Title: Rational algebraic K-theory of topological K-theory. Authors: Christian Ausoni and John Rognes. MSC-class: 19D55; 55N99 arXiv:0708.2160v1 [math.KT] Christian Ausoni Mathematical Institute University of Bonn Beringstrasse 1 DE - 53113 Bonn ausoni@math.uni-bonn.de John Rognes Department of Mathematics University of Oslo Box 1053, Blindern NO - 0316 Oslo rognes@math.uio.no Abstract: We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their associated anomaly bundles over the free loop space LX. We also rationally compute K(KU) by using the localization sequence, and K(MU) by a method that applies to all connective S-algebras.