C∗-tensor categories and subfactors for totally disconnected groups

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.