C^*$-probability spaces are purely infinite, simple. This completes the dichotomy: Every selfless $C^*$-algebra is either purely infinite or stably finite. Notably, this shows that every selfless $C^*$-algebra is pure. To accomplish this, we show that infinite reduced free products of $C^*$-probability spaces with nonfaithful states inducing faithful GNS representations are often purely infinite, simple. Having resolved the selfless dichotomy, we improve existing permanence properties of selfless $C^*$-probability spaces, make progress on a conjecture of Choda and Dykema, and produce several new isomorphisms arising from reduced free products.