JournalsjncgVol. 6, No. 4pp. 665–719

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

  • Claudia Pinzari

    Università di Roma La Sapienza, Italy
  • John E. Roberts

    Università di Roma Tor Vergata, Italy
A theory of induction and classification of tensor C*-categories cover
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Abstract

This paper addresses the problem of describing the structure of tensor C*-categories M\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 ≥ 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\mathcal{M} to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the GG-equivariant Hermitian bundles over compact homogeneous GG-spaces, with GG a compact group. Our structural results shed light on the problem of whether there is an embedding functor of M\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 M\mathcal{M} is generated by a pseudoreal object of dimension 2.

Cite this article

Claudia Pinzari, John E. Roberts, A theory of induction and classification of tensor C*-categories. J. Noncommut. Geom. 6 (2012), no. 4, pp. 665–719

DOI 10.4171/JNCG/102