Claudia Pinzari

THE REPRESENTATION CATEGORY OF THE WORONOWICZ QUANTUM GROUP SμU(d) AS A BRAIDED TENSOR C*-CATEGORY

An abstract characterization of the representation category of the Woronowicz twisted SU(d) group is given, generalizing analogous results known in the classical case.

AN ALGEBRAIC DUALITY THEORY FOR MULTIPLICATIVE UNITARIES

Multiplicative Unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki-Tatsuuma Duality Theorem characterizes abstractly C*-algebras acted on by unital endomorphisms that are intrinsically related to the regular representation of a multiplicative unitary. The relevant C*-algebras turn out to be simple and indeed separable if the corresponding multiplicative unitaries act on a separable Hilbert space. A categorical analogue provides internal characterizations of minimal representation categories of a multiplicative unitary. Endomorphisms of the Cuntz algebra related algebraically to the grading are discussed as is the notion of braided symmetry in a tensor C*-category.

EMBEDDING ERGODIC ACTIONS OF COMPACT QUANTUM GROUPS ON C*-ALGEBRAS INTO QUOTIENT SPACES

The notion of compact quantum subgroup is revisited and an alternative definition is given. Induced representations are considered and a Frobenius reciprocity theorem is obtained. A relationship between ergodic actions of compact quantum groups on C*-algebras and topological transitivity is investigated. A sufficient condition for embedding such actions in quantum quotient spaces is obtained.

Ergodic Actions of Compact Quantum Groups from Solutions of the Conjugate Equations

AbstractWe use a tensor C ∗ –category with conjugates and two quasitensor func-tors into the category of Hilbert spaces to define a C ∗ –algebra dependingfunctorially on this data. A particular case of this construction allows usto begin with solutions of the conjugate equations and associate ergodicactions of quantum groups on the C ∗ –algebra in question. The quantumgroups involved are A u (Q) and B u (Q). 1 2 1 Introduction The theory of ergodic actions of compact quantum groups on unital C ∗ –algebrashas recently attracted interest. In the group case, one of the first results was thetheorem by Hoegh-Krohn, Landstad and Stormer asserting that the multiplicityof an irreducible representation is always bounded by its dimension and thatthe unique G–invariant state is a trace [9].Ergodic theory for group actions was later investigated by Wassermann in aseries of papers [19], [20], [21], who, among other results, classified all ergodicactions of SU(2) on von Neumann algebras. In particular, he proved the im-portant result that SU(2) cannot act ergodically on the hyperfinite II

A Rigidity Result for Extensions of Braided Tensor C*-Categories Derived from Compact Matrix Quantum Groups

Let G be a classical compact Lie group and Gμ the associated compact matrix quantum group deformed by a positive parameterxa0μ (or

REGULAR OBJECTS, MULTIPLICATIVE UNITARIES AND CONJUGATION

{muin{mathbb R}setminus{0}}

Ergodic actions of SμU(2) on C∗-algebras from II1 subfactors

in the type A case). It is well known that the category of unitary representations of Gμ is a braided tensor C*–category. We show that any braided tensor *–functor