On the Relation between K- and L-Theory of $C^*$-Algebras

TL;DR

This paper establishes a natural map between connective topological K-theory and algebraic L-theory spectra of complex $C^*$-algebras, showing their equivalence after inverting 2 and relating major conjectures in the field.

Contribution

It constructs a natural spectrum map linking K- and L-theory of $C^*$-algebras and proves their equivalence after inverting 2, connecting the Baum-Connes and Farrell-Jones conjectures.

Findings

Existence of a natural spectrum map between K- and L-theory

Equivalence of K- and L-theory spectra after inverting 2

Relation between Baum-Connes and Farrell-Jones conjectures

Abstract

We prove the existence of a map of spectra $\tau_A \colon kA \to lA$ between connective topological K-theory and connective algebraic L-theory of a complex $C^*$-algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural equivalence $KA[1/2] \to LA[1/2]$ of periodic K- and L-theory spectra after inverting 2. We show that this equivalence extends to K- and L-theory of real $C^*$-algebras. Using this we give a comparison between the real Baum-Connes conjecture and the L-theoretic Farrell-Jones conjecture. We conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in L-theory is true.