Searcharxiv⌕ Search

arXiv subjects

Kazem Haghnejad Azar

Publications and source records attributed to Kazem Haghnejad Azar.

At least 37 records · Page 2Linked to original sources
arXiv2019

A generalization of order convergence

Let $E$ be a sublattice of a vector lattice $F$. $\left( x_α\right)\subseteq E$ is said to be $ F $-order convergent to a vector $ x $ (in symbols $ x_α\xrightarrow{Fo} x $), whenever there exists another net $ \left(y_α\right) $ in $F $ with the some index set satisfying $ y_α\downarrow 0 $ in $F$ and $ | x_α- x | \leq y_α$ for all indexes $ α$. If $F=E^{\sim\sim}$, this convergence is called $b$-order convergence and we write $ x_α\xrightarrow{bo} x$. In this manuscript, first we study some properties of $Fo$-convergence nets and we extend some results to the general case. In the second part, we introduce $b$-order continuous operators and we invistegate some properties of this new concept. An operator $T$ between two vector lattices $E$ and $F$ is said to be $b$-order continuous, if $ x_α\xrightarrow{bo} 0 $ in $E$ implies $ Tx_α\xrightarrow{bo} 0$ in $F$.

math.FA↗
arXiv2019

Extension of order bounded operators

Assume that a normed lattice $E$ is order dense majorizing of a vector lattice $E^t$. There is an extension norm $\Vert.\Vert_t$ for $E^t$ and we extend some lattice and topological properties of normed lattice $(E,\Vert.\Vert)$ to new normed lattice $(E^t,\Vert.\Vert_t$). For a Dedekind complete Banach lattice $F$, $T^t$ is an extension of $T$ from $E^t$ into $F$ whenever $T$ is an order bounded operator from $E$ into $F$. For each positive operator $T$, we have $\Vert T\Vert=\Vert T^t\Vert$ and we show that $T^t$ is a lattice homomorphism from $E^t$ into $F$ and moreover $T^t\in \mathcal{L}_n(E^t,F)$ whenever $0\leq T\in \mathcal{L}_n(E,F)$ and $T(x\wedge y)=Tx \wedge Ty$ for each $0\leq x,y\in E$. We also extend some lattice and topological properties of $T\in \mathcal{L}_b(E,F)$ to the extension operator $T^t\in \mathcal{L}_b(E^t,F)$.

math.FA↗
arXiv2019

A generalization of $b$-weakly compact operators

A. Bahramnezhad and K. Haghnejad Azar introduced the classes of $KB$-operators and $WKB$-operators, and they studied some of theirs properties. In the present paper, we give answer for an open problem from that paper, which two classifications of operators, $b$-weakly compact operators and $KB$-operators are different. A continuous operator $T$ from a normed vectoe lattice $E$ into a normed space $X$ is said to be $KB$-operator (respectively, $WKB$-operator) if $\{Tx_n\}_n$ has a norm (respectively, weak) convergent subsequence in $X$ for every positive increasing sequence $\{x_n\}_n$ in the closed unit ball $B_E$ of $E$. We investigate some other properties of $KB$-operators and its relationships with $b-$weakly compact operators.

math.FA↗
arXiv2019

Some notes on $b$-weakly compact operators

In this paper, we will study some properties of b-weakly compact operators and we will investigate their relationships to some variety of operators on the normed vector lattices. With some new conditions, we show that the modulus of an operator $T$ from Banach lattice $E$ into Dedekind complete Banach lattice $F$ exists and is $b$-weakly operator whenever $T$ is a $b$-weakly compact operator. We show that every Dunford-Pettis operator from a Banach lattice $E$ into a Banach space $X$ is b-weakly compact, and the converse holds whenever $E$ is an $AM$-space or the norm of $E^\prime$ is order continuous and $E$ has the Dunford-Pettis property. We also show that each order bounded operator from a Banach lattice into a $KB$-space admits a $b$-weakly compact modulus.

math.FA↗
arXiv2019

Order-to-topology continuous operators

An operator $T$ from vector lattice $E$ into vector topology $(F,τ)$ is said to be order-to-topology continuous whenever $x_α\xrightarrow{o}0$ implies $Tx_α\xrightarrowτ0$ for each $(x_α)_α\subset E$. The collection of all order-to-topology continuous operators will be denoted by $L_{oτ}(E,F)$. In this paper, we will study some properties of this new classification of operators. We will investigate the relationships between order-to-topology continuous operators and others classes of operators such as order continuous, order weakly compact and $b$-weakly compact operators.

math.FA↗
arXiv2017

Strongly order continuous operators on Riesz spaces

In this paper we introduce two new classes of operators that we call strongly order continuous and strongly $σ$-order continuous operators. An operator $T:E\rightarrow F$ between two Riesz spaces is said to be strongly order continuous (resp. strongly $σ$-order continuous), if $x _α\xrightarrow{uo}0$ (resp. $x _n \xrightarrow{uo}0$) in $E$ implies $Tx _α\xrightarrow{o}0$ (resp. $Tx _n \xrightarrow{o}0$) in $F$. We give some conditions under which order continuity will be equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all $so$-continuous linear functionals on a Riesz space $E$ is a band of $E^\sim$.

math.FA↗
arXiv2010

Topological centers of $n-$th dual of module actions

In this paper, we will study the topological centers of $n-th$ dual of Banach $A-module$ and we extend some propositions from Lau and Ülger into $n-th$ dual of Banach $A-modules$ where $n\geq 0$ is even number. Let $B$ be a Banach $A-bimodule$. By using some new conditions, we show that ${{Z}^\ell}_{A^{(n)}}(B^{(n)})=B^{(n)}$ and ${{Z}^\ell}_{B^{(n)}}(A^{(n)})=A^{(n)}$. We also have some conclusions in group algebras.

math.FA↗
arXiv2010

Arens Regularity of Tensor Products and Weak Amenability of Banach Algebras

In this note, we study the Arens regularity of projective tensor product $A\hat{\otimes}B$ whenever $A$ and $B$ are Arens regular. We establish some new conditions for showing that the Banach algebras $A$ and $B$ are Arens regular if and only if $A\hat{\otimes}B$ is Arens regular. We also introduce some new concepts as left-weak$^*$-weak convergence property [$Lw^*wc-$property] and right-weak$^*$-weak convergence property [$Rw^*wc-$property] and for Banach algebra $A$, suppose that $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property. Then if $A^{**}$ is weakly amenable, it follows that $A$ is weakly amenable. We also offer some results concerning the relation between these properties with some special derivation $D:A\rightarrow A^*$. We obtain some conclusions in the Arens regularity of Banach algebras.

math.FA↗
arXiv2010

The topological centers and factorization properties of module actions and $\ast-involution$ algebras

For Banach left and right module actions, we extend some propositions from Lau and $\ddot{U}lger$ into general situations and we establish the relationships between topological centers of module actions. We also introduce the new concepts as $Lw^*w$-property and $Rw^*w$-property for Banach $A-bimodule$ $B$ and we obtain some conclusions in the topological center of module actions and Arens regularity of Banach algebras. we also study some factorization properties of left module actions and we find some relations of them and topological centers of module actions. For Banach algebra $A$, we extend the definition of $\ast-involution$ algebra into Banach $A-bimodule$ $B$ with some results in the factorizations of $B^*$. We have some applications in group algebras.

math.FA↗
arXiv2010

The Arens Regularity and Weak Topological Center of Module Actions

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We define $\tilde{Z}_1(A^{**})$ as a weak topological center of $A^{**}$ with respect to first Arens product and find some relations between it and the topological center of $A^{**}$. We also extend this new definition into module actions and find relationship between weak topological center of module actions and reflexivity or Arens regularity of some Banach algebras, and we investigate some applications of this new definition in the weak amenability of some Banach algebras.

math.FA↗
arXiv2010

Factorization property and Arens regularity

In this paper, we study the Arens regularity properties of module actions and we extend some proposition from Baker, Dales, Lau and others into general situations. For Banach $A-bimodule$ $B$, let $Z_1(A^{**})$, ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$ be the topological centers of second dual of Banach algebra $A$, left module action $π_\ell:~A\times B\rightarrow B$ and right module action $π_r:~B\times A\rightarrow B$, respectively. We establish some relationships between them and factorization properties of $A^*$ and $B^*$. We search some necessary and sufficient conditions for factorization of $A^*$, $B$ and $B^*$ with some results in group algebras. We extend the definitions of the left and right multiplier for module actions.

math.FA↗
arXiv2010

Derivations And Cohomological Groups Of Banach Algebras

Let $B$ be a Banach $A-bimodule$ and let $n\geq 0$. We investigate the relationships between some cohomological groups of $A$, that is, if the topological center of the left module action $π_\ell:A\times B\rightarrow B$ of $A^{(2n)}$ on $B^{(2n)}$ is $B^{(2n)}$ and $H^1(A^{(2n+2)},B^{(2n+2)})=0$, then we have $H^1(A,B^{(2n)})=0$, and we find the relationships between cohomological groups such as $H^1(A,B^{(n+2)})$ and $H^1(A,B^{(n)})$, spacial $H^1(A,B^*)$ and $H^1(A,B^{(2n+1)})$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude that $H^1_{w^*}(L^\infty(G)^*,L^\infty(G)^{**})=0$. Let $G$ be an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^\infty(G),.)$ such that $Z^1(L^1(G),L^\infty(G))=\{L_{f}:~f\in L^\infty(G)\}.$ We also obtain some conclusions in the Arens regularity of module actions and weak amenability of Banach algebras. We introduce some new concepts as $left-weak^*-to-weak$ convergence property [$=Lw^*wc-$property] and $right-weak^*-to-weak$ convergence property [$=Rw^*wc-$property] with respect to $A$ and we show that if $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property and $A^{**}$ is weakly amenable, then $A$ is weakly amenable. We also show to relations between a derivation $D:A\rightarrow A^*$ and this new concepts.

math.FA↗
arXiv2010

Arens Regularity And Factorization Property

In this paper, we will study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$ be the topological centers of the left module action $π_\ell:~A\times B\rightarrow B$ and the right module action $π_r:~B\times A\rightarrow B$, respectively. In this paper, we will extend some problems from topological center of second dual of Banach algebra $A$, $Z_1(A^{**})$, into spaces ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$. We investigate some relationships between ${Z}_1({A^{**}})$ and topological centers of module actions. For an unital Banach $A-module$ $B$ we show that ${Z}^\ell_{A^{**}}(B^{**}){Z}_1({A^{**}})={Z}^\ell_{A^{**}}(B^{**})$ and as results in group algebras, for locally compact group $G$, we have ${Z}^\ell_{{L^1(G)}^{**}}(M(G)^{**})M(G)={Z}^\ell_{{L^1(G)}^{**}}(M(G)^{**})$ and ${Z}^\ell_{M(G)^{**}}({L^1(G)}^{**})M(G)={Z}^\ell_{M(G)^{**}}({L^1(G)}^{**})$. For Banach $A-bimodule$ $B$, if we assume that $B^*B^{**}\subseteq A^*$, then $~B^{**}{Z}_1(A^{**})\subseteq {Z}^\ell_{A^{**}}(B^{**})$ and moreover if $B$ is an unital as Banach $A-module$, then we conclude that $B^{**}{Z}_1({A^{**}})={Z}^\ell_{A^{**}}(B^{**})$. Let ${Z}^\ell_{A^{**}}(B^{**})A\subseteq B$ and suppose that $B$ is $WSC$, so we conclude that ${Z}^\ell_{A^{**}}(B^{**})=B$. If $\overline{B^{*}A}\neq B^*$ and $ B^{**}$ has a left unit $A^{**}-module$, then $Z^\ell_{B^{**}}(A^{**})\neq A^{**}$. We will also establish some relationships of Arens regularity of Banach algebras $A$, $B$ and Arens regularity of projective tensor product $A\hat{\otimes}B$.

math.FA↗
arXiv2010

Arens regularity and weak topological center of module actions

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We define $\tilde{Z}_1(A^{**})$ as a weak topological center of $A^{**}$ with respect to first Arens product and we will find some relations between this concept and the topological center of $A^{**}$. We also extend this new definition into the module actions and find relationship between weak topological center of module actions and reflexivity or Arens regularity of some Banach algebras, and we investigate some applications of this new definition in the weak amenability of some Banach algebras.

math.FA↗
arXiv2010

Factorization properties and topological centers of left module actions

For a Banach left module action, we will extend some propositions from Lau and $\ddot{U}$lger and others into general situations and we establish the relationships between topological centers of the left module action with the multiplier and factorization properties of left module actions. We have some applications in the dual groups.

math.FA↗
arXiv2010

A generalization of the weak amenability of some Banach algebra

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We show that by some new conditions, $A$ is weakly amenable whenever $A^{**}$ is weakly amenable. We will study this problem under generalization, that is, if $(n+2)-th$ dual of $A$, $A^{(n+2)}$, is $T-S-$weakly amenable, then $A^{(n)}$ is $T-S-$weakly amenable where $T$ and $S$ are continuous linear mappings from $A^{(n)}$ into $A^{(n)}$.

math.FA↗