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Ali Rejali

Publications and source records attributed to Ali Rejali.

At least 19 recordsLinked to original sources
arXiv2023

Homological propeties of Bimeasure algebras and their BSE properties

Let $G$ and $H$ be locally compact groups. $BM(G, H)$ denoted the Banach algebra of bounded bilinear forms on $C_{0}(G)\times C_{0}(H)$.In this paper, the homological properties of Bimeasure algebras are investigated. It is found and approved that the Bimeasure algebras $BM(G, H)$ is amenable if and only if $G$ and $H$ are discrete. The correlation between the weak amenability of $BM(G, H)$ and $M(G\times H)$ is assessed. It is found and approved that the biprojectivity of the bimeasure algebra $BM(G, H)$ is equivalent to the finiteness of $G$ and $H$. Furthermore, we show that the bimeasure group algebra $BM_{a}(G, H)$ is a BSE algebra. It will be concluded that $BM(G, H)$ is a BSE- algebra if and only if $G$ and $H$ are discrete groups.

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arXiv2023

Weighted group algebras

Let $G$ be a locally compact Abelian group, and $w: G\to (0, \infty)$ be a Borel measurable weighted function. In this paper, the algebraic and topological properties of group algebra are studied and assessed. We show that the weighted group algebra $L^{1}(G, w)$ is regular if and only if $w$ is a nonquasianalytic weight function. Also $L^{1}(G, w)$ is Tauberian, for any Borel measurable weight function $w$ on the group $G$.

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arXiv2022

BSE-property of tensor product Banach algebras

Let $A$ and $B$ be commutative semisimple Banach algebras. In this paper, the $BSE$ and $BED$- property of tensor Banach algebra $A{\otimes}_{\gamma} B$ with respect to the Banach algebras $A$ and $B$ are assesed. In particular, $BSE$ and $BED$- structure of vector-valued group algebra $L_{1}(G, A)$ and $C^{*}$- algebra $C_{0}(X, A)$ characterized.

math.FA↗
arXiv2022

$L_{1}$- Properties of vector-valued Banach algebras

Let $G$ be a locally compact group and $A$ be a commutative semisimple Banach algebra over the scalar field $\mathbb{C}$. The correlation between different types of $BSE$- Banach algebras $A$, and the Banach algebras $L^{1}(G, A)$ are assessed. It is found and approved that $M(G, A) = L^{1}(G, A)$ if and only if $G$ is discrete. Furthermore, some properties of vector-valued measure algebras on groups are given, so that $M(G, A)$ is a convolution measure algebra.

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arXiv2022

$C^{*}$- properties of vector-valued Banach algebras

Let $X$ be a locally compact Hausdorff space, and $A$ be a commutative semisimple Banach algebra over the scalar field $\mathbb{C}$. The correlation between different types of BSE- Banach algebras $A$, and the Banach algebra $C_{0}(X, A)$ are assessed. It is found and approved that $C_{0}(X, A)$ is a $C^{*}$- algebra if and only if $A$ is so. Furthermore, $C_{b}(X, A)= C_{0}(X, A)$ if and only if $X$ is compact.

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arXiv2022

An Approach to Development: Turning Education from a Service Duty to a Productive Tool

Recent economic developments of countries like Japan, Korea, and Singapore, as a result of improvement in the quality of their education, show that having a high-quality education may lead to economic growth. In this article, using some statistical methods, we argue that high quality education can change the economy towards higher growth. Therefore, for the development of the country, one should think about how to improve its education. One of the effective ways to improve the quality of education is to increase the efficiency of teachers and attract talented people to teaching positions. Research shows that raising teachers' salaries, along with a proper quality improvement program, can help facilitate this process.

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arXiv2022

BSE-properties of Vector-valued group algebras

Let $ \mathcal{A} $ be a commutative and semisimple Banach algebra with identity norm one and $ G $ be an abelian locally compact Hausdorff group. In this paper, we study BSE-Property for $L^1(G,\mathcal A)$ and show that $L^1(G,\mathcal A)$ is a BSE algebra if and only if $\mathcal A$ is so.

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arXiv2022

The BSE-property for vector-valued $L^p-$algebras

Let $\mathcal A$ be a separable Banach algebra, $G$ be a locally compact Hausdorff group and $1< p<\infty$. In this paper, we first provide a necessary and sufficient condition, for which $L^p(G,\mathcal A)$ is a Banach algebra, under convolution product. Then we characterize the character space of $L^p(G,\mathcal A)$, in the case where $\mathcal A$ is commutative and $G$ is abelian. Moreover, we investigate the BSE-property for $L^p(G,\mathcal A)$ and prove that $L^p(G,\mathcal A)$ is a BSE-algebra if and only if $\mathcal A$ is a BSE-algebra and $G$ is finite. Finally, we study the BSE-norm property for $L^p(G,\mathcal A)$ and show that if $L^p(G,\mathcal A)$ is a BSE-norm algebra then $\mathcal A$ is so. We prove the converse of this statement for the case where $G$ is finite and $\mathcal A$ is unital.

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arXiv2022

BSE-Properties of Second Dual of Banach Algebras

Let $A$ be a commutative semisimple Arens regular unital Banach algebra. The correlation between the BSE-property of the Banach algebra $A$ and its second duals are assessed. It is found and approved that if $A$ is a BSE-algebra, then so is $A^{**}$. The opposite correlation will hold in certain conditions. The correlation of the BSE-norm property of the Banach algebra and its second dual are assessed and examined. It is revealed that, if $A$ is a commutative Arens regular unital Banach algebra where $A^{**}$ is semisimple, then, $A$ and $A^{**}$ are BSE-norm algebra.

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arXiv2022

A new class of ideal Connes amenability

In this paper, we introduce a new notion of amenability, $σ-$Connes ideal, say, for a large class of dual Banach algebras. We extend the concept of ideal Connes amenability and study their properties. Let $σ$ be a $weak^{*}$-continuous endomorphism on a dual Banach algebra $\mathcal{A}$ with dense range. Then the concept of ideal Connes-amenability and $σ-$ ideally Connes amenability are the same. We gave some general results and hereditary properties with some examples for this new notion of amenability.

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arXiv2022

On the Arens regularity of Frechet algebras and their biduals

In This paper, we study the concept of weakly almost periodic functions on Frechet algebras. For a Frechet algebra A, we show that WAP(A)=wap(A). We also show that A** is Arens regular if and only if both A and WAP(A)* are Arens regular. Finally, for a sequence of Frechet algebras (An), we prove that the Frechet algebra \ell^1-\prod An is Arens regular if and only if each An is Arens regular.

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arXiv2022

$C^{\ast}-$algebra structure on vector valued-Banach algebras

Let $A$ be a commutative semisimple Banach algebra, $X$ be a locally compact Hausdorff topological space and $G$ be a locally compact topological group. In this paper, we investigate several properties of vector valued Banach algebras $C_0(X,A)$, $L^p(G,A)$, $l^p(X,A)$ and $l^{\infty}(X,A)$. We prove that these algebras are isomorphic with a $C^*-$algebra if and only if $A$ is so.

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arXiv2021

Regularity and amenability of weighted Banach algebras and their second dual on locally compact groups

Let $ω$ be a weight function on a locally compact group G mand let $ M_* (G, ω) $ be the subspace of $ M(G , ω)^* $ consisting of all functionals that vanish at infinity. In this paper, we first investigate the Arens regularity of $ M_* (G, ω)^* $ and show that $ M_* (G, ω)^* $ is Arnes regular if and only if G is finite or $ ω$ is zero cluster. This result is an answer to the question posed and it improves some well-known results. We also give necessary and sufficient criteria for the weight function spaces $ Wap(G , 1/ ω) $ and $ Wap(G , 1/ ω) $ to be equal to $ C_b (G , 1/ ω) $. We prove that for non-compact group G, the Banach algebra $ M_* (G, ω)^* $ is Arnes regular if and only if $ Wap(G , 1/ ω) = C_b (G , 1/ ω) $. We then investigate amenability of $ M_* (G, ω)^* $ and prove that $ M_* (G, ω)^* $ is amenable and Arnes regular if and only if G is finite.

math.FA↗
arXiv2021

Urysohn and Tietze extensions of Lipschitz functions

Let (X,d) be a metric space and $ α> 0 $. In this paper, we study extensions of some complex-valued Lipschitz functions, from some special subset $ X_0 $ to X. These extensions are with no-increasing Lipschitz number or the smallest Lipschitz number. Moreover, we show that under some conditions, Tietze extension theorem can be generalized for Lipschitz functions and call it Tietze-Lipschitz extension. Furthermore, we generalize Urysohn-lemma for Lipschitz functions. In fact we present a necessary and sufficient condition for that Lipschitz functions separate subsets of X.

math.FA↗
arXiv2021

The BSE property for vector-valued Frechet Lipschitz algebras

Let $( X,d )$ be a metric space with at least two elements and $( A , p_l )$ be a commutative semisimple Frechet algebra over the scalar field of complex numbers. The correlation between the BSE-property of the Frechet algebra $( A , p_l )$ and $\Lip_{d}( X , A )$ is assessed. It is found and approved that if $\Lip_{d}( X , A )$ is a BSE-Frechet algebra, then so is $A$. The opposite correlation will hold if $( A , p_l )$ is unital.

math.FA↗
arXiv2021

Ideal and weak Amenability of Fréchet locally $C^*$-algebra

The notion of Fréchet locally $C^*$-algebra generalizes the notion of $C^*$-algebra. In this paper, we first present some definitions and basic facts about locally $C^*$-algebra, and then we introduce and study the notion of ideal and weak amenability for these algebras. Also, we show that every Fréchet locally $C^*$-algebra is ideally amenable.

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arXiv2021

$n$-ideal and $n$-weak amenability of Frechet algebras

In this paper, we introduce and study some notations of amenability such as $n$-ideal amenability and $n$-weak amenability for Frechet algebra and we examine how these concepts in Banach algebra can be generalized and defined for Frechet algebra. We will also examine some inherited properties of $n$-ideal and $n$-weak amenable Frechet algebra and determine the relations between $m$ and $n$-weak amenability and $m$ and $n$-ideal amenability for Frechet algebra. Also, the relation of these new concepts of amenability of a Frechet algebra and its unitization is investigated.

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