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Journal articles on the topic 'Amenable action'

Author: Grafiati

Published: 15 December 2023

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1

POPA, SORIN. "CLASSIFICATION OF ACTIONS OF DISCRETE AMENABLE GROUPS ON AMENABLE SUBFACTORS OF TYPE II." International Journal of Mathematics 21, no. 12 (December 2010): 1663–95. http://dx.doi.org/10.1142/s0129167x10006343.

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We prove a classification result for properly outer actions σ of discrete amenable groups G on strongly amenable subfactors of type II, N ⊂ M, a class of subfactors that were shown to be completely classified by their standard invariant [Formula: see text], in [27]. The result shows that the action σ is completely classified in terms of the action it induces on [Formula: see text]. As an application of this, we obtain that inclusions of type III λ factors, 0 < λ < 1, having discrete decomposition and strongly amenable graph, are completely classified by their standard invariant.

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2

BOWEN, LEWIS, and AMOS NEVO. "Pointwise ergodic theorems beyond amenable groups." Ergodic Theory and Dynamical Systems 33, no. 3 (April 16, 2012): 777–820. http://dx.doi.org/10.1017/s0143385712000041.

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AbstractWe prove pointwise and maximal ergodic theorems for probability-measure-preserving (PMP) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable typeIII$_1$. We show that this class contains all irreducible lattices in connected semi-simple Lie groups without compact factors. We also establish similar results when the stable type isIII$_\lambda $,$0 \lt \lambda \lt 1$, under a suitable hypothesis. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of PMP actions of amenable groups to include PMP amenable equivalence relations. Secondly, we show that it is possible to reduce the proof of ergodic theorems for PMP actions of a general group to the proof of ergodic theorems in an associated PMP amenable equivalence relation, provided the group admits an amenable action with the properties stated above.

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3

BOWEN, LEWIS. "Sofic entropy and amenable groups." Ergodic Theory and Dynamical Systems 32, no. 2 (June 13, 2011): 427–66. http://dx.doi.org/10.1017/s0143385711000253.

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AbstractIn previous work, the author introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here, it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit changeσ-algebra.

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4

EXEL, RUY, and CHARLES STARLING. "Amenable actions of inverse semigroups." Ergodic Theory and Dynamical Systems 37, no. 2 (October 6, 2015): 481–89. http://dx.doi.org/10.1017/etds.2015.60.

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We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup ${\mathcal{S}}$, the action of ${\mathcal{S}}$ on its spectrum is amenable if and only if every action of ${\mathcal{S}}$ is amenable.

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5

Downarowicz, Tomasz, Dawid Huczek, and Guohua Zhang. "Tilings of amenable groups." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 747 (February 1, 2019): 277–98. http://dx.doi.org/10.1515/crelle-2016-0025.

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Abstract We prove that for any infinite countable amenable group G, any {\varepsilon>0} and any finite subset {K\subset G} , there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

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6

MEYEROVITCH, TOM. "Pseudo-orbit tracing and algebraic actions of countable amenable groups." Ergodic Theory and Dynamical Systems 39, no. 9 (January 24, 2018): 2570–91. http://dx.doi.org/10.1017/etds.2017.126.

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Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

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7

Kida, Yoshikata. "Inner amenable groups having no stable action." Geometriae Dedicata 173, no. 1 (December 1, 2013): 185–92. http://dx.doi.org/10.1007/s10711-013-9936-0.

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8

Ren, Xiankun, and Wenxiang Sun. "Local Entropy, Metric Entropy and Topological Entropy for Countable Discrete Amenable Group Actions." International Journal of Bifurcation and Chaos 26, no. 07 (June 30, 2016): 1650110. http://dx.doi.org/10.1142/s0218127416501108.

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Let [Formula: see text] be a compact metric space and [Formula: see text] a countable infinite discrete amenable group acting on [Formula: see text]. Like in the [Formula: see text]-action cases we define the notion of local entropy and by it we bound the difference between metric entropy and that of a partition, and bound the difference between topological entropy and that of a separated set, which generalize Theorems 1(1) and 1(2) in [Newhouse, 1989] from [Formula: see text]-actions to amenable group actions. We further prove that the entropy function [Formula: see text] is upper semi-continuous on [Formula: see text] for an asymptotic entropy expansive amenable group action.

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9

Mycielski, Jan. "Non-amenable groups with amenable action and some paradoxical decompositions in the plane." Colloquium Mathematicum 75, no. 1 (1998): 149–57. http://dx.doi.org/10.4064/cm-75-1-149-157.

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10

DONG, Z., and Y. Y. WANG. "FIXED POINT CHARACTERISATION FOR EXACT AND AMENABLE ACTION." Bulletin of the Australian Mathematical Society 92, no. 2 (June 16, 2015): 228–32. http://dx.doi.org/10.1017/s0004972715000520.

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Let $G$ be a finitely generated group acting on a compact Hausdorff space ${\mathcal{X}}$. We give a fixed point characterisation for the action being amenable. As a corollary, we obtain a fixed point characterisation for the exactness of $G$.

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11

Spatzier, R. J. "An example of an amenable action from geometry." Ergodic Theory and Dynamical Systems 7, no. 2 (June 1987): 289–93. http://dx.doi.org/10.1017/s0143385700004016.

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AbstractLet M be a compact manifold of not necessarily constant negative curvature. We observe that π1(M) acts amenably on the sphere at infinity of the universal cover of M with respect to a natural measure class. We also note that this action is of type III1.

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12

Drimbe, Daniel. "W∗-superrigidity for coinduced actions." International Journal of Mathematics 29, no. 04 (April 2018): 1850033. http://dx.doi.org/10.1142/s0129167x18500337.

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We prove W[Formula: see text]-superrigidity for a large class of coinduced actions. We prove that if [Formula: see text] is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group [Formula: see text], the coinduced action [Formula: see text] from an arbitrary probability measure preserving action [Formula: see text] is W[Formula: see text]-superrigid. We also prove a similar statement if [Formula: see text] is an icc non-amenable group which is measure equivalent to a product of two infinite groups. In particular, we obtain that any Bernoulli action of such a group [Formula: see text] is W[Formula: see text]-superrigid.

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13

Alpeev, Andrei, Tom Meyerovitch, and Sieye Ryu. "Predictability, topological entropy, and invariant random orders." Proceedings of the American Mathematical Society 149, no. 4 (February 9, 2021): 1443–57. http://dx.doi.org/10.1090/proc/15158.

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We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. We investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.

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14

HARTMAN, YAIR, KATE JUSCHENKO, OMER TAMUZ, and POOYA VAHIDI FERDOWSI. "Thompson’s group is not strongly amenable." Ergodic Theory and Dynamical Systems 39, no. 4 (June 28, 2017): 925–29. http://dx.doi.org/10.1017/etds.2017.49.

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15

BRODZKI, JACEK, GRAHAM A. NIBLO, PIOTR W. NOWAK, and NICK WRIGHT. "AMENABLE ACTIONS, INVARIANT MEANS AND BOUNDED COHOMOLOGY." Journal of Topology and Analysis 04, no. 03 (September 2012): 321–34. http://dx.doi.org/10.1142/s1793525312500161.

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We show that amenability of an action of a discrete group on a compact space X is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, that can be viewed as analogs of continuous bundles of dual modules over the G-space X. In the case when the compact space is a point, our result reduces to a classic theorem of Johnson, characterising amenability of groups. In the case when the compact space is the Stone–Čech compactification of the group, we obtain a cohomological characterisation of exactness, or equivalently, Yu's property A for the group, answering a question of Higson.

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16

Kechris, Alexander S. "Amenable versus hyperfinite Borel equivalence relations." Journal of Symbolic Logic 58, no. 3 (September 1993): 894–907. http://dx.doi.org/10.2307/2275102.

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LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

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17

Schneider, Friedrich Martin, and Andreas Thom. "On Følner sets in topological groups." Compositio Mathematica 154, no. 7 (May 16, 2018): 1333–61. http://dx.doi.org/10.1112/s0010437x1800708x.

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We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.

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18

Mohari, Anilesh. "A mean ergodic theorem of an amenable group action." Infinite Dimensional Analysis, Quantum Probability and Related Topics 17, no. 01 (March 2014): 1450003. http://dx.doi.org/10.1142/s0219025714500039.

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We consider a sequence of weak Kadison–Schwarz maps τn on a von-Neumann algebra ℳ with a faithful normal state ϕ sub-invariant for each (τn, n ≥ 1) and use a duality argument to prove strong convergence of their pre-dual maps when their induced contractive maps (Tn, n ≥ 1) on the GNS space of (ℳ, ϕ) are strongly convergent. The result is applied to deduce improvements of some known ergodic theorems and Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space ℳ*.

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19

Lee, Jong Bum. "Nilpotent action by an amenable group and Euler characteristic." Proceedings of the Edinburgh Mathematical Society 42, no. 1 (February 1999): 77–82. http://dx.doi.org/10.1017/s0013091500020022.

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20

Yang, Zhuocheng. "Action of amenable groups and uniqueness of invariant means." Journal of Functional Analysis 97, no. 1 (April 1991): 50–63. http://dx.doi.org/10.1016/0022-1236(91)90015-w.

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21

DRIMBE, DANIEL. "Cocycle and orbit equivalence superrigidity for coinduced actions." Ergodic Theory and Dynamical Systems 38, no. 7 (April 3, 2017): 2644–65. http://dx.doi.org/10.1017/etds.2016.134.

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We prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $\unicode[STIX]{x1D6EC}$ is a subgroup of a countable group $\unicode[STIX]{x1D6E4}$, we consider a probability measure preserving action $\unicode[STIX]{x1D6EC}\curvearrowright X_{0}$ and let $\unicode[STIX]{x1D6E4}\curvearrowright X$ be the coinduced action. Assume either that $\unicode[STIX]{x1D6E4}$ has property (T) or that $\unicode[STIX]{x1D6EC}$ is amenable and $\unicode[STIX]{x1D6E4}$ is a product of non-amenable groups. Using Popa’s deformation/rigidity theory we prove $\unicode[STIX]{x1D6E4}\curvearrowright X$ is ${\mathcal{U}}_{\text{fin}}$-cocycle superrigid, that is any cocycle for this action to a ${\mathcal{U}}_{\text{fin}}$ (e.g. countable) group ${\mathcal{V}}$ is cohomologous to a homomorphism from $\unicode[STIX]{x1D6E4}$ to ${\mathcal{V}}.$

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22

Tessera, Romain, and Alain Valette. "Locally compact groups with every isometric action bounded or proper." Journal of Topology and Analysis 12, no. 02 (October 5, 2018): 267–92. http://dx.doi.org/10.1142/s1793525319500547.

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A locally compact group [Formula: see text] has property PL if every isometric [Formula: see text]-action either has bounded orbits or is (metrically) proper. For [Formula: see text], say that [Formula: see text] has property BPp if the same alternative holds for the smaller class of affine isometric actions on [Formula: see text]-spaces. We explore properties PL and BPp and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix provides new examples of groups with property PL, including nonlinear ones.

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23

ANDEREGG, MARTIN, and PHILIPPE HENRY. "Actions of amenable equivalence relations on CAT(0) fields." Ergodic Theory and Dynamical Systems 34, no. 1 (October 30, 2012): 21–54. http://dx.doi.org/10.1017/etds.2012.122.

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AbstractWe present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

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24

Longo, Roberto. "Restricting a compact action to an injective subfactor." Ergodic Theory and Dynamical Systems 9, no. 1 (March 1989): 127–35. http://dx.doi.org/10.1017/s0143385700004855.

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Suppose we are given an action α: G → Aut (M) of a group G on a factor M; α possible way to analyse α may be to look at the invariant components where the action becomes more tractable. This point of view naturally leads to the study of the injective invariant subalgebras (recall for instance the good properties shared by amenable discrete or compact actions in the hyperfinite case [14]).

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25

Zhou, Yunhua. "Tail variational principle for a countable discrete amenable group action." Journal of Mathematical Analysis and Applications 433, no. 2 (January 2016): 1513–30. http://dx.doi.org/10.1016/j.jmaa.2015.08.058.

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26

Huang, Wen, Xiangdong Ye, and Guohua Zhang. "Local entropy theory for a countable discrete amenable group action." Journal of Functional Analysis 261, no. 4 (August 2011): 1028–82. http://dx.doi.org/10.1016/j.jfa.2011.04.014.

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27

Feldman, J., C. E. Sutherland, and R. J. Zimmer. "Subrelations of ergodic equivalence relations." Ergodic Theory and Dynamical Systems 9, no. 2 (June 1989): 239–69. http://dx.doi.org/10.1017/s0143385700004958.

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AbstractWe introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a groupQwhich serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) ofQon it; in the case whereQis amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

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28

Li, Hanfeng, and Bingbing Liang. "Mean dimension, mean rank, and von Neumann–Lück rank." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 739 (June 1, 2018): 207–40. http://dx.doi.org/10.1515/crelle-2015-0046.

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AbstractWe introduce an invariant, called mean rank, for any module{\mathcal{M}}of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of{\mathcal{M}}, the mean rank of{\mathcal{M}}, and the von Neumann–Lück rank of{\mathcal{M}}all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin–Schnirelmann theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.

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29

Frej, Bartosz, and Dawid Huczek. "A Comment on Ergodic Theorem for Amenable Groups." Canadian Mathematical Bulletin 63, no. 2 (July 29, 2019): 257–61. http://dx.doi.org/10.4153/s0008439519000110.

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AbstractWe prove a version of the ergodic theorem for an action of an amenable group, where a Følner sequence need not be tempered. Instead, it is assumed that a function satisfies certain mixing conditions.

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30

ELEK, GÁBOR. "Amenable purely infinite actions on the non-compact Cantor set." Ergodic Theory and Dynamical Systems 40, no. 6 (November 20, 2018): 1619–33. http://dx.doi.org/10.1017/etds.2018.121.

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We prove that any countable non-amenable group $\unicode[STIX]{x1D6E4}$ admits a free, minimal, amenable, purely infinite action on the non-compact Cantor set. This answers a question of Kellerhals, Monod and Rørdam [Non-supramenable groups acting on locally compact spaces. Doc. Math.18 (2013), 1597–1626].

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31

DOOLEY, ANTHONY H., and GUOHUA ZHANG. "Co-induction in dynamical systems." Ergodic Theory and Dynamical Systems 32, no. 3 (May 24, 2011): 919–40. http://dx.doi.org/10.1017/s0143385711000083.

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AbstractIf a countable amenable group G contains an infinite subgroup Γ, one may define, from a measurable action of Γ, the so-called co-induced measurable action of G. These actions were defined and studied by Dooley, Golodets, Rudolph and Sinelsh’chikov. In this paper, starting from a topological action of Γ, we define the co-induced topological action of G. We establish a number of properties of this construction, notably, that the G-action has the topological entropy of the Γ-action and has uniformly positive entropy (completely positive entropy, respectively) if and only if the Γ-action has uniformly positive entropy (completely positive entropy, respectively). We also study the Pinsker algebra of the co-induced action.

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32

DOOLEY, A. H., V. YA GOLODETS, D. J. RUDOLPH, and S. D. SINEL’SHCHIKOV. "Non-Bernoulli systems with completely positive entropy." Ergodic Theory and Dynamical Systems 28, no. 1 (February 2008): 87–124. http://dx.doi.org/10.1017/s014338570700034x.

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AbstractA new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable groupG, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any$h \in (0, \infty ]$, an uncountable family of cpe actions of entropyh, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that ifαGis co-induced from an actionαΓof a subgroup Γ, thenh(αG)=h(αΓ). We also prove that ifαΓis a non-Bernoulli cpe action of Γ, thenαGis also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of$\mathbb Z$, which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

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33

Abbott, Carolyn, David Hume, and Denis Osin. "Extending group actions on metric spaces." Journal of Topology and Analysis 12, no. 03 (October 1, 2018): 625–65. http://dx.doi.org/10.1142/s1793525319500584.

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We address the following natural extension problem for group actions: Given a group [Formula: see text], a subgroup [Formula: see text], and an action of [Formula: see text] on a metric space, when is it possible to extend it to an action of the whole group [Formula: see text] on a (possibly different) metric space? When does such an extension preserve interesting properties of the original action of [Formula: see text]? We begin by formalizing this problem and present a construction of an induced action which behaves well when [Formula: see text] is hyperbolically embedded in [Formula: see text]. Moreover, we show that induced actions can be used to characterize hyperbolically embedded subgroups. We also obtain some results for elementary amenable groups.

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34

Suzuki, Yuhei. "Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products." International Mathematics Research Notices 2020, no. 19 (August 14, 2018): 6007–41. http://dx.doi.org/10.1093/imrn/rny187.

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Abstract We extend Matui’s notion of almost finiteness to general étale groupoids and show that the reduced groupoid C$^{\ast }$-algebras of minimal almost finite groupoids have stable rank 1. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result: (1) for any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank 1; (2) any countable amenable group admits a minimal action on the Cantor set, all whose minimal extensions form the crossed product of stable rank 1; and (3) for any amenable group, the crossed product of the universal minimal action has stable rank 1.

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35

Łącka, Martha, and Marta Straszak. "Quasi-uniform convergence in dynamical systems generated by an amenable group action." Journal of the London Mathematical Society 98, no. 3 (July 22, 2018): 687–707. http://dx.doi.org/10.1112/jlms.12157.

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36

Xie, Huo'an. "ON DENSITY OF SMOOTH ELEMENTS FOR AN ACTION OF AN AMENABLE GROUP." Acta Mathematica Scientia 13, no. 3 (1993): 261–65. http://dx.doi.org/10.1016/s0252-9602(18)30215-7.

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37

Pan, Juan, and Yunhua Zhou. "Some Results on Bundle Systems for a Countable Discrete Amenable Group Action." Acta Mathematica Scientia 43, no. 3 (April 29, 2023): 1382–402. http://dx.doi.org/10.1007/s10473-023-0322-1.

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38

RØRDAM, MIKAEL, and ADAM SIERAKOWSKI. "Purely infinite C*-algebras arising from crossed products." Ergodic Theory and Dynamical Systems 32, no. 1 (April 5, 2011): 273–93. http://dx.doi.org/10.1017/s0143385710000829.

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AbstractWe study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.

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39

HAYES, BEN. "Sofic entropy of Gaussian actions." Ergodic Theory and Dynamical Systems 37, no. 7 (May 12, 2016): 2187–222. http://dx.doi.org/10.1017/etds.2016.6.

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Associated to any orthogonal representation of a countable discrete group, is a probability measure-preserving action called the Gaussian action. Using the Polish model formalism we developed before, we compute the entropy (in the sense of Bowen [J. Amer. Math. Soc.23(2010) 217–245], Kerr and Li [Invent. Math.186(2011) 501–558]) of Gaussian actions when the group is sofic. Computation of entropy for Gaussian actions has only been done when the acting group is abelian and thus our results are new, even in the amenable case. Fundamental to our approach are methods of non-commutative harmonic analysis and$C^{\ast }$-algebras which replace the Fourier analysis used in the abelian case.

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SEWARD, BRANDON. "A subgroup formula for f-invariant entropy." Ergodic Theory and Dynamical Systems 34, no. 1 (November 30, 2012): 263–98. http://dx.doi.org/10.1017/etds.2012.128.

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AbstractWe study a measure entropy for finitely generated free group actions called f-invariant entropy. The f-invariant entropy was developed by L. Bowen and is essentially a special case of his measure entropy theory for actions of sofic groups. In this paper we relate the f-invariant entropy of a finitely generated free group action to the f-invariant entropy of the restricted action of a subgroup. We show that the ratio of these entropies equals the index of the subgroup. This generalizes a well-known formula for the Kolmogorov–Sinai entropy of amenable group actions. We then extend the definition of f-invariant entropy to actions of finitely generated virtually free groups. We also obtain a numerical virtual measure conjugacy invariant for actions of finitely generated virtually free groups.

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Yan, Kesong, and Fanping Zeng. "Mean Proximality, Mean Sensitivity and Mean Li–Yorke Chaos for Amenable Group Actions." International Journal of Bifurcation and Chaos 28, no. 02 (February 2018): 1850028. http://dx.doi.org/10.1142/s0218127418500281.

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We consider mean proximality and mean Li–Yorke chaos for [Formula: see text]-systems, where [Formula: see text] is a countable discrete infinite amenable group. We prove that if a countable discrete infinite abelian group action is mean sensitive and there is a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li–Yorke chaotic. Moreover, we give some characterizations of mean proximal systems for general countable discrete infinite amenable groups.

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SZŐKE, NÓRA GABRIELLA. "A Tits alternative for topological full groups." Ergodic Theory and Dynamical Systems 41, no. 2 (August 27, 2019): 622–40. http://dx.doi.org/10.1017/etds.2019.54.

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We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.

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Echterhoff, Siegfried, and John Quigg. "Full duality for coactions of discrete groups." MATHEMATICA SCANDINAVICA 90, no. 2 (June 1, 2002): 267. http://dx.doi.org/10.7146/math.scand.a-14374.

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Using the strong relation between coactions of a discrete group $G$ on $C^*$-algebras and Fell bundles over $G$ we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of $G$ as opposed to the usual theory of [16], [11] which uses the spatially defined reduced crossed products and normal subgroups of $G$. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel [7] if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions.

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DEPREZ, TOBE, and STEFAAN VAES. "Inner amenability, property Gamma, McDuff factors and stable equivalence relations." Ergodic Theory and Dynamical Systems 38, no. 7 (March 14, 2017): 2618–24. http://dx.doi.org/10.1017/etds.2016.135.

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We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.

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Sun, Michael. "Strongly outer group actions on UHF algebras." Journal of Topology and Analysis 10, no. 03 (August 30, 2018): 701–21. http://dx.doi.org/10.1142/s1793525318500231.

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We show that for any countable discrete maximally almost periodic group [Formula: see text] and any UHF algebra [Formula: see text], there exists a strongly outer product type action [Formula: see text] of [Formula: see text] on [Formula: see text]. When [Formula: see text] is also elementary amenable, Matui–Sato have shown that such actions have their tracial Rokhlin property. Consequently, the class of crossed products [Formula: see text] satisfy Elliott’s classification conjecture. We also show the existence of the “Rokhlin” property for countable discrete almost abelian group actions on the universal UHF algebra.

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Benitez, Federico, and Diego Maltrana. "Dispositions and the Least Action Principle." Disputatio 14, no. 65 (November 1, 2022): 91–104. http://dx.doi.org/10.2478/disp-2022-0006.

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Abstract This work deals with obstacles hindering a metaphysics of laws of nature in terms of dispositions, i.e., of fundamental properties that are causal powers. A recent analysis of the principle of least action has put into question the viability of dispositionalism in the case of classical mechanics, generally seen as the physical theory most easily amenable to a dispositional ontology. Here, a proper consideration of the framework role played by the least action principle within the classical image of the world allows us to build a consistent metaphysics of dispositions as charges of interactions. In doing so we develop a general approach that opens the way towards an ontology of dispositions for fundamental physics also beyond classical mechanics.

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FRISCH, JOSHUA, TOMER SCHLANK, and OMER TAMUZ. "Normal amenable subgroups of the automorphism group of the full shift." Ergodic Theory and Dynamical Systems 39, no. 5 (September 7, 2017): 1290–98. http://dx.doi.org/10.1017/etds.2017.72.

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We show that every normal amenable subgroup of the automorphism group of the full shift is contained in its center. This follows from the analysis of this group’s Furstenberg topological boundary, through the construction of a minimal and strongly proximal action. We extend this result to higher dimensional full shifts. This also provides a new proof of Ryan’s theorem and of the fact that these groups contain free groups.

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Deaconu, Valentin, and Leonard Huang. "Group Actions on Product Systems." New Zealand Journal of Mathematics 54 (October 19, 2023): 33–47. http://dx.doi.org/10.53733/311.

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We introduce the concept of a crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We generalize a theorem of Hao and Ng about the crossed product of the Cuntz-Pimsner algebra of a $C^{\ast}$-correspondence by a group action to the context of product systems. We present examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.

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Huang, Xiaojun, Yuan Lian, and Changrong Zhu. "A Billingsley-type theorem for the pressure of an action of an amenable group." Discrete & Continuous Dynamical Systems - A 39, no. 2 (2019): 959–93. http://dx.doi.org/10.3934/dcds.2019040.

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Sharma, Shilpa, and Devendra Gupta. "Stem-cell therapy for neurologic diseases." Journal of Neuroanaesthesiology and Critical Care 02, no. 01 (April 2015): 015–22. http://dx.doi.org/10.4103/2348-0548.148379.

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AbstractWith the advent of research on stem cell therapy for various diseases, an important need was felt in the field of neurological diseases. While congenital lesion may not be amenable to stem cell therapy completely, there is a scope of partial improvement in the lesions and halt in further progression. Neuro degenerative lesions like Parkinson's disease, multiple sclerosis and amyotrophic lateral sclerosis have shown improvement with stem cell therapy. This article reviews the available literature and summarizes the current evidence in the various neurologic diseases amenable to stem cell therapy, the plausible mechanism of action, ethical concerns with insights into the future of stem cell therapy.

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