Noncommutative Choquet simplices

TL;DR

This paper introduces noncommutative Choquet simplices, generalizing classical simplices within the framework of C*-algebras, and explores their geometric, structural, and dynamical properties, including applications to noncommutative dynamics and group properties.

Contribution

It defines and characterizes noncommutative Choquet and Bauer simplices, extending classical results, and applies these concepts to noncommutative dynamics and group theory, including a new characterization of property (T).

Findings

Noncommutative simplices generalize classical simplices.

Characterization of nc Bauer simplices via C*-algebras.

Invariant nc states form nc simplices, leading to ergodic decomposition.

Abstract

We introduce a notion of noncommutative Choquet simplex, or briefly an nc simplex, that generalizes the classical notion of a simplex. While every simplex is an nc simplex, there are many more nc simplices. They arise naturally from C*-algebras and in noncommutative dynamics. We characterize nc simplices in terms of their geometry and in terms of structural properties of their corresponding operator systems. There is a natural definition of nc Bauer simplex that generalizes the classical definition of a Bauer simplex. We show that a compact nc convex set is an nc Bauer simplex if and only if it is affinely homeomorphic to the nc state space of a unital C*-algebra, generalizing a classical result of Bauer for unital commutative C*-algebras. We obtain several applications to noncommutative dynamics. We show that the set of nc states of a C*-algebra that are invariant with respect to the action of a discrete group is an nc simplex. From this, we obtain a noncommutative ergodic decomposition theorem with uniqueness. Finally, we establish a new characterization of discrete groups with Kazhdan's property (T) that extends a result of Glasner and Weiss. Specifically, we show that a discrete group has property (T) if and only if for every action of the group on a unital C*-algebra, the set of invariant states is affinely homeomorphic to the state space of a unital C*-algebra.