L-theory of $C^{\ast }$-algebras

Abstract

We establish a formula for the L-theory spectrum of real $C^{\ast }$-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 $\tau :\mathrm{k}\to \mathrm{L}$ obtained in previous work by the first two authors to real $C^{\ast }$-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.

43 pages, published version