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

TL;DR

This paper systematically explores various models of KK-theory for real C*-algebras, establishing isomorphisms, universal properties, and applying E-theory to analyze the approximation of almost commuting matrices.

Contribution

It develops the universal properties of KK-theory for real C*-algebras, proves natural isomorphisms between different models, and applies E-theory to problems involving almost commuting matrices.

Findings

Proves natural isomorphisms between different pictures of KK-theory for real C*-algebras.

Establishes universal properties of KK-theory in the real setting.

Provides new negative results on approximating almost commuting matrices by commuting ones.

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.