L-theory of $C^*$-algebras
Abstract
We establish a formula for the L-theory spectrum of real $C^*$-algebras from which we deduce a presentation of the L-groups in terms of the topological K-groups, extending all previously known results of this kind. Along the way, we extend the integral comparison map $\tau\colon \mathrm{k} \to \mathrm{L}$ obtained in previous work by the first two authors to real $C^*$-algebras and interpret it using topological Grothendieck-Witt theory. Finally, we use our results to give an integral comparison between the Baum-Connes conjecture and the L-theoretic Farrell-Jones conjecture, and discuss our comparison map $\tau$ in terms of the signature operator on oriented manifolds.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2022
- DOI:
- 10.48550/arXiv.2208.10556
- arXiv:
- arXiv:2208.10556
- Bibcode:
- 2022arXiv220810556L
- Keywords:
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- Mathematics - K-Theory and Homology;
- Mathematics - Algebraic Topology;
- Mathematics - Operator Algebras
- E-Print:
- 39 pages