scholarly journals Ergodic multiplier properties

2014 ◽  
Vol 36 (3) ◽  
pp. 794-815 ◽  
Author(s):  
ADI GLÜCKSAM
Keyword(s):  
Irreducible Representation ◽  
Heisenberg Group ◽  
Weak Mixing ◽  
Locally Compact ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Dimensional Irreducible Representation ◽  
Weakly Mixing

In this article we will extend ‘the weak mixing theorem’ for certain locally compact Polish groups (Moore groups and minimally weakly mixing groups). In addition, we will show that the Gaussian action associated with the infinite-dimensional irreducible representation of the continuous Heisenberg group,$H_{3}(\mathbb{R})$, is weakly mixing but not mildly mixing.

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

scholarly journals Weak mixing for locally compact quantum groups

2016 ◽  
Vol 37 (5) ◽  
pp. 1657-1680 ◽  
Author(s):  
AMI VISELTER
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebras ◽  
Unitary Representations ◽  
Discrete Groups ◽  
Weak Mixing ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Weakly Mixing

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

Infinitesimal Generators on the Quantum Group SUq(2)

1998 ◽  
Vol 01 (04) ◽  
pp. 573-598 ◽  
Author(s):  
Michael Schürmann ◽  
Michael Skeide
Keyword(s):  
Irreducible Representation ◽  
Quantum Group ◽  
Lie Group ◽  
Lévy Processes ◽  
Levy Processes ◽  
Infinite Dimensional ◽  
Infinitesimal Generators ◽  
Dimensional Irreducible Representation

Quantum Lévy processes on a quantum group are, like classical Lévy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SU q(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus we obtain a quantum Lévy–Khintchine formula.

Irreducible representations of finitely generated nilpotent groups

1977 ◽  
Vol 81 (2) ◽  
pp. 201-208 ◽  
Author(s):  
Daniel Segal
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Closed Field ◽  
Finitely Generated ◽  
Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation

1. Introduction. It is well known that every finite-dimensional irreducible representation of a nilpotent group over an algebraically closed field is monomial, that is induced from a 1-dimensional representation of some subgroup. However, even a finitely generated nilpotent group in general has infinite-dimensional irreducible representations, and as a first step towards an understanding of these one wants to discover whether they too are necessarily monomial. The main point of this note is to show how far they can fail to be so.

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

scholarly journals MODELS FOR THE IRREDUCIBLE REPRESENTATION OF A HEISENBERG GROUP

2004 ◽  
Vol 07 (04) ◽  
pp. 527-546 ◽  
Author(s):  
TROND DIGERNES ◽  
V. S. VARADARAJAN
Keyword(s):  
Irreducible Representation ◽  
Heisenberg Group ◽  
Configuration Space ◽  
Mechanical Model ◽  
Quantum Systems ◽  
Quantum Mechanical ◽  
Heisenberg Groups ◽  
Quantum Mechanical Model ◽  
Isotropic Subgroup ◽  
Uniform Treatment

In its most general formulation a quantum kinematical system is described by a Heisenberg group; the "configuration space" in this case corresponds to a maximal isotropic subgroup. We study irreducible models for Heisenberg groups based on compact maximal isotropic subgroups. It is shown that if the Heisenberg group is 2-regular, but the subgroup is not, the "vacuum sector" of the irreducible representation exhibits a fermionic structure. This will be the case, for instance, in a quantum mechanical model based on the 2-adic numbers with a suitably chosen isotropic subgroup. The formulation in terms of Heisenberg groups allows a uniform treatment of p-adic quantum systems for all primes p, and includes the possibility of treating adelic systems.

scholarly journals Point transitivity, -transitivity and multi-minimality

2014 ◽  
Vol 35 (5) ◽  
pp. 1423-1442 ◽  
Author(s):  
ZHIJING CHEN ◽  
JIAN LI ◽  
JIE LÜ
Keyword(s):  
Dynamical System ◽  
Open Subset ◽  
Topological Dynamical System ◽  
Property A ◽  
Weak Mixing ◽  
Furstenberg Family ◽  
Strongly Mixing ◽  
Transitive Point ◽  
Weakly Mixing

Let $(X,f)$ be a topological dynamical system and ${\mathcal{F}}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an ${\mathcal{F}}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}:f^{n}(x)\in U\}$, is in ${\mathcal{F}}$; the system $(X,f)$ is called ${\mathcal{F}}$-point transitive if there exists some ${\mathcal{F}}$-transitive point. In this paper, we first discuss the connection between ${\mathcal{F}}$-point transitivity and ${\mathcal{F}}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by ${\mathcal{F}}$-point transitivity, completing results in [Transitive points via Furstenberg family. Topology Appl. 158 (2011), 2221–2231]. We also show that multi-transitivity, ${\rm\Delta}$-transitivity and multi-minimality can be characterized by ${\mathcal{F}}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates. Ergod. Th. & Dynam. Sys. 32 (2012), 1661–1672].

scholarly journals Analog of selfduality in dimension nine

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Anna Fino ◽  
Paweł Nurowski
Keyword(s):  
Irreducible Representation ◽  
Riemannian Geometry ◽  
Four Dimensions ◽  
Dimensional Irreducible Representation

AbstractWe introduce a type of Riemannian geometry in nine dimensions, which can be viewed as the counterpart of selfduality in four dimensions. This geometry is related to a 9-dimensional irreducible representation of

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