𝐶*-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

Adamo, Maria; Archey, Dawn; Forough, Marzieh; Georgescu, Magdalena; Jeong, Ja; Strung, Karen; Viola, Maria

doi:10.1090/tran/8900

$\mathrm {C}^*$-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces
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by Maria Stella Adamo, Dawn E. Archey, Marzieh Forough, Magdalena C. Georgescu, Ja A. Jeong, Karen R. Strung and Maria Grazia Viola;

Trans. Amer. Math. Soc. 377 (2024), 1597-1640

DOI: https://doi.org/10.1090/tran/8900

Published electronically: January 18, 2024

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Abstract:

In this paper we study Cuntz–Pimsner algebras associated to 𝐶*-correspondences

over commutative $\mathrm {C}^*$-algebras from the point of view of the $\mathrm {C}^*$-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite dimensional infinite compact metric space $X$ twisted by a vector bundle, the resulting Cuntz–Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these $\mathrm {C}^*$-algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz–Pimsner algebra has stable rank one. Otherwise, it is purely infinite.

For a Cuntz–Pimsner algebra of a minimal homeomorphism of an infinite compact metric space $X$ twisted by a line bundle over $X$, we introduce orbit-breaking subalgebras. With no assumptions on the dimension of $X$, we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of $X$ is finite, they are furthermore $\mathcal {Z}$-stable and hence classified by the Elliott invariant.

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