arXiv subjects

M. J. Mehdipour

Publications and source records attributed to M. J. Mehdipour.

18 recordsLinked to original sources

arXiv2025

Jordan Left $α$-centralizers on Algebras with Applications to Group Algebras

We prove that every Jordan left $α$-centralizer from an algebra $A$ with a right identity into an arbitrary algebra $B$ is a left $α$-centralizer. This implies all Jordan homomorphisms between such algebras are homomorphisms. We extend this result to continuous Jordan left $α$-centralizers when $A$ has a bounded left approximate identity. For the group algebra $L^1(G)$, we characterize weakly compact Jordan left $α$-centralizers when $α$ is continuous and surjective, showing $L^1(G)$ admits a weakly compact epimorphism if and only if $G$ is finite. Consequently, the existence of a non-zero $α$-derivation on $L^1(G)$ is equivalent to $G$ being compact and non-abelian.

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arXiv2024

Jordan left $α$-centralizer on certain algebras

In this paper, we investigate Jordan left $α$-centralizer on algebras. We show that every Jordan left $α$-centralizer on an algebra with a right identity is a left $α$-centralizer. We also investigate this result for Banach algebras with a bounded approximate identity. Finally, we study Jordan left $α$-centralizer on group algebra $L^1(G)$.

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arXiv2024

On $θ$-centralizing $θ$-generalized derivations on convolution algebras

Let $θ$ be an isomorphism on $L_0^{\infty} (w)^*$. In this paper, we investigate $θ$-generalized derivations on $L_0^{\infty} (w)^*$. We show that every $θ$-centralizing $θ$-generalized derivation on $L_0^{\infty} (w)^*$ is a $θ$-right centralizer. We also prove that this result is true for $θ$-skew centralizing $θ$-generalized derivations.

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arXiv2024

Left $θ$-derivations on weighted convolution algebras

Let $θ$ be a homomorphism on $L_0^\infty({\Bbb R}^+, ω)^*$. In this paper, we study left $θ$-derivations on $L_0^\infty({\Bbb R}^+, ω)^*$. We show that every left $θ$-derivation on $L_0^\infty({\Bbb R}^+, ω)^*$ is always a $θ$-derivation, and if $θ$ is isomorphism, then $L_0^\infty({\Bbb R}^+, ω)^*$ has no non-zero left $θ$-derivation. We also investigate automatic continuity, Singer-Wermer's conjecture and Posner's first theorem for left $θ-$derivations on $L_0^\infty({\Bbb R}^+, ω)^*$.

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arXiv2024

On scalable $K$-frames and a version of Lax-Milgram theorem

In this paper, we first prove a theorem by a little modification on the Lax-Milgram theorem. Then, using $K$-frames, we obtain lower and upper bounds for the results obtained from this theorem. Also, we present some methods for the characterization of scalable $K$-frames. Finally, we introduce piecewise scalable $K$-frames and give necessary and sufficient conditions for a $K$-frame to be piecewise scalable.

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arXiv2023

On the Phase Retrievable Sequences

In this paper, we study phase retrievable sequences and give a characterization of phase retrievability of a sequence of bounded linear operators on a Hilbert space $H$; in particular, for $H=\ell_2^d(\Bbb{C})$. We also give several approaches for constructing phase retrievable sequences. Then, we investigate the property of phase retrieval for $g$-frames and frames.

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arXiv2023

On $(p, q)-$centralizers of certain Banach algebras

Let $A$ be an algebra with a right identity. In this paper, we study $(p, q)-$centralizers of $A$ and show that every $(p, q)-$centralizer of $A$ is a two-sided centralizer. In the case where, $A$ is normed algebra, we also prove that $(p, q)-$centralizers of $A$ are bounded. Then, we apply the results for some group algebras and verify that $L^1(G)$ has a nonzero weakly compact $(p, q)-$centralizer if and only if $G$ is compact and the center of $L^1(G)$ is non-zero. Finally, we investigate $(p, q)-$Jordan centralizers of $A$ and determine them.

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arXiv2023

Jordan derivations on certain Banach algebras

In this paper, we study the types of Jordan derivations of a Banach algebra $A$ with a right identity $e$. We show that if $eA$ is commutative and semisimple, then every Jordan derivation of $ A $ is a derivation. In this case, Jordan derivations map $A$ into the radical of $A$. We also prove that every Jordan triple left (right) derivation of $ A $ is a Jordan left (right) derivation. Finally, we investigate the range of Jordan left derivations and establish that every Jordan left derivation of $ A $ maps $ A $ into $eA$.

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arXiv2023

On $g-$Fusion Frames Representations via Linear Operators

Let $\{\frak{M} _k \} _{ k \in \mathbb{Z}} $ be a sequence of closed subspaces of Hilbert space $H$, and let $\{Θ_k\}_{k \in \mathbb{Z}}$ be a sequence of linear operators from $H$ into $\frak{M}_k$, $k \in \mathbb{Z}$. In the definition of fusion frames, we replace the orthogonal projections on $\frak{M} _k$ by $Θ_k$ and find a slight generalization of fusion frames. In the case where, $Θ_k$ is self-adjoint and $Θ_k(\frak{M} _k)= \frak{M} _k$ for all $k \in \mathbb{Z}$, we show that if a $g-$fusion frame $\{(\frak{M} _k, Θ_k)\}_{k \in \mathbb{Z}}$ is represented via a linear operator $T$ on $\hbox{span} \{\frak{M} _k\}_{ k \in \mathbb{Z}}$, then $T$ is bounded; moreover, if $\{(\frak{M} _k, Θ_k)\}_{k \in \mathbb{Z}}$ is a tight $g-$fusion frame, then $T$ is not invertible. We also study the perturbation and the stability of these fusion frames. Finally, we give some examples to show the validity of the results.

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arXiv2023

Jordan derivations on the $θ-$Lau products of Banach algebras

In this paper, we study Jordan derivation-like maps on the $θ-$Lau products of algebras. We characterize them and prove that under certain condition any Jordan derivation-like maps on the $θ-$Lau products is a derivation-like map. Moreover, we investigate the concept of centralizing for Jordan derivation-like maps on the $θ-$Lau products of algebras.

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arXiv2023

Products of generalized derivations on rings

In this paper, we show that if the product $(D_1D_2, d_1d_2)$ of generalized derivations $(D_1, d_1)$ and $(D_2, d_2)$ on an algebra $A$ is a generalized derivation, then $d_1D_2$ and $d_2D_1$ map $A$ into $\hbox{rad}(A)$. Also, for generalized derivations $(D_1, d_1)$ and $(D_2, d_2)$ on a prime ring with characteristic different from two, we give necessary and sufficient conditions under which $(D_1^2+D_1D_2,d_1^2+d_1d_2)$ is a generalized derivation as well.

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arXiv2022

Cohomological properties of vector-valued Lipschitz algebras and their second duals

Let $\frak{F}(X, A)$ be one of the Banach algebras $\hbox{Lip}(X, A)$ or $\hbox{lip}(X, A)$. In this paper, we show that $\frak{F}(X, A)$ is amenable if and only if $X$ is uniformly discrete and $A$ is amenable. We also prove that the result holds for $\hbox{lip}^\circ(X, A)$ instead of $\frak{F}(X, A)$. In the case where $A^*$ is separable, we establish that $\frak{F}(X, A)^{**}$ is amenable if and only if $X$ is uniformly discrete and $A^{**}$ is amenable, however, amenability of $\hbox{lip}^\circ(X, A)^{**}$ is equivalent to amenability of $A^{**}$ and finiteness of $X$. We prove that if $\hbox{Lip}(X, A)$ is point (respectively, weakly) amenable, then $X$ is uniformly discrete and $A$ is point (respectively, weakly) amenable. In particular, $\hbox{Lip}X$ is weakly amenable if and only if $X$ is discrete. We then investigate cohomological properties for vector-valued Banach algebras $C_0(X, A)$ and $L^1(G, A)$. Finally, we prove that biprojectivity (respectively, cyclically weak amenability) of $A^{**}$ implies biprojectivity (respectively, cyclically weak amenability) of $A$. This result holds for weak amenability and cyclic amenability when $A$ is commutative.

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arXiv2022

Homological and cohomological properties of Banach algebras and their second duals

In this paper, we investigate homological properties of Banach algebras. We show that retractions Banach algebras preserve biprojectivity, contractibility and biflatness. We also prove that contractibility of second dual of a Banach algebra implies contractibility of the Banach algebra. For a Banach algebra $A$ with $Δ(A)\neq\emptyset$, let $\frak{F}(X, A)$ be one of the Banach algebras $C_b(X, A)$, $C_0(X, A)$, $\hbox{Lip}_α(X, A)$ or $\hbox{lip}_α(X, A)$. In the following, we study homological properties of Banach algebra $\frak{F}(X, A)$, especially contractibility of it. We prove that contractibility of $\frak{F}(X, A)$ is equivalent to finiteness of $X$ and contractibility of $A$. In the case where, $A$ is commutative, we show that $\frak{F}(X, A)$ is contractible if and only if $A$ is a $C^*-$algebra and both $X$ and $Δ(A)$ are finite. In particular, $\hbox{lip}_α^0(X, A)$ is contractible if and only if $X$ is finite. We also investigate contractibility of $L^1(G, A)$ and establish $L^1(G, A)$ is contractible if and only if $G$ finite and $A$ is contractible. Finally, we show that biprojectivity of the Beurling algebra $L^1(G, ω)$ is equivalent to compactness of $G$, however, biprojectivity of the Banach algebras $L^1(G, ω)^{**}$ is equivalent to finiteness of $G$. This result holds for the Banach algebra $M(G, ω)^{**}$ instead of $L^1(G, ω)^{**}$.

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arXiv2022

Weak amenability of weighted measure algebras and their second duals

In this paper, we study the weak amenability of weighted measure algebras and prove that $M(G, ω)$ is weakly amenable if and only if $G$ is discrete and every bounded quasi-additive function is inner. We also study the weak amenability of $L^1(G, ω)^{**}$ and $M(G, ω)^{**}$ and show that the weak amenability of theses Banach algebras are equivalent to finiteness of $G$. This gives an answer to the question concerning weak amenability of $L^1(G, ω)^{**}$ and $M(G, ω)^{**}$.

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arXiv2022

Cohomological properties of different types of weak amenability

In this paper, we deal with cohomological properties of weak amenability, cyclic amenability, cyclic weak amenability and point amenability of Banach algebras. We look at some hereditary properties of them and show that continuous homomorphisms with dense range preserve cyclically weak amenability, however, weak amenability and cyclically amenability are preserved under certain conditions. We also study these cohomological properties of the $θ-$Lau product $A\times_θB$ and the projective tensor product $A\hat{\otimes} B$. Finally, we investigate the cohomological properties of $A^{**}$ and establish that cyclically weak amenability of $A^{**}$ implies cyclically weak amenability of $A$. This result is true for point amenability instead of cyclically weak amenability.

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arXiv2022

Different types of weak amenability for Banach algebras

In this paper, we introduce and investigate the concepts of cyclically weakly amenable and point amenable. Then, we compare these concepts with the concepts of weakly amenable and cyclically amenable and find the relation between them. For example, we prove that a Banach algebra is weakly amenable if and only if it is both cyclically amenable and cyclically weakly amenable. In the case where $A$ is commutative, the weak amenability and cyclically weak amenability of $A$ are equivalent. We also show that if $A$ is a Banach algebra with $Δ(A)\neq\emptyset$, then $A$ is cyclically weakly amenable if and only if $A$ is point amenable and essential. For a unital, commutative Banach algebra $A$, the notions of weakly amenable, cyclically weakly amenable and point amenable coincide. In this case, these are equivalent to the fact that every maximal ideal of $A$ is essential.

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arXiv2022

Weak amenability of weighted group algebras

In this paper, we study weak amenability of Beurling algebras. To this end, we introduce the notion inner quasi-additive functions and prove that for a locally compact group $G$, the Banach algebra $L^1(G, ω)$ is weakly amenable if and only if every non-inner quasi-additive function in $L^\infty(G, 1/ω)$ is unbounded. This provides an answer to the question concerning weak amenability of $L^1(G, ω)$ and improve some known results in connection with it.

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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.

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