## Kyoto Journal of Mathematics

### Twisted ${K}$-theory, ${K}$-homology, and bivariant Chern–Connes type character of some infinite dimensional spaces

Snigdhayan Mahanta

#### Abstract

We study the twisted K-theory and K-homology of some infinite dimensional spaces, like $\operatorname {SU}(\infty)$, in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable $\sigma$-$C^{*}$-algebras that generalizes both the twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern–Connes-type character taking values in a bivariant local cyclic homology. We analyze 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).

#### Article information

Source
Kyoto J. Math. Volume 54, Number 3 (2014), 597-640.

Dates
First available in Project Euclid: 14 August 2014

https://projecteuclid.org/euclid.kjm/1408020880

Digital Object Identifier
doi:10.1215/21562261-2693460

Mathematical Reviews number (MathSciNet)
MR3263554

Zentralblatt MATH identifier
1309.19006

#### Citation

Mahanta, Snigdhayan. Twisted ${K}$ -theory, ${K}$ -homology, and bivariant Chern–Connes type character of some infinite dimensional spaces. Kyoto J. Math. 54 (2014), no. 3, 597--640. doi:10.1215/21562261-2693460. https://projecteuclid.org/euclid.kjm/1408020880

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