The spectrum of equivariant Kasparov theory for cyclic groups of prime order

Abstract: In 2006, Ralf Meyer and Ryszard Nest proved that the G-equivariant Kasparov category of a locally compact group G carries the structure of a tensor-triangulated category. This structure conveniently handles the usual homological algebra, bootstrap constructions and assembly maps involved in many KK-theoretical calculations, e.g. in connection with the Baum-Connes conjecture. As with any tensor triangulated category, we can also associate to the G-equivariant Kasparov category its spectrum in the sense of Paul Balmer. This is a topological space (similar to the Zariski spectrum of a commutative ring) which allows us, as it were, to re-inject some genuinely geometric ideas in non-commutative geometry. It turns out that the spectrum contains enough information to prove the Baum-Connes conjecture for G, hence we should expect the question of its computation to be very hard. In this talk, after discussing such preliminaries and motivation, I will present joint work with Ralf Meyer providing the state of the art on this subject. Although more general partial results are known, a complete answer is only known so far for finite groups of prime order and for algebras in the bootstrap category.

category theoryoperator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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