L-theory of C∗-algebras

Land, Markus; Nikolaus, Thomas; Schlichting, Marco

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 τ:k→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 τ in terms of the signature operator on oriented manifolds.