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

TL;DR

This paper constructs an operator algebraic model for the first group of the unit spectrum of complex topological K-theory using bundles of stabilized infinite Cuntz C*-algebras, connecting operator algebras with algebraic topology.

Contribution

It develops a new operator algebraic model for the units of K-theory, extending to localizations at primes, based on the $ ext{I}$-monoid framework and linking automorphism groups of C*-algebras with classical spectra.

Findings

Model for $gl_1(KU)$ using bundles of $O_{ extinfty} ensor ext{K}$.

Extended models for localizations at prime $p$ and away from $p$.

Connections established between automorphism groups of C*-algebras and classical spectra.

Abstract

We give an operator algebraic model for the first group of the unit spectrum $gl_1(KU)$ of complex topological K-theory, i.e. $[X, BGL_1(KU)]$, by bundles of stabilized infinite Cuntz C*-algebras $O_{\infty} \otimes \K$. 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*-algebras that arose in our generalization of the Dixmier-Douady theory and classical spectra from algebraic topology.