Pictures of KK-theory for real C*-algebras and almost commuting matrices
  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. 
Jeffrey L. Boersema, Terry A. Loring, and Efren Ruiz
math.OA