Axiomatic $KK$-theory for Real C*-algebras

Abstract

We establish axiomatic characterizations of $K$-theory and $KK$-theory for real C*-algebras. In particular, let $F$ be an abelian group-valued functor on separable real C*-algebras. We prove that if $F$ is homotopy invariant, stable, and split exact, then $F$ factors through the category $KK$. Also, if $F$ is homotopy invariant, stable, half exact, continuous, and satisfies an appropriate dimension axiom, then there is a natural isomorphism $K(A)\to F(A)$ for a large class of separable real C*-algebras $A$. Furthermore, we prove that a natural transformation $F(A)\to G(A)$ of homotopy invariant, stable, half-exact functors which is an isomorphism when $A$ is complex is necessarily an isomorphism when $A$ is real.

This paper has been replaced by "Pictures of KK-theory for real C*-algebras" which is posted in the arXiv at arXiv:1210.3133. Several different pictures of KK-theory for real C*-algebras is given there and the results are used to obtain new results for the problem of approximating a set of almost commuting matrices over the field of real numbers

Cited by in corpus (1)

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