A theory of induction and classification of tensor C*-categories

TL;DR

This paper develops a framework for understanding tensor C*-categories with conjugates and irreducible units, linking their structure to quantum groups and Hilbert bimodule representations, and explores conditions for embedding into Hilbert spaces.

Contribution

It introduces a functorial correspondence between tensor C*-categories and Hilbert bimodule representations of quantum groups, generalizing classical group actions.

Findings

Existence of a full and faithful tensor functor to bimodule representations

Connection between embedding functors and ergodic actions of Lie groups

Embedding exists for categories generated by a pseudoreal object of dimension 2

Abstract

This paper addresses the problem of describing the structure of tensor C*-categories M with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley--Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension bigger than 2. The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from M to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G-equivariant Hermitian bundles over compact homogeneous G-spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of M into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if M is generated by a pseudoreal object of dimension 2.