## Banach Journal of Mathematical Analysis

### 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 $F(A)\rightarrow G(A)$ 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.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 1 (2016), 27-47.

Dates
Accepted: 6 April 2015
First available in Project Euclid: 15 October 2015

https://projecteuclid.org/euclid.bjma/1444913861

Digital Object Identifier
doi:10.1215/17358787-3163312

Mathematical Reviews number (MathSciNet)
MR3453522

Zentralblatt MATH identifier
1352.46064

#### Citation

Boersema, Jeffrey L.; Loring, Terry A.; Ruiz, Efren. Pictures of $KK$ -theory for real $C^{*}$ -algebras and almost commuting matrices. Banach J. Math. Anal. 10 (2016), no. 1, 27--47. doi:10.1215/17358787-3163312. https://projecteuclid.org/euclid.bjma/1444913861

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