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The nuclear dimension of \(C^\ast\)-algebras associated to homeomorphisms. (English) Zbl 1376.46042

The paper under review studies the nuclear dimension of \(C^\ast\)-algebras associated to groups and dynamical systems. The main result is Theorem 5.1: Given a locally compact Hausdorff space \(X\) with finite covering dimension and an arbitrary homeomorphism \(\alpha\) of \(X\), the crossed product of \(C_0(X)\) by the automorphism induced by \(\alpha\) has finite nuclear dimension.
The literature contains many results estimating the nuclear dimension of a crossed product in terms of the Rokhlin dimension of the action (see [I. Hirshberg et al., Comm. Math. Phys. 335, No. 2, 637–670 (2015; Zbl 1333.46055)], [G. Szabó, Proc. Lond. Math. Soc. (3) 110, No. 3, 673–694 (2015; Zbl 1330.54047)]). However, such results are limited to free homeomorphisms. By removing this assumption, the paper provides a substantial generalization.
The authors deduce estimates for the nuclear dimension of certain group \(C^\ast\)-algebras. It follows in particular that the \(C^\ast\)-algebra of the lamplighter group has finite nuclear dimension, although it has infinite decomposition rank.

MSC:

46L05 General theory of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
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