Jeffrey L. Boersema

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C*-algebras, generalizing Bousfields topological united K-theory. United K-theory incorporates three functors -- real K-theory, complex K-theory, and self-conjugate K-theory -- and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Kunneth-type, non-splitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C*-algebras A and B which holds as long as the complexification of A is in the bootstrap category. Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product O_{k+1} otimes O_{l+1} is not determined solely by the greatest common divisor of k and l. Hence we have examples of non-isomorphic, simple, purely infinite, real C*-algebras whose complexifications are isomorphic.

Pictures of

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.