14.45-15.45. MARCO ZUNINO, Strasbourg. "Quasitriangular Crossed Structures" We discuss the definition of both crossed group coalgebras and crossed group categories recently introduced by Turaev for topological motivation. The first is a generalisation of the standard notion of a Hopf algebra, the second of a tensor category. Quasitriagular structures have an analog in this context. For crossed group coalgebras, we provide an analog of Drinfeld quantum double construction. In that way, starting from a crossed coalgebra H, we obtain a quasitriangular crossed coalgebra D(H). For crossed group categories, we provide an analog of Drinfeld and Joyal - Street center construction: starting from a crossed category C, we obtain a braided crossed category Z(C). We consider the case C=Rep(H). By introducing the category YD(H) of Yetter-Drinfeld modules over H, we prove the isomorphism Rep(D(H))=YD(H)=Z(Rep(H)). We conclude the talk by shortly discussing some open problems.

15.45-16.30. Break

16.30-17.30. EZIO VASSELLI, Roma. "Crossed Products by Endomorphisms, Vector Bundles and Group Duality" We construct the crossed product of a C*-algebra A with centre Z by an endomorphism ρ, which is special in a weaker sense w.r.t. the notion introduced by Doplicher and Roberts. We assign to ρ a geometrical invariant, representing a cohomological obstruction to be special in the usual sense, and determining rank and first Chern class of a vector bundle E, whose module of sections induces ρ on A. We prove that A is the fixed point algebra w.r.t. a (noncompact) G-action, and characterize the category of powers of ρ as the category of tensor powers of a suitable G-Hilbert Z-bimodule (a so-called 'noncommutative pullback' of E).

17.45-18.45. CLAUDIA PINZARI, Roma. "On the equivalence between Jones index theory and conjugation theory for categories of Hilbert bimodules over C*-algebras" We introduce the notion of finite right (respectively left) numerical index on a C*-bimodule AXB equipped with a bi-Hilbertian structure. This notion is based only on a Pimsner-Popa-type inequality. The right (respectively left) index element of X can be constructed in the centre of the enveloping von Neumann algebra of A (respectively B). X is called of finite right index if the right index element lies in the multiplier algebra of A. In this case we can perform the Jones basic construction. The C*-algebra of right adjointable bimodules mappings over a bimodule of finite right index has a natural structure of continuous field of finite dimensional C*-algebras over a compact Hausdorff space. We show that if A is unital, the right index element belongs to A if and only if X is finitely generated as a right module. A finite index bimodule is a bi-Hilbertian C*-bimodule which is at the same time of finite right and left index. We next compare the notion of finite index with the notion of conjugation in a tensor 2-C*-category of right Hilbert bimodules over C*-algebras, in the sense of Longo and Roberts. We show that finite index C*-bimodules, when regarded as objects of the tensor 2-C*-category of Hilbert C*-bimodules with right inner products, have a conjugate in the same category, and conversely, objects of that category with a conjugate can be made into bi-Hilbertian bimodules with finite index.