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

We define united K-theory for real C*-algebras, generalizing Bousfield's 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 Künneth-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 N .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 R k+1 ⊗ O R 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. United K-theoryWe begin by recording the definition of united K-theory, even though the terms contained in it are as yet undefined.It is the task of this chapter to make full sense of the definition.Definition 1.1.Let A be a real C*-algebra.The united K-theory of A is the triple of Z-graded abelian groupstogether with the eight natural transformations {r, c, ε, ζ, ψ U , ψ T , γ, τ }.The K-groups comprising united K-theory will be defined in Section 1.1 and the natural transformations among the three graded groups will be described in Section 1.2.In Section 1.3 we will record some important properties of the category CRT which is the target category of United K-theory.Finally in Section 1.4 we will prove the existence of three long exact sequences involving the three graded groups comprising united K-theory.