On the relation between K- and L-theory of C∗-algebras

We prove the existence of a map of spectra τ A : k A → ℓ A 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 K A 1 2 → ≃ L A 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.