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

Abstract

We extend our previous results on generalized Dixmier-Douady theory to graded $C^*$-algebras, as means for explicit computations of the invariants arising for bundles of ungraded $C^*$-algebras. For a strongly self-absorbing $C^*$-algebra $D$ and complex Clifford algebras $\mathbb{C}\ell_{n}$ we show that the classifying spaces of the groups of graded automorphisms $\mathrm{Aut}_{\text{gr}}(\mathbb{C}\ell_{n}\otimes \mathcal{K }\otimes D)$ admit compatible infinite loop space structures giving rise to a cohomology theory $\hat{E}^*_D(X)$. For $D$ stably finite and $X$ a finite CW-complex, we show that the tensor product operation defines a group structure on the isomorphism classes of locally trivial bundles of graded $C^*$-algebras with fibers $ \mathbb{C}\ell_{k}\otimes D \otimes \mathcal{K}$ and that this group is isomorphic to $H^0(X,\mathbb{Z}/2)\oplus \hat{E}^1_{D}(X)$. Moreover, we establish isomorphisms $\hat{E}^1_{D}(X)\cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^1_{D}(X)$ and $\hat{E}^1_{D}(X)\cong E^1_{D\otimes \mathcal{O}_\infty}(X)$, where $E^1_{D}(X)$ is the group that classifies the locally trivial bundles with fibers $D\otimes \mathcal{K}$. 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 and the multiplication on the last two factors is twisted similarly to the Brauer theory for bundles with fibers the graded compact operators on a finite and respectively infinite dimensional Hilbert space.