Abstract

A G -kernel is a group homomorphism from a (discrete) group G to Out(A), the outer automorphism group of a C-algebra A. There are cohomological obstructions to lifting such a G-kernel to a group action. In the setting of von Neumann algebras, G-kernels on the hyperfinite II1 -factor have been completely understood via deep results of Connes, Jones and Ocneanu. In the talk I will explain how G-kernels on C-algebras and the lifting obstructions can be interpreted in terms cohomology with coefficients in crossed modules. G-kernels, group actions and cocycle actions then give rise to induced maps on classifying spaces. For strongly self-absorbing C*-algebras these classifying spaces turn out to be infinite loop spaces creating a bridge to stable homotopy theory. Not only does this make the invariants computable, it also gives rise to equivariant refinements. The first part is a joint project with S. Giron Pacheco and M. Izumi, the second with my PhD student V. Bianchi.