Disintegration of positive isometric group representations on $$\varvec{\mathrm {L}^{p}}$$ L p -spaces

Jeu, Marcel de; Rozendaal, Jan

doi:10.1007/s11117-017-0499-4

Cited by 10 publications

(17 citation statements)

References 16 publications

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“…Some spectral theoretic results about positive group representations can be found in [53,24] and some structural results about groups of positive operators in [48,Sec 3]. A systematic investigation of positive group representations was just initiated in the recent series of papers [11,12,10]. While all of these references focus on strongly continuous representations of certain topological groups, we do not require any topological assumptions in the following.…”

Section: Group Representations On Atomic Banach Latticesmentioning

confidence: 99%

Convergence of positive operator semigroups

Gerlach

Glück

2019

Trans. Amer. Math. Soc.

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We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations.Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive C 0 -semigroup containing or dominating a kernel operator converges strongly as t → ∞. We gain new insights into the structure theoretical background of those theorems and generalise them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter.

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Section: Group Representations On Atomic Banach Latticesmentioning

confidence: 99%

Convergence of positive operator semigroups

Gerlach

Glück

2019

Trans. Amer. Math. Soc.

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“…The parts (2) and (3) are immediate from (1). As for (4), we know from (2) that The matter is usually considered in the context of a locally compact Hausdorff group and a locally compact Hausdorff space in [14], but more can be said.…”

Section: Preliminaries On Group Representationsmentioning

confidence: 99%

“…their separability is known to be equivalent to the separability of j µ , (see e.g. [1]). As a second consequence, when combined with Lemma 5.7 and with the observed fact that a lattice homomorphism of ( ) ( )…”

Section: Disintegration: General Casementioning

confidence: 99%

“…Apart from an intrinsic interest and mathematical relevance, the wish to understand such representations originates from quantum theory, where the unitary representations of the symmetry group of a physical system have a natural role. However, whereas symmetry yields a unitary representation of the pertinent symmetry group, there is also a family of canonical representations on Banach lattices in [1]. Likewise, for all 0 ≤ < ∞  , the motion group of ( )…”

Section: Introduction and Overviewmentioning

confidence: 99%

“…To see that (2) implies(1), note that the equality in (1) is just the one in (2) if0 j f ≥ (note that 0 j j T f ≥ then. For general j f , then have R E T R A C T E D S. Joseph et al DOI: 10.4236/apm.2019.911044 915 Advances in Pure Mathematics…”

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confidence: 99%

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Disintegration of Group Representations on Direct Integrals of Banach Spaces

Joseph¹,

Saeed²,

Suoliman³

et al. 2019

APM

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Retraction type (multiple responses allowed):  Unreliable findings   Lab error  Inconsistent data  Analytical error  Biased interpretation   Other:  Irreproducible results  Failure to disclose a major competing interest likely to influence interpretations or recommendations  Unethical research   Fraud   Data fabrication  Fake publication  Other: X Plagiarism  Self plagiarism  Overlap  Redundant publication *  Copyright infringement  Other legal concern:  Editorial reasons   Handling error  Unreliable review(s)  Decision error  Other:  Other: Results of publication (only one response allowed):  are still valid. Xwere found to be overall invalid.

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Measure-preserving semiflows and one-parameter Koopman semigroups

2018

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For a finite measure space X, we characterize strongly continuous Markov lattice semigroups on L p (X) by showing that their generator A acts as a derivation on the dense subspace D(A) ∩ L ∞ (X). We then use this to characterize Koopman semigroups on L p (X) if X is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space.In this article we address mainly the following two issues. First, we characterize strongly continuous Markov lattice semigroups (T (t)) t≥0 on L p (X) by properties of their generators for a finite measure space X = (X, Σ, µ). We will show that a strongly continuous semigroup on L p (X) is a Markov lattice semigroup if and only if its generator A acts as a derivation on D(A) ∩ L ∞ (X), 1 ∈ D(A) and the semigroup is locally bounded on L ∞ (X). Similar results have been established by R. Nagel and R. Derndinger in [2, Satz 2.5] for semigroups on C(K), see also [8,, and recently by T. ter Elst and M. Lemańczyk in [4] for unitary groups on L 2 (X). Second, we show that such semigroups are always similar to a semigroup of Koopman operators. More precisely, we construct a compact space K and a Borel measure ν such that L 1 (X, Σ, µ) is isometrically Banach lattice isomorphic to L 1 (K, ν) and, via this isomorphism, the semigroup (T (t)) t≥0 is similar to a semigroup of Koopman operators on L 1 (K, ν) induced by a continuous semiflow (ϕ t ) t≥0 on K. Furthermore, in case that the space L 1 (X, Σ, µ) is separable, we show that K can be chosen to be metrizable. Similar results have been already obtained for strongly continuous representations of locally compact groups on L p (X) as bi-Markov embeddings, see [7, Theorem 5.14]. The article is organized as follows. In the second part of the introduction, we specify our notation and recall some basic facts we use throughout the article. In Section 1, we prove our main result, Theorem 1.1, that characterizes strongly Markov semigroups of lattice homomorphisms by the condition that their generator acts as a derivation, followed by a version for semigroups that are not necessarily Markov. In Section 2 we then use these results to characterize Koopman semigroups and in particular obtain [4, Theorem 1.1] in which ter Elst and Lemańczyk proved a corresponding result for unitary operator groups as Corollary 2.6. In Section 3, we turn to the construction of topological models. Finally, in Section 4, we consider ergodic, measure-preserving flows and give a new proof for the fact that they contain at most countably many non-ergodic mappings, provided that their induced group of Koopman operators is strongly continuous on L 2 (X). This has previously been proven in [9, Theorem 1] for R k -actions.

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Disintegration of positive isometric group representations on $$\varvec{\mathrm {L}^{p}}$$ L p -spaces

Cited by 10 publications

References 16 publications

Convergence of positive operator semigroups

Convergence of positive operator semigroups

Disintegration of Group Representations on Direct Integrals of Banach Spaces

Measure-preserving semiflows and one-parameter Koopman semigroups

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