Bundles of strongly self-absorbing $C^{*}$-algebras with a Clifford grading

Let $D$ be a strongly self-absorbing $C^{*}$-algebra. In previous work, we showed that locally trivial bundles with fibers $\mathcal{K} \otimes D$ over a finite CW-complex $X$ are classified by the first group $E^{1}_{D}(X)$ in a generalized cohomology theory $E^{*}_{D}(X)$. In this paper, we establish a natural isomorphism $ E^{1}_{D \otimes \mathcal{O}_{\infty}}(X) \cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^{1}_{D}(X)$ for stably-finite $D$. In particular, $E^{1}_{\mathcal{O}_{\infty}}(X) \cong H^{1}(X;\mathbb{Z}/2) \times_{_{tw}} E^{1}_{\mathcal{Z}}(X)$, where $\mathcal{Z}$ is the Jiang–Su algebra. The multiplication operation on the last two factors is twisted in a manner similar to Brauer theory for bundles with fibers consisting of graded compact operators. The proof of the isomorphism described above made it necessary to extend our previous results on generalized Dixmier–Douady theory to graded $C^{*}$-algebras. More precisely, for complex Clifford algebras $\mathbb{C}\ell_{n}$, we show that the classifying spaces of the groups of graded automorphisms of $\mathbb{C}\ell_{n} \otimes \mathcal{K} \otimes D$ possess compatible infinite loop space structures. These structures give rise to a cohomology theory $\hat{E}^{*}_{D}(X)$. We establish isomorphisms $\hat{E}^{1}_{D}(X) \cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^{1}_{D}(X)$ and $\hat{E}^{1}_{D}(X) \cong E^{1}_{D \otimes \mathcal{O}_{\infty}}(X)$ for stably finite $D$. Together, these isomorphisms represent a crucial step in the integral computation of $E^{1}_{D \otimes \mathcal{O}_{\infty}}(X)$.