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 $C^{\ast }$-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 $F\left(A\right)\to G\left(A\right)$ between homotopy equivalent, stable, half-exact functors defined on real $C^{\ast }$-algebras is an isomorphism, provided it is an isomorphism on the smaller class of $C^{\ast }$-algebras. Finally, we develop $E$-theory for real $C^{\ast }$-algebras and use that to obtain new negative results regarding the problem of approximating almost commuting real matrices by exactly commuting real matrices.