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

We associate a $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-algebras associated to homeomorphisms twisted by vector bundles of Adamo, Archey, Forough, Georgescu, Jeong, Strung and Viola.