Rokhlin dimension: duality, tracial properties, and crossed products

Gardella, Eusebio; Hirshberg, Ilan; Santiago, Luis

doi:10.1017/etds.2019.68

Cited by 32 publications

(44 citation statements)

References 72 publications

(173 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Corollary 4.49 is a significant generalization of previously known results concerning Jiang-Su absorption: for finite groups this was shown by Hirshberg-Winter-Zacharias in [40,Theorem 5.9], and in [28,Theorem 5.4.4] by the first-named author for compact groups. Similar results have been independently obtained with different methods in [32].…”

Section: 7supporting

confidence: 87%

“…For separable unital A, the following first appeared as [30,Theorem 3.3]. The particular case of commuting towers has also been independently obtained in [32], using completely different methods.…”

Section: Relationship Between Order Zero Dimension and D-containmentmentioning

confidence: 99%

“…Proof. Again, the proof given in [15] requires only minor changes: the isomorphisms θ (31) and (32) are automatically G-invariant. Finally, the maps ψ : A → C(X) ⊗ D and ϕ : C(X) ⊗ D → A are readily checked to be equivariant (observe that the structure map of a G-C(X)-algebra is equivariant when C(X) is endowed with the trivial G-action).…”

Section: 5mentioning

confidence: 99%

See 2 more Smart Citations

Applications of model theory to C*-dynamics

Gardella

Lupini

2018

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. Finiteness of the order zero dimension implies that the dimension of the target algebra can be bounded by the dimension of the domain. The dimension can be, among others, decomposition rank, nuclear dimension, or Rokhlin dimension. As a consequence, we obtain new inequalities for these quantities.As a third application we obtain the following result: if a C*-algebra A absorbs a strongly self-absorbing C*algebra D, and α is an action of a compact group G on A with finite Rokhlin dimension with commuting towers, then α absorbs any strongly self-absorbing action of G on D. This has a number of interesting consequences, already in the case of the trivial action on D. For example, we deduce that D-stability passes from A to the crossed product. Additionally, in many cases of interest, our result restricts the possible values of the Rokhlin dimension to 0, 1 and ∞, showing a striking parallel to the behavior of the nuclear dimension for simple C*-algebras. We also show that an action of a finite group with finite Rokhlin dimension with commuting towers automatically has the Rokhlin property if the algebra is UHF-absorbing.2000 Mathematics Subject Classification. Primary 03C98, 46L55; Secondary 28D05, 46L40, 46M07. 1 2 EUSEBIO GARDELLA AND MARTINO LUPINIproperty. In the context of C*-algebras, relative commutants were used in connection with the Rokhlin property for group actions in the work of Herman-Jones [37], Kishimoto [51], Izumi [42], , and the first-named author [28]. The study of Rokhlin dimension has also made extensive use of these tools, for example in [28] and [38], as well as the more recent work on strongly self-absorbing actions [64]. As is clear from these works, the use of sequence algebras in the equivariant setting becomes even more delicate when the acting group is not discrete, since a continuous action on an operator algebra may induce a discontinuous action on its relative commutant. As such, equivariant (central) sequence algebras are interesting objects whose systematic study is justified by their wide application in the literature.The present work takes up this task. For a given compact second countable group G, we consider actions of G on C*-algebras (G-C*-algebras) as structures in the framework of continuous model theory. When the group G is finite, one can regard a G-action as a usual metric structure by adding a function symbol for every element of the group. This does not work for a general compact g...

show abstract

Section: 7supporting

confidence: 87%

Section: Relationship Between Order Zero Dimension and D-containmentmentioning

confidence: 99%

See 1 more Smart Citation

Applications of model theory to C*-dynamics

Gardella

Lupini

2018

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…We will repeatedly use the following fact, which is probably standard. Its proof can be found, for example, in [GHS16]. For compact G, we identify C(G, A) and C(G) ⊗ A in the usual way.…”

Section: An Averaging Processmentioning

confidence: 99%

Crossed products by compact group actions with the Rokhlin property

Gardella

2017

J. Noncommut. Geom.

Self Cite

View full text Add to dashboard Cite

Abstract. We present a systematic study of the structure of crossed products and fixed point algebras by compact group actions with the Rokhlin property on not necessarily unital C * -algebras. Our main technical result is the existence of an approximate homomorphism from the algebra to its subalgebra of fixed points, which is a left inverse for the canonical inclusion. Upon combining this with results regarding local approximations, we show that a number of classes characterized by inductive limit decompositions with weakly semiprojective building blocks, are closed under formation of crossed products by such actions. Similarly, in the presence of the Rokhlin property, if the algebra has any of the following properties, then so do the crossed product and the fixed point algebra: being a Kirchberg algebra, being simple and having tracial rank zero or one, having real rank zero, having stable rank one, absorbing a strongly self-absorbing C * -algebra, satisfying the Universal Coefficient Theorem (in the simple, nuclear case), and being weakly semiprojective. The ideal structure of crossed products and fixed point algebras by Rokhlin actions is also studied.The methods of this paper unify, under a single conceptual approach, the work of a number of authors, who used rather different techniques. Our methods yield new results even in the well-studied case of finite group actions with the Rokhlin property.

show abstract

“…Indeed, this would imply the existence of a unital * -homomorphism Q → Z ∞ via Lemma 3.7, so it is impossible. More concretely,[25, Example 3.32] exhibits an example of a Z 2 -action α on a Kirchberg algebra with dim Rok (α) = 1 and dim c Rok (α) = 2.…”

mentioning

confidence: 99%

Rokhlin dimension: absorption of model actions

Szabó

2019

Analysis & PDE

View full text Add to dashboard Cite

In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly selfabsorbing C * -algebras. Namely, as to be made precise in the paper, let G be a well-behaved locally compact group. If D is a strongly selfabsorbing C * -algebra, and α : G A is an action on a separable, Dabsorbing C * -algebra that has finite Rokhlin dimension with commuting towers, then α tensorially absorbs every semi-strongly self-absorbing Gactions on D. In particular, this is the case when α satisfies any version of what is called the Rokhlin property, such as for G = R or G = Z k . This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some nontrivial applications. Most notably it is shown that for any k ≥ 1 and on any strongly self-absorbing Kirchberg algebra, there exists a unique R k -action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.

show abstract

Rokhlin dimension: duality, tracial properties, and crossed products

Cited by 32 publications

References 72 publications

Applications of model theory to C*-dynamics

Applications of model theory to C*-dynamics

Crossed products by compact group actions with the Rokhlin property

Rokhlin dimension: absorption of model actions

Contact Info

Product

Resources

About