#### Volume 15, issue 1 (2015)

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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
##### Abstract

We give an operator algebraic model for the first group of the unit spectrum ${\mathit{gl}}_{1}\left(KU\right)$ of complex topological $K\phantom{\rule{0.3em}{0ex}}$–theory, ie $\left[X,{BGL}_{1}\left(KU\right)\right]$, by bundles of stabilized infinite Cuntz ${C}^{\ast }$–algebras ${\mathsc{O}}_{\infty }\otimes \mathbb{K}$. We develop similar models for the localizations of $KU$ at a prime $p$ and away from $p$. Our work is based on the $\mathsc{ℐ}$–monoid model for the units of $K\phantom{\rule{0.3em}{0ex}}$–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 ${C}^{\ast }$–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
##### 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