Loop group actions on Cuntz algebras and geometric higher twists

TL;DR

This paper constructs faithful actions of the analytic loop group on Cuntz algebras and uses these actions to explicitly build geometric twists in higher twisted K-theory.

Contribution

It introduces a new faithful action of the loop group on Cuntz algebras and applies it to construct geometric twists in higher twisted K-theory.

Findings

Faithful action of $L_{an}U(n)$ on ${ m O}_N$ for $N \\geq n$

Extension of the action to ${ m O}_\infty$

Explicit construction of ${ m O}_\infty \otimes \mathcal K$-bundles as geometric twists

Abstract

We consider representations of the Cuntz algebras ${\mathcal O}_N$ as constructed by Bratteli-Jorgensen and use these to define a faithful action of the analytic loop group $L_{\text{an}}U(n)$ on ${\mathcal O}_N$ for $N \geq n$. This extends to a faithful action on the infinite Cuntz algebra ${\mathcal O}_{\infty}$, and we show that this action can be used to explicitly construct ${\mathcal O}_\infty \otimes \mathcal K$-bundles (where $ \mathcal K$ is the algebra of compact operators) over spaces, which are geometric twists in higher twisted $K$-theory.