Finite group actions on C*-algebras with the Rohlin property, I

Izumi, Masaki

doi:10.1215/s0012-7094-04-12221-3

Cited by 157 publications

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“…, d, respectively. Notice that if d = 0 then we obtain the usual Rokhlin property for actions of finite groups (see [10,11]). …”

Section: Rokhlin Dimension For Actions Of Finite Groupsmentioning

confidence: 92%

“…See, for instance, [9,16] and references therein for actions of Z, and [4,10,11,23], for the finite group case. In fact, Rokhlin properties, especially for the single automorphism case, are quite prevalent, and indeed as a byproduct we establish in this paper that they are generic for automorphisms of unital C * -algebras which absorb a UHF algebra of infinite type.…”

Section: Introductionmentioning

confidence: 99%

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Rokhlin Dimension and C*-Dynamics

Hirshberg

Winter

Zacharias

2015

Commun. Math. Phys.

159

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Abstract:We develop the concept of Rokhlin dimension for integer and for finite group actions on C * -algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C * -algebras, where Z denotes the Jiang-Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z-stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic Z d -actions on compact metrizable and finite dimensional spaces.

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“…, d, respectively. Notice that if d = 0 then we obtain the usual Rokhlin property for actions of finite groups (see [10,11]). …”

Section: Rokhlin Dimension For Actions Of Finite Groupsmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Rokhlin Dimension and C*-Dynamics

Hirshberg

Winter

Zacharias

2015

Commun. Math. Phys.

159

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“…The following is Definition 1.1 of [26]. It is essentially Definition 3.1 of [11], although that definition is stated in terms of central sequences. See the discussion after Definition 1.1 of [26] for the equivalence of the two definitions.…”

Section: The Tracial Rokhlin Property and Tracial Approximate Represementioning

confidence: 99%

“…The purpose of this paper is to provide other examples of actions of finite cyclic groups with the tracial Rokhlin property on C*-algebras with tracial rank zero. We demonstrate by example the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability (Definition 3.6(2) of [11]) and its tracial analog, tracial approximate representability (Definition 3.2 of [26]). (To emphasize the distinction with their tracial analogs, in this paper we refer to the strict Rokhlin property and to strict approximate representability.)…”

Section: Introductionmentioning

confidence: 99%

Finite cyclic group actions with the tracial Rokhlin property

Phillips

2015

Trans. Amer. Math. Soc.

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Abstract. We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of Z/2Z on a UHF algebra has the tracial Rokhlin property; in particular, unlike for the strict Rokhlin property, every UHF algebra admits such an action. We prove that Blackadar's action of Z/2Z on the 2 ∞ UHF algebra, whose crossed product is not AF because it has nontrivial K 1 -group, has the tracial Rokhlin property, and we give an example of an action of Z/2Z on a simple unital AF algebra which has the tracial Rokhlin property and such that the K 0 -group of the crossed product has torsion. In particular, the crossed product of a simple unital AF algebra by an action of Z/2Z with the tracial Rokhlin property need not be AF. We give examples of a tracially approximately representable action of Z/2Z on a simple unital AF algebra which is nontrivial on K 0 , and of a tracially approximately representable action of Z/2Z on a simple unital AT algebra with real rank zero which is nontrivial on K 1 . IntroductionThe tracial Rokhlin property for actions of finite groups on C*-algebras was introduced in [26] for the purpose of proving that every simple higher dimensional noncommutative torus is an AT algebra (done in [27]), and proving that certain crossed products of such algebras by finite cyclic groups are AF algebras (done in [5]). The purpose of this paper is to provide other examples of actions of finite cyclic groups with the tracial Rokhlin property on C*-algebras with tracial rank zero. We demonstrate by example the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability (Definition 3.6(2) of [11]) and its tracial analog, tracial approximate representability (Definition 3.2 of [26]). (To emphasize the distinction with their tracial analogs, in this paper we refer to the strict Rokhlin property and to strict approximate representability.) In [4], Blackadar constructed an action of Z/2Z on the 2 ∞ UHF algebra such that the crossed product has nontrivial K 1 -group, and is hence not AF. As one of our examples, we prove that this action has the tracial Rokhlin property. Earlier, in one of the exercises (10.11.3) of his book [2], Blackadar gave an example of an order two automorphism of K 0 (A) for a simple separable AF algebra A such that, if this automorphism could be implemented by an order two automorphism of A, then the resulting crossed product by Z/2Z would have

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“…Take α to be as in Lemma 4.7 of [Izu04] or, more generally, as in Theorem 4.8(3) of [Izu04] with the groups Γ 0 and Γ 1 chosen so that at least one of them is not finitely generated, and also such that O 2 ⋊ α Z 2 satisfies the Universal Coefficient Theorem. The action α is outer, so B is simple by Theorem 3.1 of [Kis81] and purely infinite by Corollary 4.6 of [JO98].…”

Section: Semiprojectivity Of the Crossed Product Algebramentioning

confidence: 99%

Semiprojectivity with and without a group action

Phillips

Sørensen

Thiel

2015

Journal of Functional Analysis

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Abstract. The equivariant version of semiprojectivity was recently introduced by the first author. We study properties of this notion, in particular its relation to ordinary semiprojectivity of the crossed product and of the algebra itself.We show that equivariant semiprojectivity is preserved when the action is restricted to a cocompact subgroup. Thus, if a second countable compact group acts semiprojectively on a C * -algebra A, then A must be semiprojective. This fails for noncompact groups: we construct a semiprojective action of Z on a nonsemiprojective C * -algebra.We also study equivariant projectivity and obtain analogous results, however with fewer restrictions on the subgroup. For example, if a discrete group acts projectively on a C * -algebra A, then A must be projective. This is in contrast to the semiprojective case.We show that the crossed product by a semiprojective action of a finite group on a unital C * -algebra is a semiprojective C * -algebra. We give examples to show that this does not generalize to all compact groups.Equivariant semiprojectivity was introduced in [Phi12], by applying the usual definition of semiprojectivity to the category of unital G-algebras (C * -algebras with actions of the group G) with unital G-equivariant * -homomorphisms. See Definition 1.1 below. The purpose of [Phi12] was to show that certain actions of compact groups on various specific C * -algebras are semiprojective. In particular, it is shown that any action of a second countable compact group on a finite dimensional C * -algebra is semiprojective, and that for n < ∞, quasifree actions of second countable compact groups on the Cuntz algebras O n are semiprojective.In this paper we study equivariant semiprojectivity more abstractly. We also introduce equivariant projectivity and carry out a parallel study of it. We extend the definition to allow actions by general locally compact groups, and we consider the nonunital version of equivariant semiprojectivity.From the work in [Phi12], it is not even clear whether a semiprojective action of a noncompact group can exist. One reason for skepticism was that the trivial

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Finite group actions on C*-algebras with the Rohlin property, I

Cited by 157 publications

References 51 publications

Rokhlin Dimension and C*-Dynamics

Rokhlin Dimension and C*-Dynamics

Finite cyclic group actions with the tracial Rokhlin property

Semiprojectivity with and without a group action

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