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

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 ≥ 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.

This paper addresses the problem of describing the structure of tensor $C^*$--categories ${\cal M}$ 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 $\geq2$. 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 ${\cal M}$ 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 ${\cal M}$ 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 ${\cal M}$ is generated by a pseudoreal object of dimension $2$.