Computing cohomology groups that classify bundles of strongly self-absorbing $C^{\ast }$-algebras

Locally trivial bundles of $C^{\ast }$-algebras with fibre $D\otimes \mathcal{K}$ for a strongly self-absorbing $C^{\ast }$-algebra $D$ over a finite CW-complex $X$ form a group $E_D^1(X)$ that is the first group of a cohomology theory $E_D^{\ast }(X)$. In this paper we compute these groups by expressing them in terms of ordinary cohomology and connective $K$-theory. To compare the $C^{\ast }$-algebraic version of $g{l}_1(KU)$ with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological $K$-theory.