arXiv subjects

Aaron Kettner

Publications and source records attributed to Aaron Kettner.

3 recordsLinked to original sources

arXiv2025

Twisted topological correspondences and Cartan subalgebras

We introduce twisted topological correspondences, which generalize both Katsura's topological correspondences as well as the twisted topological graphs introduced by Li. We show that, up to isomorphism, they are in bijection with certain principal bundles. This makes it possible to study topological correspondences using the machinery of principal and fiber bundles. We show how to associate a $C^*$-correspondence to a twisted topological correspondence, and give two different characterizations of the $C^*$-correspondences arising that way. The first one is the existence of an atlas of the vector bundle associated to the $C^*$-correspondence whose transition functions take values in a certain subgroup of the unitary group $U(n)$, and which is in some sense compatible with the left action. The other characterization is in terms of Cartan subalgebras in the compact operators on the $C^*$-correspondence. We use our findings to prove rigidity results of the $C^*$-correspondences associated to twisted topological correspondences.

math.OA↗

arXiv2025

Cuntz--Pimsner algebras of partial automorphisms twisted by vector bundles II: Nuclear dimension

We show that Cuntz--Pimsner algebras associated to partial automorphisms twisted by vector bundles are classifiable in the sense of the Elliott program whenever the action is minimal and the base space is compact, infinite and has finite covering dimension. We also investigate the tracial state space of our algebras, and show that traces are in bijection to certain conformal measures. This generalizes results about partial crossed products by Geffen and complements results about the $C^*$-algebras associated to homeomorphisms twisted by vector bundles of Adamo, Archey, Forough, Georgescu, Jeong, Strung and Viola. We use our findings to generalize various existing statements about orbit-breaking subalgebras.

math.OA↗

arXiv2024

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

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.

math.OA↗