Nuclear dimension of crossed products associated to partial dynamical systems

Geffen, Shirly

doi:10.1016/j.jfa.2021.109031

Cited by 4 publications

(4 citation statements)

References 26 publications

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“…In other words, there exists a free global action of on some locally compact Hausdorff space Y containing X as an open subset and such that X is Morita equivalent to Y . Note that dim(Y ) = 0 as well by [11,Proposition 3.2]. Hence Proposition 2.5 implies that dad( X ) = dad( Y ) and by [6, Theorem 10.7], the latter is bounded above by asdim( ).…”

Section: Applications To Partial Actionsmentioning

confidence: 86%

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On the dynamic asymptotic dimension of étale groupoids

Bönicke

2024

Math. Z.

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We investigate the dynamic asymptotic dimension for étale groupoids introduced by Guentner, Willett and Yu. In particular, we establish several permanence properties, including estimates for products and unions of groupoids. We also establish invariance of the dynamic asymptotic dimension under Morita equivalence. In the second part of the article, we consider a canonical coarse structure on an étale groupoid, and compare the asymptotic dimension of the resulting coarse space with the dynamic asymptotic dimension of the underlying groupoid.

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Section: Applications To Partial Actionsmentioning

confidence: 86%

“…be the associated transformation groupoid. Using [11,Corollary 3.6] there exist free partial actions θ (k) for k ∈ N such that each θ (k) admits a Hausdorff globalisation, and such that Z θ X can be written as an increasing union…”

Section: Applications To Partial Actionsmentioning

confidence: 99%

On the dynamic asymptotic dimension of étale groupoids

Bönicke

2024

Math. Z.

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“…Created by cleverly fusing the completely positive approximation property for C*algebras together with Lebesgue's covering dimension, it is an intriguing concept that has played a prominent role in the classification of C*-algebras, particularly through its appearance in the Toms-Winter conjecture (see [26,Section 5]). It has received a lot of attention in recent years (see [3,7,8,10,9,11,12,13,15,16,20,21,24,25], for example).…”

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confidence: 99%

The nuclear dimension of extensions of commutative C*-algebras by the compact operators

Gardner¹,

Tikuisis²

2022

Preprint

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“…(See [16] for previously known results regarding nuclear dimension of C ⋆ -algebras attached to partial dynamical systems. )…”

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confidence: 99%

Partial C*-dynamics and Rokhlin dimension

Abadie¹,

Gardella²,

Geffen

2021

Ergod. Th. Dynam. Sys.

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We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product $A\rtimes G$ by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing $C^*$ -algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.

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Nuclear dimension of crossed products associated to partial dynamical systems

Cited by 4 publications

References 26 publications

On the dynamic asymptotic dimension of étale groupoids

On the dynamic asymptotic dimension of étale groupoids

The nuclear dimension of extensions of commutative C*-algebras by the compact operators

Partial C*-dynamics and Rokhlin dimension

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