Operator Theoretic Aspects of Ergodic Theory

Eisner, Tanja; Farkas, Bálint; Haase, Markus; Nagel, Rainer

doi:10.1007/978-3-319-16898-2

Cited by 240 publications

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“…In Koopman's theory, as it is well known, the operator algebras have many applications to study of dynamical systems and ergodic theory ( [3]). In this direction, the study of algebraic properties of M X,m may be useful: Let G be a discrete group of measure preserving homeomorphisms of X.…”

Section: The Main Definitionsmentioning

confidence: 99%

A class of Banach algebras of generalized matrices

Sadr

2017

Funct Anal Its Appl

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Abstract. We introduce a class of Banach algebras of generalized matrices and study the existence of approximate units, ideal structure, and derivations of them. IntroductionLet X be a compact metrizable space and m be a Borel probability measure on X. In this note we study some aspects of the algebraic structure of a Banach algebra M of generalized complex matrices whose their arrays are indexed by elements of X 2 and vary continuously. The multiplication of M is defined similar to the ordinary matrix multiplication and uses m as the weight for arrays. See Section 2 for exact definition. In the case that m has full support, M is isometric isomorphic to a subalgebra of compact operators acting on the Banach space of continuous functions on X. Indeed any element of M defines an integral operator in a canonical way. Thus M can be interpreted as a Banach algebra of integral operators or kernels ([2]). In Section 3 we investigate the existence of approximate units of M. In Section 4 we show that if X is infinite then the center of M is zero. In Section 5 we study ideal structure of M. In Section 6 we consider some classes of representations of M. In Section 7 we show that under some mild conditions bounded derivations on M are approximately inner.Notations. For a compact space X and a Banach space E we denote by C(X; E) the Banach space of continuous E-valued functions on X with supremum norm. We also let C(X) := C(X; C). There is a canonical isometric isomorphism C(X; E) ∼ = C(X)⊗E wherě ⊗ denotes the completed injective tensor product. The phrase "point-wise convergence topology" is abbreviated to "pct". By pct on C(X; E) we mean the vector topology under which a net (f λ ) λ ∈ C(X; E) converges to f if and only if f λ (x) → f (x) in the norm of E for every x ∈ X. If f and f ′ are complex functions on spaces X andThe support of a Borel measure m is denoted by Spm. B x,δ denotes the open ball with center at x and radius δ. The main definitionsLet X be a compact metrizable space and m be a Borel probability measure on X. By analogy with matrix multiplication we let the convolution of f, g ∈ C(X 2 ) be defined by f ⋆ g(x, y) = X f (x, z)g(z, y)dm(z). Also by analogy with matrix adjoint we let f * ∈ C(X 2 ) be defined by f * (x, y) =f (y, x). It is easily verified that ⋆ is an associative multiplication, * is an involution, and also, f ⋆ g ∞ ≤ f ∞ g ∞ and f * ∞ = f ∞ ; thus C(X 2 ) becomes 2010 Mathematics Subject Classification. 46H05; 46H35; 46H10; 47B48; 46H25.

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Section: The Main Definitionsmentioning

confidence: 99%

A class of Banach algebras of generalized matrices

Sadr

2017

Funct Anal Its Appl

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“…See e.g. , Theorem 17.13. Theorem If

(Ω, Σ, μ; φ)

is an ergodic system with discrete spectrum, then it is isomorphic to a rotation by a generating element on a compact monothetic group endowed with the normalised Haar measure.…”

Section: Preliminariesmentioning

confidence: 99%

“…As a consequence, the point spectrum of an ergodic system with discrete spectrum is a complete isomorphism invariant. See , Corollary 17.14. Corollary Two ergodic dynamical systems whose Koopman operators have discrete spectrum are isomorphic if and only if their Koopman operators have the same point spectrum.…”

Section: Preliminariesmentioning

confidence: 99%

“…Compact monothetic groups can be characterized by their character groups. See , Proposition 17.7. Theorem Let G be a compact monothetic group with generating element

e \in G

.…”

Section: Preliminariesmentioning

confidence: 99%

“…For this construction we need realizations of rotations on the a ‐adic integers

Z_{a}

(see , Theorem 2.6) based on the von Neumann‐Kakutani map. See, e.g., , 17.4. Lemma Let

φ_{a}

be the bijective, piecewise linear transformation on the measure space

([0, 1), B, λ)

defined by

φ_{a} (x) : = x - 1 + \frac{1 + a_{m + 1}}{a_{0} a_{1} \dots a_{m + 1}}; x \in [1 - \frac{1}{a_{0} a_{1} \dots a_{m}}, 1 - \frac{1}{a_{0} a_{1} \dots a_{m + 1}})

for

m \in {double-struckN}_{0}

, where

a = (a_{1}, \dots, a_{n}, \dots)

is an infinite sequence with

1 < a_{n} \in N

and

a_{0} : = 1

.…”

Section: Realizations Of Rotations On the Indecomposable Compact Monomentioning

confidence: 99%

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Realizations of rotations on an indecomposable compact monothetic group

Maier

2015

Mathematische Nachrichten

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By the classical Halmos‐von Neumann theorem, each compact monothetic group corresponds to an ergodic dynamical system with discrete spectrum. For such groups we prove two results. We first construct a compact monothetic group which does not split into a direct product of a connected and a totally disconnected compact monothetic group. Then we present a measure preserving dynamical system on the unit square being isomorphic to a rotation on this indecomposable group.

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A note on entropy of Delone sets

Hauser

2022

Mathematische Nachrichten

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In this note we present that the patch counting entropy can be obtained as a limit and investigate which sequences of compact sets are suitable to define this quantity. We furthermore present a geometric definition of patch counting entropy for Delone sets of infinite local complexity and that the patch counting entropy of a Delone set equals the topological entropy of the corresponding Delone dynamical system. We present our results in the context of (non-compact) locally compact abelian groups that contain Meyer sets.

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Operator Theoretic Aspects of Ergodic Theory

Cited by 240 publications

References 0 publications

A class of Banach algebras of generalized matrices

A class of Banach algebras of generalized matrices

Realizations of rotations on an indecomposable compact monothetic group

A note on entropy of Delone sets

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