Amenable actions of unitary tensor categories (FWOTM1064)

Amenability is an important concept in the theory of groups and
group actions on (compact) topological spaces. It has many
fascinating connections to the theory of operator algebras and their
approximation properties. The categorification of this setup is gaining
attention, where actions of unitary tensor categories on C*-categories
are considered. We will introduce the notion of amenability for such
actions.
We will also build further on recent work by Kalantar and Kennedy.
They showed that an amenable action of a group on a certain
topological space, the Furstenberg boundary, can be related to the
purely operator-algebraic concept of exactness. More recently, the
Furstenberg boundary was generalised to the non-commutative
setting and has led to important advances in the theory of discrete
quantum groups. Our aim is to take this to the next level, and to
construct a Furstenberg boundary associated with a unitary tensor
category as a certain action on a C*-category.
As an application, we will provide new examples of C*-algebras
satisfying Ozawa’s conjecture. This conjecture broadly states that
any exact C*-algebra can be embedded in a larger nuclear C*-
algebra that is ‘not too far removed’ from the original one.

Keywords:functional analysis, C*-tensor category,, amenability, quantum group, compact quantum group, discrete quantum group, geometric group theory, group action, operator algebras, C*-algebra, von Neumann algebra, amenable action, tensor category, nuclear C*, algebra, exact C*-algebra, Ozawa's conjecture, exact group, Furstenberg boundary

Disciplines:Category theory, homological algebra, Functional analysis, Operator theory