2026/06/08 by Miles Gould · 1 citation
#math.OA
The purpose of this note is to address the gap in the stable rank one/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 either has stable rank one or is purely infinite. 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 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.