Finite cyclic group actions with the tracial Rokhlin property

Phillips, N. Christopher

doi:10.1090/tran/5566

Cited by 19 publications

(36 citation statements)

References 30 publications

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“…The proof of Proposition 3.5 in [55] shows that there exists x ∈ K 1 (A ⋊ α Z 2 ) with x = 0 such that K 1 ( α)(x) = −x. It is shown in Proposition 5.4.1 in [5] that A ⋊ α Z 2 is isomorphic to the tensor product of the Bunce-Deddens algebra of type 2 ∞ with A, so in particular K 1 (A ⋊ α Z 2 ) does not have any 2-torsion.…”

Section: Definition 22 Letmentioning

confidence: 99%

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Rokhlin dimension: duality, tracial properties, and crossed products

Gardella¹,

Hirshberg²,

Santiago³

2019

Ergod. Th. Dynam. Sys.

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We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of C * -algebras, including: D-absorbing C * -algebras, where D is a strongly self-absorbing C * -algebra, stable C * -algebras, C * -algebras with finite nuclear dimension (or decomposition rank), C * -algebras with finite stable rank (or real rank), and C * -algebras whose K-theory is either trivial, rational, or n-divisible for n ∈ N. The combination of nuclearity and the UCT is also shown to be preserved by these actions. Some of these results are new even in the wellstudied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property.At the core of our arguments is a certain local approximation of the crossed product by a continuous C(X)-algebra with fibers that are stably isomorphic to the underlying algebra. The space X is computed in some cases of interest, and we use its description to construct a Z 2 -action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

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Section: Definition 22 Letmentioning

confidence: 99%

“…is torsion free by the claim above, part (7) in Proposition 3.16 implies that K 0 (A ⋊ α Z 2 ) is also torsion free. However, this contradicts part (6) of Proposition 4.2 in [55], where it is shown that K 0 (A ⋊ α Z 2 ) has torsion isomorphic to Z 2 . This contradiction implies that dim c Rok (α) = 2, as desired.…”

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confidence: 93%

Rokhlin dimension: duality, tracial properties, and crossed products

Gardella¹,

Hirshberg²,

Santiago³

2019

Ergod. Th. Dynam. Sys.

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“…5. (See also[35, Example 2.9].) Therefore α is not cocycle conjugate to ν Z2 because W 2 ⋊ α Z 2 is not isomorphic to W 2 .…”

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confidence: 99%

Finite group actions on certain stably projectionless C*-algebras with the Rohlin property

Nawata

2015

Trans. Amer. Math. Soc.

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Abstract. We introduce the Rohlin property and the approximate representability for finite group actions on stably projectionless C * -algebras and study their basic properties. We give some examples of finite group actions on the Razak-Jacelon algebra W 2 and show some classification results of these actions. This study is based on the work of Izumi, Robert's classification theorem and Kirchberg's central sequence C * -algebras. Introduction A C* -algebra A is said to be stably projectionless if A ⊗ K has no non-zero projections where K is the C * -algebra of compact operators on an infinite-dimensional separable Hilbert space. Note that every stably projectionless C * -algebra does not have a unit. In this paper we study finite group actions on stably projectionless C * -algebras. In the theory of operator algebras, the study of group actions on operator algebras is one of the most fundamental subjects and has a long history. We refer the reader to [14]

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“…In [27] it was shown that for G = Z/2Z the positive cone in K 0 (B θ ) coincides with the preimage of the positive cone in K 0 (A θ ) under the natural homomorphism K 0 (B θ ) → K 0 (A θ ) (in other words, it consists of all elements x ∈ K 0 (B θ ) such that tr * (x) > 0, where tr * : K 0 (B θ ) → R is the homomorphism induced by the trace). As was pointed to us by Chris Phillips, similar statement is also known to hold for other groups G. Namely, it follows from the fact that the corresponding crossed products are simple AH algebras with slow dimension growth and real rank zero (see Theorems 8.11 and 9.10 of [20]). …”

Section: 4mentioning

confidence: 58%

Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

Polishchuk¹

Noncommutative Geometry and Number Theory

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Abstract. We define the notion of a holomorphic bundle on the noncommutative toric orbifold T θ /G associated with an action of a finite cyclic group G on an irrational rotation algebra. We prove that the category of such holomorphic bundles is abelian and its derived category is equivalent to the derived category of modules over a finite-dimensional algebra Λ. As an application we finish the computation of K 0 -groups of the crossed product algebras describing the above orbifolds initiated in [17], [28], [29], [12] and [13]. Also, we describe a torsion pair in the category of Λ-modules, such that the tilting with respect to this torsion pair gives the category of holomorphic bundles on T θ /G.

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Finite cyclic group actions with the tracial Rokhlin property

Cited by 19 publications

References 30 publications

Rokhlin dimension: duality, tracial properties, and crossed products

Rokhlin dimension: duality, tracial properties, and crossed products

Finite group actions on certain stably projectionless C*-algebras with the Rohlin property

Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

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