### Abstract

We extend the notion of Rokhlin dimension from topological dynamical systems to C^{∗}-correspondences. We show that in the presence of finite Rokhlin dimension and a mild quasidiagonal-like condition (which, for example, is automatic for finitely generated projective correspondences), finite nuclear dimension passes from the scalar algebra to the associated Toeplitz–Pimsner and (hence) Cuntz–Pimsner algebras. As a consequence we provide new examples of classifiable C^{∗}-algebras: if A is simple, unital, has finite nuclear dimension and satisfies the UCT, then for every finitely generated projective H with finite Rokhlin dimension, the associated Cuntz–Pimsner algebra O(H) is classifiable in the sense of Elliott’s Program.

Original language | English (US) |
---|---|

Pages (from-to) | 613-643 |

Number of pages | 31 |

Journal | Houston Journal of Mathematics |

Volume | 44 |

Issue number | 2 |

State | Published - Jan 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

^{∗}-correspondences.

*Houston Journal of Mathematics*,

*44*(2), 613-643.

}

^{∗}-correspondences',

*Houston Journal of Mathematics*, vol. 44, no. 2, pp. 613-643.

**Rokhlin dimension for C ^{∗}-correspondences.** / Brown, Nathanial; Tikuisis, Aaron; Zelenberg, Aleksey M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Rokhlin dimension for C∗-correspondences

AU - Brown, Nathanial

AU - Tikuisis, Aaron

AU - Zelenberg, Aleksey M.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We extend the notion of Rokhlin dimension from topological dynamical systems to C∗-correspondences. We show that in the presence of finite Rokhlin dimension and a mild quasidiagonal-like condition (which, for example, is automatic for finitely generated projective correspondences), finite nuclear dimension passes from the scalar algebra to the associated Toeplitz–Pimsner and (hence) Cuntz–Pimsner algebras. As a consequence we provide new examples of classifiable C∗-algebras: if A is simple, unital, has finite nuclear dimension and satisfies the UCT, then for every finitely generated projective H with finite Rokhlin dimension, the associated Cuntz–Pimsner algebra O(H) is classifiable in the sense of Elliott’s Program.

AB - We extend the notion of Rokhlin dimension from topological dynamical systems to C∗-correspondences. We show that in the presence of finite Rokhlin dimension and a mild quasidiagonal-like condition (which, for example, is automatic for finitely generated projective correspondences), finite nuclear dimension passes from the scalar algebra to the associated Toeplitz–Pimsner and (hence) Cuntz–Pimsner algebras. As a consequence we provide new examples of classifiable C∗-algebras: if A is simple, unital, has finite nuclear dimension and satisfies the UCT, then for every finitely generated projective H with finite Rokhlin dimension, the associated Cuntz–Pimsner algebra O(H) is classifiable in the sense of Elliott’s Program.

UR - http://www.scopus.com/inward/record.url?scp=85052924264&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052924264&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85052924264

VL - 44

SP - 613

EP - 643

JO - Houston Journal of Mathematics

JF - Houston Journal of Mathematics

SN - 0362-1588

IS - 2

ER -

^{∗}-correspondences. Houston Journal of Mathematics. 2018 Jan 1;44(2):613-643.