Relative Amenability, Amenability, and Coamenability of Coideals

Abstract: Amenability is a deeply studied property of groups, with many interesting reformulations and connections to the operator algebraic aspects of groups. For example, the reduced $C^*$-algebra $C^*_r(G)$ of a discrete group has a unique tracial state if and only if there are no non-trivial amenable normal subgroups. This, among other related results, makes it apparent that the structure of the amenable subgroups of $G$ contains important information about $C^*_r(G)$. For a quantum group $\mathbb{G}$, an appropriate analogue of a subgroup is a coideal $N\subseteq L^\infty(\mathbb{G})$. We will present notions of relative amenability, amenability, and coamenability for coideals of discrete and compact quantum groups motivated by "relativizations" of amenability and coamenability of a subgroup of a group. We will discuss the known relationships between these formally distinct notions and their relevance to certain properties of the reduced $C^*$-algebras of discrete quantum groups.

operator 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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