Topological Boundaries of Representations and Coideals

Abstract: We will introduce and study quantum analogues of Furstenberg-Hamana boundaries of representations of discrete quantum groups, where the Furstenberg boundary is the Furstenberg-Hamana boundary of the left regular representation. Our focus is on the GNS representations of idempotent states, or to put it differently, the quasi-regular representations coming from coideals associated to compact quasi-subgroups. We use their Furstenberg-Hamana boundaries to study (co)amenability properties of such coideals. Then, we combine our work with recent work of Hataishi and De Ro to settle open problems of Kalantar, Kasprzak, Skalski, and Vergnioux for wide classes of quantum groups, including unimodular discrete quantum groups and C*-exact discrete quantum groups. For example, we prove that a unimodular discrete quantum group has the unique trace property iff it acts faithfully on its Furstenberg boundary.

This is joint work with Fatemeh Khosravi.

category theoryoperator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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