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

Abstract

We extend our previous results on generalized Dixmier-Douady theory to graded $C^{\ast }$-algebras, as means for explicit computations of the invariants arising for bundles of ungraded $C^{\ast }$-algebras. For a strongly self-absorbing $C^{\ast }$-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 ${\hat{E}}_D^{\ast }(X)$. 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 $C^{\ast }$-algebras with fibers $\mathbb{C}{\ell }_k\otimes D\otimes \mathcal{K}$ and that this group is isomorphic to $H^0(X,\mathbb{Z}/2)\oplus {\hat{E}}_D^1(X)$. Moreover, we establish isomorphisms ${\hat{E}}_D^1(X)\cong H^1(X;\mathbb{Z}/2){\times }_{{}_{tw}}{E}_D^1(X)$ and ${\hat{E}}_D^1(X)\cong E_{D\otimes {\mathcal{O}}_{\infty }}^1(X)$, where $E_D^1(X)$ is the group that classifies the locally trivial bundles with fibers $D\otimes \mathcal{K}$. In particular $E_{{\mathcal{O}}_{\infty }}^1(X)\cong H^1(X;\mathbb{Z}/2){\times }_{{}_{tw}}{E}_{\mathcal{Z}}^1(X)$ 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.

arXiv comment30 pages

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