Noncommutative Choquet simplices

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NONCOMMUTATIVE CHOQUET SIMPLICESMATTHEW KENNEDY AND ELI SHAMOVICH Abstract We introduce a notion of noncommutative Choquetsimplex, or briefly an nc simplex, that generalizes the classicalnotion of a simplex. While every simplex is an nc simplex, thereare many more nc simplices. They arise naturally from C*-algebrasand in noncommutative dynamics. We characterize nc simplices interms of their geometry and in terms of structural properties oftheir corresponding operator systems.There is a natural definition of nc Bauer simplex that general-izes the classical definition of a Bauer simplex. We show that acompact nc convex set is an nc Bauer simplex if and only if it isaffinely 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 invariantwith respect to the action of a discrete group is an nc simplex. Fromthis, we obtain a noncommutative ergodic decomposition theoremwith uniqueness.Finally, we establish a new characterization of discrete groupswith Kazhdan’s property (T) that extends a result of Glasner andWeiss. 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 statespace of a unital C*-algebra.arxiv.org/pdf/1911.01023.pdf