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.

arXiv commentv2: minor changes, final version, to appear in the proceedings of the 2015 Abel Symposium

References in corpus (7)

• On a Class of Type II${}_1$ Factors with Betti Numbers Invariants
• Examples of groups which are not weakly amenable
• CCAP for universal discrete quantum groups
• Skein theory for the D_{2n} planar algebras
• Representation theory for subfactors, $\lambda$-lattices and C*-tensor categories
• Drinfeld center and representation theory for monoidal categories
• Finitely presentable, non-Hopfian groups with Kazhdan's Property (T) and infinite outer automorphism group

Cited by in corpus (6)

• Property (T) discrete quantum groups and subfactors with triangle presentations
• The Fourier algebra of a rigid $C^{\ast }$-tensor category
• Some remarks on free products of rigid $C^{\ast }$-2-categories
• Standard $\lambda$-lattices, rigid ${\mathrm{C}}^{\ast }$ tensor categories, and (bi)modules
• $L^2$-Betti numbers of $C^{\ast }$-tensor categories associated with totally disconnected groups
• On fixed point planar algebras