Kazem Haghnejad Azar

University of Mohaghegh Ardabili | UMA · Department of Mathematics and Applications

We introduce the class of unbounded M-weakly compact operators and the class of unbounded L-weakly compact operators. We investigate some properties for this new classification of operators, and we study the relation between them and M-weakly compact and L-weakly compact operators. We also present an operator characterization of Banach lattices wit...

Let E be a sublattice of a vector lattice F. In this paper, we will introduce and study some properties of this new class of operators as F-order-norm continuousoperators and its relationships with some known classifications of operators. We also define the new class of operators that named order weakly compact operators. A continuous operator T :...

In this paper, we introduce and study new classifications of operators as name unbounded L-weakly and generalized M-weakly compact operators. We study some lattice properties of these classifications of operators and we investigate their relationships with others known operators such as M-weakly compact operators. We show that each M-weakly compact...

We investigate on the analysing of systems including machines and living organisms. We investigate to understand that how living organisms or machines react to their internal and external events and how do we explain the performance of a function of a system. We need to explain how a system central processes can bring information together and use i...

Let $E$ be a sublattice of a vector lattice $F$. A continuous operator $T$ from the vector lattice $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous whenever $x_\alpha\xrightarrow{Fo}0$ implies $Tx_\alpha\xrightarrow{\Vert.\Vert}0$ for each $(x_\alpha)_\alpha\subseteq E$. Our mean from the convergence $ x_\alpha\stack...

We investigate on the analysing of systems including machines and living organisms. We investigate to understand that how living organisms or machines react to their internal and external events and how do we explain the performance of a function of a system. Here we will discuss the consciousness of a system and we will study the comparisons betwe...

A functional f on a Banach lattice E is un-continuous, if \(x_\alpha \xrightarrow {un}0\) implies \(f(x_\alpha )\rightarrow 0\) for each norm bounded net \((x_\alpha )\subseteq E\). We denote the vector space of all un-continuous functionals on E by \({{E}^{\diamond }}\) and we call it un-dual of Banach lattice E. In this paper, we study the un-dua...

Let $E$ be a sublattice of a vector lattice $F$.$\left( x_\alpha \right)\subseteq E$ is said to be $ F $-order convergent to a vector $ x $ (in symbols $ x_\alpha \xrightarrow{Fo} x $), whenever there exists another net $ \left(y_\alpha\right) $ in $F $ with the some index set satisfying $ y_\alpha\downarrow 0 $ in $F$ and $ \vert x_\alpha - x \ver...

Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and $f$ is Aron-Berener regular when all natural extensions are equal. In this manuscript, we have some results on the Aron-Berner regular maps. We investigate the relation between Arens regulari...

A continuous operator $T$ between two Banach lattices $E$ and $F$ is called almost order-weakly compact, whenever for each almost order bounded subset $A$ of $E$, $T(A)$ is a relatively weakly compact subset of $F$. In Theorem 4, we show that the positive operator $T$ from $E$ into Dedekind complete $F$ is almost order-weakly compact if and only if...

Let $X$ be an ordered vector space. The net $\{x_\alpha\}\subseteq X$ is semi unbounded order convergent to $x$ (in symbol $x_\alpha\xrightarrow{suo}x$), if there is a net $\{y_\beta\}$, possibly over a different index set, such that $y_\beta \downarrow 0$ and for every $\beta$ there exists $\alpha_0$ such that $\{\{\pm(x_\alpha - x)\}^u,y\}^l\subs...

We introduce the class of unbounded $M$-weakly operators and the class of unbounded $L$-weakly compact operators. We investigate some properties for these new classification of operators and we study relation between them and $M$-weakly compact and $L$-weakly compact operators. We also present an operator characterization of Banach lattices with or...

An operator T from vector lattice E into topological vector space (F,τ) is said to be order-to-topology continuous whenever xα→o0 implies Txα→τ0 for each (xα)α⊂E. The collection of all order-to-topology continuous operators will be denoted by Loτ(E,F). In this paper, we will study some properties of this new class of operators. We will investigate...

Let f : X × Y × Z −→ W be a bounded tri-linear map on normed spaces. We say that f is close-to-regular when f t * * * * s = f s * * * * t and we say that f is Aron-Berner regular when all natural extensions are equal. In this manuscript, we give a simple criterion for the close-to-regularity of tri-linear maps.

In this manuscript, we will study o~-convergence in (partially) ordered vector spaces and we will study a kind of convergence in a vector space V. A vector space V is called semi-order vector space (in short semi-order space), if there exist an ordered vector space W and an operator T from V into W. In this way, we say that V is semi-order space wi...

In this paper, we study approximate identity properties, some propositions from Baker, Dales, Lau in general situations and we establish some relationships between the topological centers of module actions and factorization properties with some results in group algebras. We consider under which sufficient and necessary conditions the Banach algebra...

Let X, Y, Z and W be normed spaces and f : X × Y × Z −→ W be a bounded tri-linear mapping. In this manuscript, we introduce the topological centers of bounded tri-linear mapping and we invistagate their properties. We study the relationships between weakly compactenss of bounded linear mappings and regularity of bounded tri-linear mappings. We exte...

A continuous operator T between two normed vector lattices E and F is called unbounded order-norm continuous whenever x? uo? 0 implies ||Tx?|| ? 0, for each norm bounded net (x?)? ? E. Let E and F be two Banach lattices. A continuous operator T : E ? F is called unbounded norm continuous, if for each norm bounded net (x?)? ? E, x? un? 0 implies Tx?...

Let $B$ be a Banach $A-bimodule$. We introduce the weak topological centers of left module action and we show it by $\tilde{{Z}}^\ell_{B^{**}}(A^{**})$. For a compact group, we show that $L^1(G)=\tilde{Z}_{M(G)^{**}}^\ell(L^1(G)^{**})$ and on the other hand we have $\tilde{Z}_1^\ell{(c_0^{**})}\neq c_0^{**}$. Thus the weak topological centers are d...

در این مقاله به خواص آرنز منظم نگاشت دو خطی کران­دار می­پردازیم و نشان می­دهیم که نگاشت دو خطی کران­دار آرنز منظم است اگر و تنها اگر نگاشت خطی با ضابطۀ ضعیف فشرده باشد. سپس قضیه‌ای را اثبات می­کنیم که ویژگی ضعیف فشردگی نگاشت دو خطی کران­دار و آرنز منظم را به یک‌دیگر مرتبط می­سازد. هم‌چنین به بررسی آرنز منظم و خاصیت ضعیف فشردگی نگاشت­های خطی کران­دا...

Introduction Let , and be Banach spaces and be a bilinear mapping. In 1951 Arens found two extension for as and from into . The mapping is the unique extension of such that from into is continuous for every , but the mapping is not in general continuous from into unless . Thus for all the mapping is continuous if and only if is Arens regular. Regar...

In this paper, we study some cohomlogical properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n,B^m)$ and $H_{w^*}^1(A^n,B^m)$, where $0\leq m,n\leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D...

In this paper, we study $un$-dual (in symbol, $\ud{E}$) of Banach lattice $E$ and compare it with topological dual $E^*$. If $E^*$ has order continuous norm, then $E^* = \ud{E}$. We introduce and study weakly unbounded norm topology ($wun$-topology) on Banach lattices and compare it with weak topology and $uaw$-topology. In the final, we introduce...

In this manuscript, we will study both $\tilde{o}$-convergence in (partially) ordered vector spaces and a kind of convergence in a vector space $V$. A vector space $V$ is called semi-order vector space (in short semi-order space), if there exist an ordered vector space $W$ and an operator $T$ from $V$ into $W$. In this way, we say that $V$ is semi-...

We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in \(E^{\sim \sim }\) and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if \(T_A\) is strongly order bounded. We show that the class of strongly order bounded o...

The main aim of this paper is studying the family \(W_b(E,F)\) of b-weakly compact operators between two Banach lattices. For an order dense sublattice G of a vector lattice E, if \(T:G\rightarrow F\) is a b-weakly compact operator between two Banach lattices, then \(T\in W_b(E,F)\) whenever the norm of E is order continuous and \(T:E\rightarrow F\...

Let $X,Y,Z$ and $W$ be normed spaces and $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear mapping. In this Article, we define the topological centers for bounded tri-linear mapping and we invistagate thier properties. We study the relationships between weakly compactenss of bounded linear mappings and regularity of bounded tri-linear...

Let $E$ be a sublattice of a vector lattice $F$. A net $\{ x_\alpha \}_{\alpha \in \mathcal{A}}\subseteq E$ is said to be $ F $-order convergent to a vector $ x \in E$ (in symbols $ x_\alpha \xrightarrow{Fo} x $ in $E$), whenever there exists a net $ \{y_\beta\}_{\beta \in \mathcal{B}} $ in $F $ satisfying $ y_\beta\downarrow 0 $ in $F$ and for eac...

In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if $f:X\times Y\times Z\longrightarrow W $ is a bounded tri-linear mapping and $h:W\longrightarrow S$ is a bounded linear mapping, then $f$ is regular if and only if $hof$ is regular. We also shall give some necessary and sufficient conditions such th...

Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and we say that $f$ is completely regular when all natural extensions are equal. In this manuscript, we have some results on the close-to-regular maps and investigate the close-to-regularity of t...

Let $\mathfrak{A}$ be a Banach algebra, and $\mathcal{X}$ a Banach $\mathfrak{A}$-bimodule. A bounded linear mapping $\mathcal{D}:\mathfrak{A}\rightarrow \mathcal{X}$ is approximately semi-inner derivation if there eixist nets $(\xi_{\alpha})_{\alpha}$ and $(\mu_{\alpha})_{\alpha}$ in $\mathcal{X}$ suh that, for each $a\in\mathfrak{A}$, $\mathcal{D...

In this paper, we will study on some topologies induced by order convergences in a vector lattice. We will investigate the relationships of them.

Let $E$ be a sublattice of a vector lattice $F$. $\left( x_\alpha \right)\subseteq E$ is said to be $ F $-order convergent to a vector $ x $ (in symbols $ x_\alpha \xrightarrow{Fo} x $), whenever there exists another net $ \left(y_\alpha\right) $ in $F $ with the some index set satisfying $ y_\alpha\downarrow 0 $ in $F$ and $ | x_\alpha - x | \leq...

An operator $T $ from a vector lattice $E$ into a normed lattice $F$ is called unbounded $\sigma$-order-to-norm continuous whenever $x_{n}\xrightarrow{uo}0$ implies $\| Tx_{n}\|\rightarrow 0$, for each sequence $(x_{n})_n\subseteq E$. For a net $(x_{\alpha})_{\alpha}\subseteq E$, if $x_{\alpha}\xrightarrow{un}0$ implies $Tx_{\alpha}\xrightarrow{un}...

In page 838, line 7, we should write: For sequences in a Dedekind complete Riesz space E, if and only if there exists a sequence such that and for each (see [1, Page 17–18]).

Let $\mathcal{A}$ be a Banach algebra, and $\mathcal{X}$ be a Banach $\mathcal{A}$-bimodule. A derivation $\mathcal{D}:\mathcal{A}\rightarrow \mathcal{X}$ from Banach algebra into Banach space is called semi-inner if there are $\eta , \xi \in \mathcal{X}$ such that $$ \mathcal{D}(a)=a.\eta-\xi.a=\delta_{\eta,\xi}(a), \;\;\;\;\; (a\in \mathcal{A}).$...

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 $T$ fro...

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 w...

An operator $T$ from vector lattice $E$ into vector topology $(F,\tau)$ is said to be order-to-topology continuous whenever $x_\alpha\xrightarrow{o}0$ implies $Tx_\alpha\xrightarrow{\tau}0$ for each $(x_\alpha)_\alpha\subset E$. The collection of all order-to-topology continuous operators will be denoted by $L_{o\tau}(E,F)$. In this paper, we will...

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 $T$ fro...

In this paper, we introduce b-order Dunford-Pettis operators, that is, an operator T from a normed Riesz space E into a Banach space X is called b-order Dunford-Pettis if T carries each b-order bounded subset of E into a Dunford-Pettis subset of X, and we investigate its relationship with order Dunford-Pettis operators. We also introduce the b-AM-c...

In this paper, using the concept of unbounded order convergence in Riesz spaces, we define new classes of operators, named unbounded order continuous (uo-continuous, for short) and boundedly unbounded order continuous operators. We give some conditions under which uo-continuity will be equivalent to order continuity of some operators on Riesz space...

Aqzzouz, Moussa and Hmichane proved that an operator T from a Banach lattice E into a Banach space X is b-weakly compact if and only if {Txn}n is norm convergent for every positive increasing sequence {xn}n of the closed unit ball BE of E. In the present paper, we introduce and study new classes of operators that we call KB-operators and WKB-operat...

In this paper we introduce two new classes of operators that we call strongly order continuous and strongly $\sigma$-order continuous operators. An operator $T:E\rightarrow F$ between two Riesz spaces is said to be strongly order continuous (resp. strongly $\sigma$-order continuous), if $x _\alpha \xrightarrow{uo}0$ (resp. $x _n \xrightarrow{uo}0$)...

Let (T1,..,Tn) be n-tuple in (Um(A))ⁿ. We investigate formula for joint (Harte) spectrum of (T1,..,Tn) with respect to upper triangular matrices algebra Um(A) and obtain condition such that joint spectrum of the n-tuple in (Um(A))n equals with respect to Um(A) and Mm(A).

Some new results concerning Arens regularity of Banach algebras are presented. n-weak amenability of module extensions of Banach algebras and w∗-continuous derivations on Banach algebras are studied.

In this paper, we extend some propositions of Banach algebras into module actions and establish the relationships between topological centers of module actions. We introduce some new concepts as Lw*w-property and Rw*w-property for Banach modules and obtain some conclusions in the topological center of module actions and Arens regularity of Banach a...

Let A and B be Banach algebras and T: B → A be a continuous homomorphism. n-weak amenability of the Banach algebra A × T B (defined in Bade, W. G., Curtis, P. C., Dales, H. G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc., 55(2), 359–377 (1987)) is studied. The new version of a Banach algebra defined...

Assume that A, B are Banach algebras and that m: A×B → B, m ′: A × A → B are bounded bilinear mappings. We study the relationships between Arens regularity of m, m ′ and the Banach algebras A, B. For a Banach A-bimodule B, we show that B factors with respect to A if and only if B ** is unital as an A ** -module. Let Ze′′ (B **) = B ** where e ′′ is...

In this paper, we extend some problems of Arens regularity and factorizations properties of Banach algebras for general structures and we establish the relationships between topological centers and factorization properties of left module actions with some conclusions in the Arens regularity of Banach algebras. To a Banach algebra A, we extend the d...

We introduce the new concepts LW * W-property and RW * W-property for a Banach algebra A. Under certain conditions, we show that, if A has the LW * W-property and the RW * W-property, then A is Arens regular. We also offer some applications of these new concepts in group algebras.

We extend the notion of Arens regularity and module Arens regularity of Banach algebras to Arens regularity of module actions. We also investigate the more general notion of Arens regularity for bilinear maps. Finally, we find necessary and sufficient conditions for module Arens regularity of the semigroup algebra of an inverse semigroup.

We introduce some new concepts as left-weak * -to-weak convergence property [Lw * wc-property] and right-weak * -weak convergence property [Rw * wc-property] for a Banach algebra A. Suppose that A * and A ** , respectively, have the Rw * wc-property and the Lw * wc-property, then if A ** is weakly amenable, it follows that A is weakly amenable. Let...

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...

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 \"{U}lger 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...

In this paper, we extend some problems from Arens regularity and module Arens regularity of Banach algebras to module actions.

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...

For Banach left and right module actions, we will establish the relationships between topological centers of module actions with some result in the 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 convergenc...

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 topo...

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 a...

In this paper, first we 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 $\pi_\ell:~A\times B\rightarrow B$ and the right module action $\pi_r:~B\times A\rightarrow B$, respectively. We i...

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 $\pi_\ell:A\times B\rightarrow B$ 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...

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 $\pi_\ell:~A\times B\rightarrow B$ and the right module action $\pi_r:~B\times A\rightarrow B$, respectively. In th...

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...

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.

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...

In this article, for Banach left and right module actions, we will 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 investigate t...

Assume that $A$, $B$ are Banach algebras and $m:A\times B\to B$, $m^\prime:A\times A\to B$ are bounded bilinear mappings. We will study the relation between Arens regularities of $m$, $m^\prime$ and the Banach algebras $A$, $B$. For Banach $A-bimodule$ $B$, we show that $B$ factors with respect to $A$ if and only if $B^{**}$ is an unital $A^{**}-mo...

In this article, for Banach left and right module actions, we will 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 investigate t...

In this note for a topological group $G$, we introduce a bounded subset of $G$ and we find some relationships of this definition with other topological properties of $G$.

Let $A$ be a Banach algebra with the second dual $A^{**}$. If $A$ has a bounded approximate identity $(=BAI)$, then $A^{**}$ is unital if and only if $A^{**}$ has a $weak^* bounded approximate $$identity(=W^*BAI)$. If $A$ is Arens regular and $A$ \noindent has a BAI, then $A^*$ factors on both sides. In this paper we introduce new concepts $LW^*W$...

Let A be Banach algebra and let A * ,A ** denote its first and second dual space, respectively. We show that if Z 1 and Z 2 are the topological center of A ** with respect to first and second Arens product, then we have Z 1 A⊆Z 1 and AZ 2 ⊆Z 2 . In particular, under some conditions, the Banach algebra A is left strongly Arens irregular.

We introduce some new concepts of topological spaces which say separable − α topological space and O-topological group, − α first axiom, − α second axiom, and we find some relations between them with some applications in normed spaces.