Noncommutative Geometry and Quantum Physics Vietri sul Mare, August 31 - September 5, 2009
Home page Speakers Participants Program Picture Lodging Travel Equipment Registration Sponsors	Speaker: Claudia Pinzari, Univ. Roma "Sapienza" Title: A theory of induction and classification of tensor C*-categories Abstract: (Joint work with J.E. Roberts) We are interested in the problem of representing tensor C*-categories with conjugation. If a permutation symmetry is present, a result by Doplicher and Roberts of the 80's shows that such categories are isomorphic to representation categories of compact groups. However, low dimensional QFTs or the theory of subfactors initiated by V. Jones, provide categories with conjugation but not permutation symmetric. Often, these can not be embedded into the category of Hilbert spaces. In the case where the object set has a distinguished generating object, we show how to associate to the given category an ergodic action of the Wang-van Daele compact quantum groups Ao(F) and Au(F) on a unital noncommutative C*-algebra. This ergodic space will be understood as a virtual subgroup in the sense of Mackey but in a noncommutative compact setting. We shall develop a theory of induction from this virtual subgroup which leads to an identification of the given category with a category of representations of the quantum group over Hilbert C*-bimodules. In particular, a new geometric interpretation of a full tensor subcategory of the category of Ocneanu's bimodules associated with a II1 subfactor will be given. Our results shed light on the problem of recognizing which tensor categories can be embedded into the category of Hilbert spaces.
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