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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

We give an operator algebraic model for the first group of the unit spectrum gl1(KU) of complex topological K–theory, ie [X,BGL1(KU)], by bundles of stabilized infinite Cuntz C–algebras O K. We develop similar models for the localizations of KU at a prime p and away from p. Our work is based on the –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.

unit spectrum, topological $K$–theory, twisted $K$–theory, strongly self-absorbing $C^*$–algebra, ring spectrum
Mathematical Subject Classification 2010
Primary: 46L80, 55P42
Received: 12 June 2013
Revised: 14 May 2014
Accepted: 21 July 2014
Published: 23 March 2015
Marius Dadarlat
Department of Mathematics
Purdue University
150 N University Street
West Lafayette, IN 47907-2067
Ulrich Pennig
Mathematisches Institut
Westfälische Wilhelms-Universität Münster
Einsteinstraße 62
D-48149 Münster