Weighted convolution algebras on subsemigroups of the real line

Dales, H. G.; Dedania, H. V.

doi:10.4064/dm459-0-1

Cited by 14 publications

(18 citation statements)

References 26 publications

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“…For a locally compact group G, L 1 (G, ω) is semisimple if G is abelian [5]; for non-abelian G, it is not known whether L 1 (G, ω) is semisimple or not (page 175 of [12]). For an abelian semigroup S, 1 (S, ω) is semisimple iff S is separating and ω is semisimple (Proposition 4.8 of [11]). This quickly gives the following.…”

Section: α(S)||α(t )| = |α(T S)| ≤ ω(T S) ≤ω(T )ω(S)mentioning

confidence: 98%

“…Let ω be a weight on S. Then ω is (i) semisimple [11] if lim n→∞ ω(s n ) 1 n > 0, s ∈ S. (ii) radical [11] if lim n→∞ ω(s n ) 1 n = 0, s ∈ S. (iii) Beurling-Domar [14] if ω ≥ 1 and n∈N log ω(s n )…”

Section: Definition 25mentioning

confidence: 99%

“…with the norm f ω , which is a convolution Banach algebra [11,14] as well as in the study of the group algebras L 1 (G, ω) on a locally compact group G [10,21]. Notice that recently the semigroup algebras 1 (S) as well as the Beurling algebra L 1 (G, ω) have acquired considerable attention [12,13].…”

Section: Introductionmentioning

confidence: 99%

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Multipliers of weighted semigroups and associated Beurling Banach algebras

2011

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Given a weighted discrete abelian semigroup (S, ω), the semigroup M ω (S) of ω-bounded multipliers as well as the Rees quotient M ω (S)/S together with their respective weightsω andω q induced by ω are studied; for a large class of weights ω, the quotient 1 (M ω (S),ω)/ 1 (S, ω) is realized as a Beurling algebra on the quotient semigroup M ω (S)/S; the Gel'fand spaces of these algebras are determined; and Banach algebra properties like semisimplicity, uniqueness of uniform norm and regularity of associated Beurling algebras on these semigroups are investigated. The involutive analogues of these are also considered. The results are exhibited in the context of several examples.

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Section: α(S)||α(t )| = |α(T S)| ≤ ω(T S) ≤ω(T )ω(S)mentioning

confidence: 98%

Section: Definition 25mentioning

confidence: 99%

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Multipliers of weighted semigroups and associated Beurling Banach algebras

2011

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“…w(λλ ′ ) ≤ w(λ)w(λ ′ ) for all λ, λ ′ ∈ Λ. Then a Banach algebra A w (Λ) can be defined, which is a weighted analogue of the Banach algebra ℓ 1 (Λ) introduced by Hewitt and Zuckerman [16] (see [3]; cf. [25, p. 70] for the case of semigroups with involution).…”

Section: Weighted Semigroup Algebras and Their Spectramentioning

confidence: 99%

“…[4, Chapter V, §3], [30, Chapter III, Exercise 23]). Now the weighted semigroup algebras are obtained as follows ( [3]; cf. [25]): Proposition 1.…”

Section: Weighted Semigroup Algebras and Their Spectramentioning

confidence: 99%

Weighted inversion of general Dirichlet series

Glöckner

Lucht

2013

Trans. Amer. Math. Soc.

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Inversion theorems of Wiener type are essential tools in analysis and number theory. We derive a weighted version of an inversion theorem of Wiener type for general Dirichlet series from that of Edwards from 1957, and we outline an alternative proof based on the duality theory of convex cones and extension techniques for characters of semigroups. Variants and arithmetical applications are described, including the case of multidimensional weighted generalized Dirichlet series.

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Lattice Homomorphisms in Harmonic Analysis

Dales

Jeu

2019

Trends in Mathematics

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Let S be a non-empty, closed subspace of a locally compact group G that is a subsemigroup of G. Suppose that X, Y , and Z are Banach lattices that are vector sublattices of the order dual Cc(S, R) ∼ of the real-valued, continuous functions with compact support on S, and where Z is Dedekind complete. Suppose that * : X × Y → Z is a positive bilinear map such that supp (x * y) ⊆ supp x • supp y for all x ∈ X + and y ∈ Y + with compact support. We show that, under mild conditions, the canonically associated map from X into the vector lattice of regular operators from Y into Z is then a lattice homomorphism. Applications of this result are given in the context of convolutions, answering questions previously posed in the literature. As a preparation, we show that the order dual of the continuous, compactly supported functions on a closed subspace of a locally compact space can be canonically viewed as an order ideal of the order dual of the continuous, compactly supported functions on the larger space. As another preparation, we show that L p -spaces and Banach lattices of measures on a locally compact space can be embedded as vector sublattices of the order dual of the continuous, compactly supported functions on that space.

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Weighted convolution algebras on subsemigroups of the real line

Cited by 14 publications

References 26 publications

Multipliers of weighted semigroups and associated Beurling Banach algebras

Multipliers of weighted semigroups and associated Beurling Banach algebras

Weighted inversion of general Dirichlet series

Lattice Homomorphisms in Harmonic Analysis

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