Twisted K-theory, K-homology and bivariant Chern-Connes type character of some infinite dimensional spaces

Abstract

We study the twisted K-theory and K-homology of some infinite dimensional spaces, like SU(\infty), in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable σ-C^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse the structure of the dual Chern--Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate the twisted periodic cyclic homology via locally convex algebras and the local cyclic homology via C^*-algebras (in the compact case).

v2. 32 pages, acknowledgements added and some minor changes made (including the title); v3. 34 pages, some changes following the referee's suggestions; v4. references updated, exposition improved, to appear in Kyoto J. Math

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Cited by in corpus (3)

• Colocalizations of noncommutative spectra and bootstrap categories
• Noncommutative stable homotopy and stable infinity categories
• Algebraic K-theory, K-regularity, and T-duality of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras