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.
We introduce the concept of Rokhlin dimension for actions of residually finite groups on C * -algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not been considered previously. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product C * -algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group C * -algebras have finite nuclear dimension. This extends an analogous result about Z m -actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing C * -algebra.
The external Kasparov product is used to construct odd and even spectral triples on crossed products of C * -algebras by actions of discrete groups which are equicontinuous in a natural sense. When the group in question is Z this gives another viewpoint on the spectral triples introduced by Belissard, Marcolli and Reihani. We investigate the properties of this construction and apply it to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set and some other crossed products of AF-algebras.2000 Mathematics Subject Classification. Primary: 46L05; Secondary: 46L87, 58B34.
Abstract. Higher-rank versions of Wold decomposition are shown to hold for doubly commuting isometric representations of product systems of C * -correspondences over N k 0 , generalising the classical result for a doubly commuting pair of isometries due to M. S lociński. Certain decompositions are also obtained for the general, not necessarily doubly commuting, case and several corollaries and examples are provided. Possibilities of extending isometric representations to fully coisometric ones are discussed.The classical notion of Wold decomposition refers to the unique decomposition of a Hilbert space isometry into a part which is unitary and a part which is isomorphic to a unilateral shift. For a simple proof and several applications of this result we refer to the classical monograph [SzF]. In analogy with the famous dilation problem for tuples of contractions, it is natural to ask whether some version of Wold decomposition is available for a tuple of commuting isometries. Indeed, M. S lociński established in [S lo] such a decomposition for a doubly commuting pair of isometries. This result (and its generalisations) was later used in [BCL] to provide models for tuples of commuting isometries and analyse the structure of C * -algebras they generate. Another example of the analysis of the structure of a pair of commuting isometries, also of relevance to our work here, may be found in [Pop].In recent years there has been an increased interest in Wold decompositions for objects of a different type. It originated from the work of G. Popescu, who in [Pope] established a result of this kind for a row contraction. Various related ideas were extended to an impressive degree in the series of papers of P. Muhly and B. Solel, who developed the theory of tensor algebras over C * -correspondences. In particular in [MS 2 ] they proved the existence of a Wold decomposition for an isometric representation of a C * -correspondence over a C * -algebra A. Another, more concrete, example of such a decomposition may be found in [JuK].In this paper we establish a higher-rank version of the main result of [MS 2 ]. The corresponding crucial concept of a product system of a C * -correspondence over N k 0 was introduced in [Fow]; the notion has been exploited in the recent work by B. Solel on dilations of commuting completely positive maps ([So 1−2 ]). Here we prove that every doubly commuting isometric representation of a product system of C * -correspondences over N k 0 decomposes uniquely into a combination of fully coisometric and induced parts (in the classical terminology they correspond respectively to a unitary and a shift part). It turns out that for isometric representations which are not doubly commuting it is still possible to characterise maximal pieces of the Permanent address of the first named author:
We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many properties formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C *algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we establish identities when tensoring with strongly self-absorbing C * -algebras. Finally, we show that the bivariant Cuntz semigroup of the present work can be used to classify all unital and stably finite C * -algebras.
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