December, 2024 Bundles of strongly self-absorbing C-algebras with a Clifford grading
Marius DADARLAT, Ulrich PENNIG
Author Affiliations +
J. Math. Soc. Japan Advance Publication 1-34 (December, 2024). DOI: 10.2969/jmsj/92259225

Abstract

Let D be a strongly self-absorbing C-algebra. In previous work, we showed that locally trivial bundles with fibers KD over a finite CW-complex X are classified by the first group ED1(X) in a generalized cohomology theory ED(X). In this paper, we establish a natural isomorphism EDO1(X)H1(X;Z/2)×twED1(X) for stably-finite D. In particular, EO1(X)H1(X;Z/2)×twEZ1(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 Cn, we show that the classifying spaces of the groups of graded automorphisms of CnKD possess compatible infinite loop space structures. These structures give rise to a cohomology theory E^D(X). We establish isomorphisms E^D1(X)H1(X;Z/2)×twED1(X) and E^D1(X)EDO1(X) for stably finite D. Together, these isomorphisms represent a crucial step in the integral computation of EDO1(X).

Funding Statement

The first author was partially supported by an NSF grant #DMS-2247334.

Citation

Download Citation

Marius DADARLAT. Ulrich PENNIG. "Bundles of strongly self-absorbing -algebras with a Clifford grading." J. Math. Soc. Japan Advance Publication 1 - 34, December, 2024. https://doi.org/10.2969/jmsj/92259225

Information

Received: 10 October 2023; Revised: 7 May 2024; Published: December, 2024
First available in Project Euclid: 12 December 2024

Digital Object Identifier: 10.2969/jmsj/92259225

Subjects:
Primary: 46L35
Secondary: 55N15 , 55N20

Keywords: -algebras , automorphism groups , Clifford algebras , stable homotopy theory

Rights: Copyright ©2024 Mathematical Society of Japan

Advance Publication
Back to Top