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

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.