• DOI:10.1007/s00208-021-02261-z
• Corpus ID: 207870960

Noncommutative Choquet simplices

@article{Kennedy2019NoncommutativeCS,
  title={Noncommutative Choquet simplices},
  author={Matthew Kennedy and Eli Shamovich},
  journal={Mathematische Annalen},
  year={2019},
  volume={382},
  pages={1591 - 1629}
}

• M. Kennedy, E. Shamovich
• Published 4 November 2019
• Mathematics
• Mathematische Annalen

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.