Rokhlin Dimension and C*-Dynamics

By Ilan Hirshberg, W. Winter, J. Zacharias. 2012

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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 $$\mathcal{Z}$$Z-stable C*-algebras, where $$\mathcal{Z}$$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 $${\mathcal{Z}}$$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 $${\mathbb{Z}^{d}}$$Zd-actions on compact metrizable and finite dimensional spaces.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e198998ab8edfbeaff7e17ee0ce83b4a4abce0a9

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Citations in the corpus

Dynamic asymptotic dimension: relation to dynamics, topology, coarse geometry, and $$C^*$$C∗-algebras (2015, result): “We should note that there are many other results in the literature that give estimates on nuclear dimension or other C˚-algebraic regularity properties based on conditions on actions: see for example [36, 19, 37, 20, 11]. Many of these results go further than ours in at least some ways: for example,”
Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions (2016, methodology): “In this section, we show that pointwise outer actions of amenable, residually finite groups on Kirchberg algebras have Rokhlin dimension at most one in the sense of [24,77].”
Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra (2012, background): “τ(aj) ≤ 1 + ε0. The latter estimate follows from Corollary 6.8 (iv), which implies that k P j ajk ≤ (1+ε0)1/2. The following lemma is almost contained in [10, Lemma 5.7]. However, as the statement in [10] formally is diﬀerent from the fact that we shall need, we provide a proof of our lemma for the ”
Tracially Z-absorbing C*-algebras (2012, background): “requires restrictive Ktheoretic constraints on the algebra and the action. While for the single automorphism case the Rokhlin property is much more common (and even generic in certain cases, see e.g. [4]), it still requires the existence of many projections. The Rokhlin property has been generalized”
Classifying crossed product C*-algebras (2013, background): “ products have the same ordered K-theory and only one state on K0, so their Elliott invariants are just distinguished by their trace spaces. 5. Genericclassifiability We now follow up on a theme from [10] to show that, in fair generality, classiﬁability is generically preserved under crossed product”
The Rokhlin dimension of topological ℤm‐actions (2013, background): “jk. We now come to the main theorem of this section. Namely, we show that Zm-actions of ﬁnite Rokhlin dimension preserve ﬁnite nuclear dimension of the underlying C∗-algebra. Theorem 1.10 (compare to [10], 4.1). Let Abe a unital C∗-algebra and α: ZmyAbe a group action via automorphisms. Then we have”
Rokhlin dimension for compact group actions (2014, methodology): “More recently, Hirshberg, Winter and Zacharias introduced in [15] the notion of finite Rokhlin dimension for finite group actions (as well as automorphisms), as a generalization of the Rokhlin property.”
Mean dimension and Jaworski‐type theorems (2012, background): “Winter and Szabó have informed me that the theorem that aperiodic finitedimensional systems admit markers have found applications in the classification theory for C*-algebras arising from dynamical systems (see [10, 21]).”
Rokhlin dimension for actions of residually finite groups (2014, background): “We remark that this fact also follows from an early unpublished result by Matui and Sato (see the introduction of [28]).”