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

Abstract

This paper addresses the problem of describing the structure of tensor C*-categories $\mathcal{M}$ 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 $\ge 2$. 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 $\mathcal{M}$ 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 $\mathcal{M}$ 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 $\mathcal{M}$ is generated by a pseudoreal object of dimension 2.