Loop group actions on Cuntz algebras and geometric higher 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.