Unit spectra of K–theory from strongly self-absorbing C∗–algebras

Dadarlat, Marius; Pennig, Ulrich

doi:10.2140/agt.2015.15.137

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Unit spectra of $K$–theory from strongly self-absorbing $C^*$–algebras

Marius Dadarlat and Ulrich Pennig

Algebraic & Geometric Topology 15 (2015) 137–168

DOI: 10.2140/agt.2015.15.137

Abstract

We give an operator algebraic model for the first group of the unit spectrum ${gl}_{1} (K U)$ of complex topological $K$ –theory, ie $[X, {BGL}_{1} (K U)]$ , by bundles of stabilized infinite Cuntz $C^{*}$ –algebras $O_{\infty} \otimes K$ . We develop similar models for the localizations of $K U$ at a prime $p$ and away from $p$ . Our work is based on the $ℐ$ –monoid model for the units of $K$ –theory by Sagave and Schlichtkrull and it was motivated by the goal of finding connections between the infinite loop space structure of the classifying space of the automorphism group of stabilized strongly self-absorbing $C^{*}$ –algebras that arose in our generalization of the Dixmier–Douady theory and classical spectra from algebraic topology.

Keywords

unit spectrum, topological $K$–theory, twisted $K$–theory, strongly self-absorbing $C^*$–algebra, ring spectrum

Mathematical Subject Classification 2010

Primary: 46L80, 55P42

References

Publication

Received: 12 June 2013

Revised: 14 May 2014

Accepted: 21 July 2014

Published: 23 March 2015

Authors

Marius Dadarlat
Department of Mathematics Purdue University 150 N University Street West Lafayette, IN 47907-2067 USA

Ulrich Pennig
Mathematisches Institut Westfälische Wilhelms-Universität Münster Einsteinstraße 62 D-48149 Münster Germany

Volume 15, issue 1 (2015)