  We show that continuous group homomorphisms between unitary groups of unital
C*-algebras induce maps between spaces of continuous real-valued affine
functions on the trace simplices. Under certain $K$-theoretic regularity
conditions, these maps can be seen to commute with the pairing between $K_0$
and traces. If the homomorphism is contractive and sends the unit circle to the
unit circle, the map between spaces of continuous real-valued affine functions
can further be shown to be unital and positive (up to a minus sign).
