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

2022/10/28 by Dadarlat, Marius, Pennig, Ulrich
#FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2210.16360

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 ℂℓ_n we show that the classifying spaces of the groups of graded automorphisms Aut_gr(ℂℓ_n⊗ K ⊗ D) admit compatible infinite loop space structures giving rise to a cohomology theory 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 ℂℓ_k⊗ D ⊗ K and that this group is isomorphic to H⁰(X,ℤ/2)⊕ E¹_D(X). Moreover, we establish isomorphisms E¹_D(X)≅ H¹(X;ℤ/2) ×__tw E¹_D(X) and E¹_D(X)≅ E¹_{D⊗ O_∞}(X), where E¹_D(X) is the group that classifies the locally trivial bundles with fibers D⊗ K. In particular E¹_{O_∞}(X)≅ H¹(X;ℤ/2) ×__tw E¹_Z(X) where 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.

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