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\mathrm{C}^*$-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

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Abstract

In this paper we study Cuntz--Pimsner algebras associated to $\mathrm{C}^*$-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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Cartan preserving isomorphisms between crossed products by Hilbert C(X)-bimodules
  • M S Adamo
  • M Forough
  • K R Strung
M. S. Adamo, M. Forough, and K. R. Strung. Cartan preserving isomorphisms between crossed products by Hilbert C(X)-bimodules. In preparation.
Nuclear dimension of simple C * -algebras
  • J Castillejos
  • S Evington
  • A Tikuisis
  • S White
  • W Winter
J. Castillejos, S. Evington, A. Tikuisis, S. White, and W. Winter. Nuclear dimension of simple C * -algebras. Invent. Math., 224(1):245-290, 2021.
C * -algebras by Example. Fields Institute Monographs
  • K R Davidson
K. R. Davidson. C * -algebras by Example. Fields Institute Monographs. Amer. Math. Soc., Providence, R.I., 1996.
Classifiable C * -algebras from minimal Z-actions and their orbit-breaking subalgebras
  • R J Deeley
  • I F Putnam
  • K R Strung
R. J. Deeley, I. F. Putnam, and K. R. Strung. Classifiable C * -algebras from minimal Z-actions and their orbit-breaking subalgebras. arXiv preprint mathOA/2012.10947.