Cuntz--Pimsner algebras of partial automorphisms twisted by vector bundles I: Fixed point algebra, simplicity and the tracial state space

TL;DR

This paper constructs and analyzes Cuntz--Pimsner algebras from partial automorphisms of vector bundles, revealing their fixed point algebra structure, ideal properties, and conditions for simplicity and tracial states.

Contribution

It generalizes previous results by describing the fixed point algebra, ideal structure, and tracial states of these algebras in the context of partial automorphisms and vector bundles.

Findings

Fixed point algebra forms a continuous field over the base space

Ideals correspond to open invariant subspaces under free actions

Cuntz--Pimsner algebra is simple if the action is free and minimal

Abstract

We associate a $C^*$-algebra to a partial action of the integers acting on the base space of a vector bundle, using the framework of Cuntz--Pimsner algebras. We investigate the structure of the fixed point algebra under the canonical gauge action, and show that it arises from a continuous field of $C^*$-algebras over the base space, generalising results of Vasselli. We also analyse the ideal structure, and show that for a free action, ideals correspond to open invariant subspaces of the base space. This shows that if the action is free and minimal, then the Cuntz--Pimsner algebra is simple. In the case of a line bundle, we establish a bijective corrrespondence between tracial states on the algebra and invariant measures on the base space. This generalizes results about the $C^*$-algebras associated to homeomorphisms twisted by vector bundles of Adamo, Archey, Forough, Georgescu, Jeong, Strung and Viola.