Banach Journal of Mathematical Analysis

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

Jeffrey L. Boersema, Terry A. Loring, and Efren Ruiz


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)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

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

Received: 24 March 2015
Accepted: 6 April 2015
First available in Project Euclid: 15 October 2015

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Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 46L87: Noncommutative differential geometry [See also 58B32, 58B34, 58J22]

$C^{*}$-algebras real $C^{*}$-algebras $KK$-theory almost commuting matrices


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.

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