KK with extra structures in the approach via qA

Joachim Cuntz (University of Münster)

Thursday 26th October, 2023 16:00-17:00 Maths 311B


An alternative approach to Kasparov's classical KK-theory uses quasihomomorphisms and the universal algebra qA associated with a given C*-algebra A. One virtue of this approach is a very simple construction of the Kasparov product KK(A,B)xKK(B,C) to KK(A,C). KK-groups have by now been defined and studied for various categories of C*-algebras with extra structures. This includes notably equivariant, nuclear and ideal related KK-theory. We extend and develop the qA-approach to cover these cases. In each case we get a simple description of the corresponding KK and of the product. A nearly automatic consequence is the fact that in each case the KK-functor can be characterized as a universal functor by the usual properties of stability, homotopy invariance and split exactness in the given category. In applications, our description of the KK-groups can be readily applied to define explicit KK-elements for each of these categories..

This is joint work with J.Gabe.

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