We give a systematic account of the various pictures of -theory for real -algebras, proving natural isomorphisms between the groups that arise from each picture. As part of this project, we develop the universal properties of -theory, and we use CRT-structures to prove that a natural transformation between homotopy equivalent, stable, half-exact functors defined on real -algebras is an isomorphism, provided it is an isomorphism on the smaller class of -algebras. Finally, we develop -theory for real -algebras and use that to obtain new negative results regarding the problem of approximating almost commuting real matrices by exactly commuting real matrices.
"Pictures of -theory for real -algebras and almost commuting matrices." Banach J. Math. Anal. 10 (1) 27 - 47, January 2016. https://doi.org/10.1215/17358787-3163312