scholarly journals Non-removable term ergodic action semigroups/groups

2015 ◽  
Vol 143 (8) ◽  
pp. 3445-3453
Author(s):  
Aliasghar Sarizadeh
Keyword(s):  
Ergodic Action

scholarly journals On ergodic foliations

1988 ◽  
Vol 8 (3) ◽  
pp. 437-457 ◽  
Author(s):  
Kyewon Park
Keyword(s):  
Quotient Space ◽  
Group Actions ◽  
Covering Space ◽  
Ergodic Action ◽  
Covering Spaces

AbstractWe define an ergodic ℤ-foliation and show that it can be realized as a quotient space of the ‘covering space’. The covering space has two actions, T and S, where T is a ℤ-action, S is a map of order two, and S and T skew-commute; that is, STS = T−1. We study the isometry between two foliations via the isomorphism between two bigger group actions in the covering spaces. Properties of an ergodic foliation are studied in a way similar to the study of an ergodic action. We construct a counterexample of a K-automorphism to show that, unlike Bernoulli automorphisms, ℤ-actions do not completely determine ℤ-foliations.

Ergodic elements for actions of Lie groups

1996 ◽  
Vol 16 (4) ◽  
pp. 703-717
Author(s):  
K. Robert Gutschera
Keyword(s):  
Invariant Measure ◽  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Group Structure ◽  
Structure Theory ◽  
Ergodic Action ◽  
Finite Invariant Measure ◽  
Connected Lie Group

AbstractGiven a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

Standard orbit factors and entropy-free Vershik equivalence

1999 ◽  
Vol 19 (2) ◽  
pp. 383-389 ◽  
Author(s):  
J. FELDMAN
Keyword(s):  
Free Action ◽  
Ergodic Action

1. Let ${\bf r}=(r_1,r_2,\dots)$ be a sequence of integers greater than one. Then every ${\bf r}$-adic reverse filtration has a maximal standard orbit factor.2. For ‘entropy-free’ ${\bf r}$, $a\geqq 0$, and any free action $A$ of the group $G=(Z/r_1Z)+(Z/r_2Z)+\dotsb$, there is a free ergodic action $A'$ of $G$ with $h(A')=a$ and such that the reverse filtration produced by $A'$ is isomorphic to that produced by $A$.

Ergodic Actions of Compact Groups on Operator Algebras II: Classification of Full Multiplicity Ergodic Actions

1988 ◽  
Vol 40 (6) ◽  
pp. 1482-1527 ◽  
Author(s):  
Antony Wassermann
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Crossed Product ◽  
Ergodic Action ◽  
Compact Groups ◽  
Type I ◽  
List Type ◽  
Von Neumann ◽  
C Algebra ◽  
Set Up

In the first paper of this series [17], we set up some general machinery for studying ergodic actions of compact groups on von Neumann algebras, namely, those actions for which . In particular we obtained a characterisation of the full multiplicity ergodic actions:THEOREM A. If α is an ergodic action of G on , then the following conditions are equivalent:(1) Each spectral subspace has multiplicity dim π for π in .(2) Each π in admits a unitary eigenmatrix in .(3) The W* crossed product is a (Type I) factor.(4) The C* crossed product of the C* algebra of norm continuity is isomorphic to the algebra of compact operators on a Hilbert space.

On dynamical systems preserving weights

2017 ◽  
Vol 38 (7) ◽  
pp. 2729-2747
Author(s):  
LAVY KOILPITCHAI ◽  
KUNAL MUKHERJEE
Keyword(s):  
Von Neumann Algebra ◽  
Bernoulli Shift ◽  
Ergodic Action ◽  
Spectral Multiplicity ◽  
Weak Mixing ◽  
Locally Compact ◽  
Separable Group ◽  
Von Neumann ◽  
Separable Predual ◽  
Normal Semifinite Trace

The canonical unitary representation of a locally compact separable group arising from an ergodic action of the group on a von Neumann algebra with separable predual preserving a faithful normal semifinite (infinite) weight is weak mixing. On the contrary, there exists a non-ergodic automorphism of a von Neumann algebra preserving a faithful normal semifinite trace such that the spectral measure and the spectral multiplicity of the induced action are respectively the Haar measure (on the unit circle) and $\infty$. Despite not even being ergodic, this automorphism has the same spectral data as that of a Bernoulli shift.

scholarly journals On ergodic actions whose self-joinings are graphs

1987 ◽  
Vol 7 (4) ◽  
pp. 531-557 ◽  
Author(s):  
A. del Junco ◽  
D. Rudolph
Keyword(s):  
Probability Space ◽  
Compact Subgroup ◽  
Common Factors ◽  
Ergodic Measure ◽  
Group Extension ◽  
Locally Compact Group ◽  
Ergodic Action ◽  
Minimal Self ◽  
Weakly Mixing ◽  
Simple Action

AbstractWe call an ergodic measure-preserving action of a locally compact group G on a probability space simple if every ergodic joining of it to itself is either product measure or is supported on a graph, and a similar condition holds for multiple self-joinings. This generalizes Rudolph's notion of minimal self-joinings and Veech's property S.Main results The joinings of a simple action with an arbitrary ergodic action can be explicitly descnbed. A weakly mixing group extension of an action with minimal self-joinings is simple. The action of a closed, normal, co-compact subgroup in a weakly-mixing simple action is again simple. Some corollaries. Two simple actions with no common factors are disjoint. The time-one map of a weakly mixing flow with minimal self-joinings is prime Distinct positive times in a -action with minimal self-joinings are disjoint.

Projective Modules over Higher-Dimensional Non-Commutative Tori

1988 ◽  
Vol 40 (2) ◽  
pp. 257-338 ◽  
Author(s):  
Marc A. Rieffel
Keyword(s):  
Ergodic Action ◽  
Identity Operator ◽  
Projective Modules ◽  
Unitary Operators ◽  
Pointwise Multiplication ◽  
Differentiable Manifolds ◽  
Higher Dimensional ◽  
Continuous Complex ◽  
Structure Constants ◽  
Complex Valued

The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary n-torus Tn with its algebra, C(Tn), of continuous complex-valued functions under pointwise multiplication. But C(Tn) is the universal C*-algebra generated by n commuting unitary operators. By definition, [15, 16, 50], a non-commutative n-torus is the universal C*-algebra generated by n unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [8, 10], these algebras have a natural differentiable structure, defined by a natural ergodic action of Tn as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [37] that the K-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are KK-equivalent to ordinary tori by Corollary 7.5 of [52].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [17].)

Approximate transitivity of the ergodic action of the group of finite permutations of on

2018 ◽  
Vol 39 (11) ◽  
pp. 2881-2895
Author(s):  
B. MITCHELL BAKER ◽  
THIERRY GIORDANO ◽  
RADU B. MUNTEANU
Keyword(s):  
Symmetric Group ◽  
Product Space ◽  
Ergodic Action ◽  
Probability Measures ◽  
Natural Action ◽  
Bernoulli Measure ◽  
Approximate Transitivity

In this paper we show that the natural action of the symmetric group acting on the product space $\{0,1\}^{\mathbb{N}}$ endowed with a Bernoulli measure is approximately transitive. We also extend the result to a larger class of probability measures.

Properties of a jointly ergodic action of the direct product of two groups

1991 ◽  
Vol 43 (5) ◽  
pp. 635-639
Author(s):  
V. Ya. Golodets ◽  
A. M. Sokhet
Keyword(s):  
Direct Product ◽  
Ergodic Action
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