Convex structures revisited

Abstract

We provide a complete characterization of extreme points of the space of sofic representations. We also show that the restriction map $\text{Sof}(G,P^{{\it\omega}})$ to $\text{Sof}(H,P^{{\it\omega}})$, where $H\subset G$, is not always surjective. The first part of the paper is a continuation of Păunescu [A convex structure on sofic embeddings. Ergod. Th. & Dynam. Sys.34(4) (2014), 1343–1352] and follows more closely the plan of Brown [Topological dynamical systems associated to $\text{II}_{1}$-factors. Adv. Math.227(4) (2011), 1665–1699].