  Let $A$ be a unital $C^*$-algebra. Its unitary group, $UA$, contains a wealth
of topological information about $A$. However, the homotopy type of $UA$ is out
of reach even for $A = M_2(\CC)$. There are two simplifications which have been
considered. The first, well-traveled road, is to pass to $\pi_*(U(A\otimes \KK
))$ which is isomorphic (with a degree shift) to $K_*(A)$. This approach has
led to spectacular success in many arenas, as is well-known.
  A different approach is to consider $\pi_*(UA)\otimes\QQ $, the rational
homotopy of $UA$. In joint work with G. Lupton and N. C. Phillips we have
calculated this functor for the cases $A = C(X)\otimes M_n(\CC)$ and $A$ a
unital continuous trace $C^*$-algebra. In this note we look at some concrete
examples of this calculation and, in particular, at the $\ZZ$-graded map \[ \pi
_*(UA)\otimes\QQ \longrightarrow K_{*+1}(A)\otimes\QQ . \]
