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

Dădărlat, Marius and Pennig, Ulrich ORCID: https://orcid.org/0000-0001-5441-6130 2024. Bundles of strongly self-absorbing C*-algebras with a Clifford grading. Journal of the Mathematical Society of Japan

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Abstract

Let D be a strongly self-absorbing C∗-algebra. In previous work, we showed that locally trivial bundles with fibers K ⊗ D over a finite CW-complex X are classified by the first group E1D (X) in a generalized cohomology theory E∗ D (X). In this paper, we establish a natural isomorphism E1D ⊗O∞(X) ∼= H1(X; Z/2) ×tw E1D (X) for stably-finite D. In particular, E1O ∞(X) ∼= H1(X; Z/2) ×tw E1Z (X), where 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 Cℓn, we show that the classifying spaces of the groups of graded automorphisms of Cℓn ⊗ K ⊗ D possess compatible infinite loop space structures. These structures give rise to a cohomology theory ˆE ∗D (X). We establish isomorphisms ˆE 1D (X) ∼= H1(X; Z/2)×tw E1D (X) and ˆE 1D (X) ∼= E1D ⊗O∞(X) for stably finite D. Together, these isomorphisms represent a crucial step in the integral computation of E1D ⊗O∞(X).

Item Type:	Article
Status:	In Press
Schools:	Mathematics
Publisher:	Mathematical Society of Japan
ISSN:	0025-5645
Date of First Compliant Deposit:	9 July 2024
Date of Acceptance:	1 July 2024
Last Modified:	08 Nov 2024 16:00
URI:	https://orca.cardiff.ac.uk/id/eprint/170456

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