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.
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Acknowledgements
The first author is grateful to Martino Lupini for enlightening conversations about the classical Poulsen simplex and the Kirchberg–Wassermann operator system. The authors thank Jamie Gabe for pointing out the reference [28].
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Communicated by Andreas Thom.
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First author supported by NSERC Grant Number 50503-10787.
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Kennedy, M., Shamovich, E. Noncommutative Choquet simplices. Math. Ann. 382, 1591–1629 (2022). https://doi.org/10.1007/s00208-021-02261-z
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DOI: https://doi.org/10.1007/s00208-021-02261-z