C*-tensor categories and subfactors for totally disconnected groups

TL;DR

This paper constructs a rigid C*-tensor category from a totally disconnected group and a compact open subgroup, characterizing properties like Haagerup and property (T), and relating it to planar algebras and subfactors.

Contribution

It introduces a new categorical framework linking totally disconnected groups with subfactor theory and planar algebras, providing concrete realizations and characterizations.

Findings

Characterization of Haagerup property and property (T) for the category

Equivalence of the category to a planar algebra from a group action

Realization of the category as bimodules generated by a hyperfinite subfactor

Abstract

We associate a rigid C*-tensor category $C$ to a totally disconnected locally compact group $G$ and a compact open subgroup $K < G$. We characterize when $C$ has the Haagerup property or property (T), and when $C$ is weakly amenable. When $G$ is compactly generated, we prove that $C$ is essentially equivalent to the planar algebra associated by Jones and Burstein to a group acting on a locally finite bipartite graph. We then concretely realize $C$ as the category of bimodules generated by a hyperfinite subfactor.