L‐theory of C∗$C^$‐algebras

We establish a formula for the L‐theory spectrum of real ‐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 obtained in previous work by the first two authors to real ‐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 in terms of the signature operator on oriented manifolds.