Convolution operators on the dual of hypergroup algebras

Ali Ghaffari

Displaying similar documents to “Convolution operators on the dual of hypergroup algebras”

An $L^{p}$ -version of a theorem of D.A. Raikov

Gero Fendler (1985)

Annales de l'institut Fourier

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Let $G$ be a locally compact group, for $p \in (1, \infty)$ let $P f_{p} (G)$ denote the closure of $L^{1} (G)$ in the convolution operators on $L^{p} (G)$ . Denote $W_{p} (G)$ the dual of $P f_{p} (G)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space $A_{p} (G)$ . It is shown that on the unit sphere of $W_{p} (G)$ the $σ (W_{p}, P f_{p})$ topology and the strong $A_{p}$ -multiplier topology coincide.

Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1

Leonardo Colzani, Peter Sjögren (1999)

Studia Mathematica

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We study convolution operators bounded on the non-normable Lorentz spaces $L^{1, q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1, q}$ . In particular, when the positions of the atoms of a discrete measure are linearly independent over...

Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska (2015)

Studia Mathematica

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Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for...

On the product formula on noncompact Grassmannians

Piotr Graczyk, Patrice Sawyer (2013)

Colloquium Mathematicae

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We study the absolute continuity of the convolution $δ_{e^{X}}^{♮} * δ_{e^{Y}}^{♮}$ of two orbital measures on the symmetric space SO₀(p,q)/SO(p)×SO(q), q > p. We prove sharp conditions on X,Y ∈ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for SO₀(p,q)/SO(p)×SO(q) also serves for the spaces SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q), q > p. We moreover apply our results to...

The $V_{a}$ -deformation of the classical convolution

Anna Dorota Krystek (2007)

Banach Center Publications

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We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $μ *_{T} ν = T^{- 1} (T μ * T ν)$ . We deal with the $V_{a}$ -deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_{a}$ -deformed classical convolution and give the orthogonal...

The type set for some measures on $ℝ^{2 n}$ with $n$ -dimensional support

E. Ferreyra, T. Godoy, Marta Urciuolo (2002)

Czechoslovak Mathematical Journal

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Let $ϕ_{1}, \dots, ϕ_{n}$ be real homogeneous functions in $C^{\infty} (ℝ^{n} - {0})$ of degree $k \geq 2$ , let $ϕ (x) = (ϕ_{1} (x), \dots, ϕ_{n} (x))$ and let $μ$ be the Borel measure on $ℝ^{2 n}$ given by $μ (E) = \int_{ℝ^{n}} χ_{E} (x, ϕ (x)) {| x |}^{γ - n} d x$ where $d x$ denotes the Lebesgue measure on $ℝ^{n}$ and $γ > 0$ . Let $T_{μ}$ be the convolution operator $T_{μ} f (x) = (μ * f) (x)$ and let $E_{μ} = {(1 / p, 1 / q) ∥ T_{μ} ∥_{p, q} < \infty, 1 \leq p, q \leq \infty} .$ Assume that, for $x \neq 0$ , the following two conditions hold: $det (d^{2} ϕ (x) h)$ vanishes only at $h = 0$ and $det (d ϕ (x)) \neq 0$ . In this paper we show that if $γ > n (k + 1) / 3$ then $E_{μ}$ is the empty set and if $γ \leq n (k + 1) / 3$ then $E_{μ}$ is the closed segment with endpoints $D = (1 - \frac{γ}{n (k + 1)}, 1 - \frac{2 γ}{n (k + 1)})$ and $D^{'} = (\frac{2 γ}{n (1 + k)}, \frac{γ}{n (1 + k)})$ . Also, we give some examples.

A property for locally convex *-algebras related to property (T) and character amenability

Xiao Chen, Anthony To-Ming Lau, Chi-Keung Ng (2015)

Studia Mathematica

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For a locally convex *-algebra A equipped with a fixed continuous *-character ε (which is roughly speaking a generalized F*-algebra), we define a cohomological property, called property (FH), which is similar to character amenability. Let $C_{c} (G)$ be the space of continuous functions with compact support on a second countable locally compact group G equipped with the convolution *-algebra structure and a certain inductive topology. We show that $(C_{c} (G), ε_{G})$ has property (FH) if and only if G has property...

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

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For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^{p} ()$ to $L^{q} ()$ . We also give a condition on p which is necessary if this operator maps $L^{p} ()$ into L²().

Compact convolution operators between $L_{p} (G)$ -spaces

G. Crombez, W. Govaerts (1978)

Colloquium Mathematicae

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Operators which commute with convolutions on subspaces of $L_{\infty} (G)$

Anthony To-Ming Lau (1978)

Colloquium Mathematicae

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Sequential closures of $σ$ -subalgebras for a vector measure

Werner J. Ricker (1996)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ be a locally convex space, $m : Σ \to X$ be a vector measure defined on a $σ$ -algebra $Σ$ , and $L^{1} (m)$ be the associated (locally convex) space of $m$ -integrable functions. Let $Σ (m)$ denote ${χ_{_{E}}; E \in Σ}$ , equipped with the relative topology from $L^{1} (m)$ . For a subalgebra $𝒜 \subseteq Σ$ , let $𝒜_{σ}$ denote the generated $σ$ -algebra and ${\bar{𝒜}}_{s}$ denote the closure of $χ (𝒜) = {χ_{_{E}}; E \in 𝒜}$ in $L^{1} (m)$ . Sets of the form ${\bar{𝒜}}_{s}$ arise in criteria determining separability of $L^{1} (m)$ ; see [6]. We consider some natural questions concerning ${\bar{𝒜}}_{s}$ and, in particular, its relation to $χ (𝒜_{σ})$ . It is shown that...