Pictures of KK-theory for real C*-algebras and almost commuting matrices

Abstract

We give a systematic account of the various pictures of KK-theory for real C*-algebras, proving natural isomorphisms between the groups that arise from each picture. As part of this project, we develop the universal properties of KK-theory, and we use CRT-structures to prove that a natural transformation from F(A) to G(A) between homotopy equivalent, stable, half-exact functors defined on real C*-algebras is an isomorphism provided it is an isomorphism on the smaller class of C*-algebras. Finally, we develop E-theory for real C*-algebras and use that to obtain new negative results regarding the problem of approximating almost commuting real matrices by exactly commuting real matrices.

Version 1: 31 pages. arXiv admin note: text overlap with arXiv:0909.0972 . Version 2: Shorten the paper to 21 pages. Added a section on unitary representations of KO-classes

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