  We show that the Hausdorffized algebraic K-theory of a 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.
