Polar decomposition in algebraic K-theory

We show that the Hausdorffized algebraic K-theory of a unital C∗-algebra decomposes naturally as a direct sum of the Hausdorffized unitary algebraic K-theory and the space of continuous affine functions on the trace simplex. Under mild regularity hypotheses, an analogous natural direct sum decomposition holds for the ordinary (non-Hausdorffized) algebraic K-theory.