## Project Details

### Description

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.

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.

Acronym | FWOTM1064 |
---|---|

Status | Active |

Effective start/end date | 1/11/21 → 31/10/23 |

### Flemish discipline codes

- Category theory, homological algebra
- Functional analysis
- Operator theory

### 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