L-theory of $C^*$-algebras

TL;DR

This paper derives a formula linking the L-theory spectrum of real C*-algebras to their topological K-groups, extending previous results and connecting to major conjectures in topology.

Contribution

It provides a new formula for the L-theory spectrum of real C*-algebras and extends the integral comparison map to this setting, linking algebraic and topological invariants.

Findings

Derived a formula for L-theory spectrum from topological K-groups.

Extended the integral comparison map to real C*-algebras.

Connected the results to the Baum-Connes and Farrell-Jones conjectures.

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.