  Locally trivial bundles of $C^*$-algebras with fibre $D \otimes \mathcal{K}$
for a strongly self-absorbing $C^*$-algebra $D$ over a finite CW-complex $X$
form a group $E^1_D(X)$ that is the first group of a cohomology theory
$E^*_D(X)$. In this paper we compute these groups by expressing them in terms
of ordinary cohomology and connective $K$-theory. To compare the
$C^*$-algebraic version of $gl_1(KU)$ with its classical counterpart we also
develop a uniqueness result for the unit spectrum of complex periodic
topological $K$-theory.
