Real C*-Algebras, United KK-Theory, and the Universal Coefficient Theorem

Abstract

We define united KK-theory for real C*-algebras A and B such that A is separable and B is σ-unital,. United KK-theory contains real, complex, and self-conjugate KK-theory; but unlike unaugmented real KK-theory, it admits a universal coefficient theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK CRT (A, B) can be written as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KKequivalence for real C*-algebras whose complexification is in the bootstrap category.

Ko-Homology and Type I String Theory

“…Again, a universal coefficient theorem exists in united KKtheory [19], giving rise to a homomorphism…”

Antisymmetries of the CAR algebra

“…In particular, the real counterpart to the Universal Coefficient Theorem of Rosenberg and Schochet [17], which was obtained by Boersema [3], uses a rather complicated theory of CRT -modules due to Bousfield [4]. Rather than attacking the real counterpart to Lin's classification in full generality, the…”

“…A real version of the Universal Coefficient Theorem of Rosenberg and Schochet [17] was obtained by Boersema in [3]. When restricted to the real parts of the CRT -modules, his theorem implies, as described in Corollary 4.9 of [3], that, for real separable C * -algebras A and B, with A ⊗ C in the bootstrap category N , there is a short exact sequence…”

“…When restricted to the real parts of the CRT -modules, his theorem implies, as described in Corollary 4.9 of [3], that, for real separable C * -algebras A and B, with A ⊗ C in the bootstrap category N , there is a short exact sequence…”

“…In this section the relevant K-theoretical tools are summarized for later use. The first set of results relates to the united K-theory used by Boersema in [3] to prove a universal coefficient theorem for real C * -algebras. The basic objects of study in united Ktheory are CRT -modules, first defined by Bousfield in [4] and applied to the Ktheory of real C * -algebras by Boersema in [2].…”