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

Abstract

We give an operator algebraic model for the first group of the unit spectrum ${\mathit{gl}}_1\left(KU\right)$ of complex topological $K$–theory, ie $\left[X,{BGL}_1\left(KU\right)\right]$, by bundles of stabilized infinite Cuntz $C^{\ast }$–algebras ${\mathcal{O}}_{\infty }\otimes \mathbb{K}$. We develop similar models for the localizations of $KU$ at a prime $p$ and away from $p$. Our work is based on the $\mathcal{ℐ}$–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 $C^{\ast }$–algebras that arose in our generalization of the Dixmier–Douady theory and classical spectra from algebraic topology.