Decomposable partial actions

Abadie, Fernando; Gardella, Eusebio; Geffen, Shirly

doi:10.1016/j.jfa.2021.109112

Cited by 1 publication

(22 citation statements)

References 21 publications

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“…We prove this by induction on k. Since A (1) α ( 1) G = A (1) by Example 2.2 and A (1) is a quotient of A, this follows from (E). Assume we have proved it for k − 1, and let us prove it for k. Since P passes to extensions by (E), the exact sequence in (4.5) implies that it suffices to show that D (k) δ (k) G satisfies P. Combining (D.2) and Theorem C in [2], it follows that D (k) δ (k) G is isomorphic to a finite direct sum of algebras of the form…”

Section: Moreover D (K)mentioning

confidence: 89%

“…In this section, we explore the structure of the crossed products and fixed point algebras of partial actions with finite Rokhlin dimension. Our approach makes use of the decomposition property introduced and studied in [2], which we recall in §4.1. For unital partial actions, a more direct argument can be given, which even yields better results (notably in the zero-dimensional case).…”

Section: Structure Of the Crossed Productmentioning

confidence: 99%

“…The decomposition property. An important ingredient in our study of partial actions with finite Rokhlin dimension is our previous work [2] on decomposable partial actions. For the convenience of the reader, we make here a small digression.…”

Section: Structure Of the Crossed Productmentioning

confidence: 99%

“…Find δ ∈ (0, 1/4) such that e δ a − a < ε for all a ∈ F ∩ C 0 (U ) and f δ a − a < ε for all a ∈ F . Set (1,2] .…”

Section: Case 1 K /mentioning

confidence: 99%

“…For the estimates in (1), (2), and (3), one combines the estimates from Corollary 4.25 in [14] and Theorem 3.20 in [13] with the known estimates for dim nuc , dr, or sr of an extension, or of A ⊗ M n , and applies these a total of |G| − 1 times to the extensions in (4.5). We omit the details.…”

Section: Moreover D (K)mentioning

confidence: 99%

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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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