Generalized Matrix Algebras Research Articles

Lie centralizing mappings on generalized matrix algebras through two-sided zero products

Let [Formula: see text] be a generalized matrix algebra defined by the Morita context [Formula: see text] and [Formula: see text] the center of [Formula: see text]. In this paper, under some certain conditions on [Formula: see text], we prove that if [Formula: see text] is an additive map satisfying [Formula: see text] for any [Formula: see text] with [Formula: see text], then [Formula: see text] has the form [Formula: see text] for all [Formula: see text], where [Formula: see text] and [Formula: see text] is an additive map from [Formula: see text] into [Formula: see text]. Finally as its applications, we characterize Lie centralizer maps and generalized Lie derivations on [Formula: see text]. Moreover we prove that the similar conclusions remain valid on full matrix algebras, triangular algebras, upper triangular matrix algebras.

Lie (Jordan) $ \sigma- $centralizer at the zero products on generalized matrix algebra

<p>Given a unital commutative ring $ \mathscr{R} $, $ (\mathscr{A}, \mathscr{B}) $ and $ (\mathscr{B}, \mathscr{A}) $ are bimodules of $ \mathscr{M} $ and $ \mathscr{N} $, respectively, where $ \mathscr{A}, \mathscr{B} $ are unitals $ \mathscr{R}- $algebras. The $ \mathscr{R}- $algebra $ \mathscr{G} = $ $ \mathscr{G}(\mathscr{A}, \mathscr{M}, \mathscr{N}, \mathscr{B}) $ is a generalized matrix algebra described by the Morita context $ (\mathscr{A}, \mathscr{B}, \mathscr{M}, \mathscr{N}, \zeta_{\mathscr{M}\mathscr{N}}, \chi_{\mathscr{N}\mathscr{M}}) $. The present study investigated the structure of Lie (Jordan) $ \sigma- $centralizers at the zero products on order two generalized matrix algebra and established that each Jordan $ \sigma- $centralizer at the zero products is a $ \sigma- $centralizer at the zero product on order two generalized matrix algebra. We also provided sufficient and necessary conditions under which a Lie $ \sigma- $centralizer at the zero product is proper on an order two generalized matrix algebra.</p>

Generalized Lie $ n $-derivations on generalized matrix algebras

Generalized Lie $ n $-derivations on generalized matrix algebras

On k-semi-centralizing maps of generalized matrix algebras

Abstract Let 𝒢 = 𝒢 (A, M, N, B) be a generalized matrix algebra over a commutative ring with unity. In the present article, we study k-semi-centralizing maps of generalized matrix algebras.

Characterizations of Lie centralizers of generalized matrix algebras

Let G be a generalized matrix algebra. A linear map ϕ : G → G is said to be a left (right) Lie centralizer at E ∈ G if ϕ ( [ S , T ] ) = [ ϕ ( S ) , T ] ( ϕ ( [ S , T ] ) = [ S , ϕ ( T ) ] ) holds for all S , T ∈ G with ST = E. ϕ is of a standard form if ϕ ( A ) = Z A + γ ( A ) for all A ∈ G , where Z is in the center of G and γ is a linear map from G into its center vanishing on each commutator [ S , T ] whenever ST = E. In this paper, we give a complete characterization of ϕ . It is shown that, under some suitable assumptions on G , ϕ has a standard form.

Lie centralizers and commutant preserving maps on generalized matrix algebras

Let [Formula: see text] be a 2-torsion free unital generalized matrix algebra with center [Formula: see text], and [Formula: see text] be a linear mapping on [Formula: see text] satisfying the condition [Formula: see text] This paper is devoted to the study of the structure of [Formula: see text] under some mild assumptions on [Formula: see text]. We provide the necessary and sufficient conditions for [Formula: see text] to be in the form [Formula: see text] ([Formula: see text]), where [Formula: see text] and [Formula: see text] is a linear mapping. Then we apply our results to characterize linear mappings on [Formula: see text] that are commutant preservers or double commutant preservers.

Additivity of nonlinear higher anti-derivable mappings on generalized matrix algebras

<abstract><p>In this article, we proved that each nonlinear higher anti-derivable mapping on generalized matrix algebras is automatically additive. As for its applications, we find a similar conclusion on triangular algebras, full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras and factor von Neumann algebras respectively.</p></abstract>

A study on generalized matrix algebras having generalized Lie derivations

Let R be a commutative ring with unity. The R-algebra G=G(A,M,N,B) is a generalized matrix algebra defined by the Morita context (A,B,M,N, ξMN,ΩNM). In this article, westudy generalized Lie derivation and show that every generalized Lie derivation on a generalized matrix algebra has the standard form under certain assumptions.

Lie $ n $-centralizers of generalized matrix algebras

<abstract><p>In this paper, we introduce the notion of Lie $ n $-centralizers. We then give a description of Lie $ n $-centralizers on a generalized matrix algebra and present the necessary and sufficient conditions for a Lie $ n $-centralizer to be proper. As applications, we determine generalized Lie $ n $-derivations on a generalized matrix algebra and Lie $ n $-centralizers of some operator algebras.</p></abstract>

Local Lie derivations of generalized matrix algebras

<abstract><p>In this paper, we investigate local Lie derivations of a certain class of generalized matrix algebras and show that, under certain conditions, every local Lie derivation of a generalized matrix algebra is a sum of a derivation and a linear central-valued map vanishing on each commutator. The main result is then applied to full matrix algebras and unital simple algebras with nontrivial idempotents.</p></abstract>

σ-centralizers of generalized matrix algebras

σ-centralizers of generalized matrix algebras

Lie triple centralizers on generalized matrix algebras

In this article, we introduce the notion of Lie triple centralizer as follows. Let be an algebra, and be a linear mapping. We say that ϕ is a Lie triple centralizer whenever for all . Then we characterize the general form of Lie triple centralizers on a generalized matrix algebra and under some mild conditions on we present the necessary and sufficient conditions for a Lie triple centralizer to be proper. As an application of our results, we characterize generalized Lie triple derivations on generalized matrix algebras.

Multiplicative Lie triple higher derivations on generalized matrix algebras

Multiplicative Lie triple higher derivations on generalized matrix algebras

Generalized Lie triple derivations on generalized matrix algebras

Let be a commutative ring with identity, be -algebras, be an -bimodule and be a -bimodule. The -algebra is a generalized matrix algebra defined by the Morita context In this article, we give the structure of generalized Lie triple derivations GL on generalized matrix algebras and prove that under certain restrictions GL can be written as where Δ is a generalized derivation and χ is a central valued mapping.

Local derivations of generalized matrix algebras

Local derivations of generalized matrix algebras

Nonlinear Jordan Derivable Mappings of Generalized Matrix Algebras by Lie Product Square-Zero Elements

The aim of the paper is to give a description of nonlinear Jordan derivable mappings of a certain class of generalized matrix algebras by Lie product square-zero elements. We prove that under certain conditions, a nonlinear Jordan derivable mapping Δ of a generalized matrix algebra by Lie product square-zero elements is a sum of an additive derivation δ and an additive antiderivation f . Moreover, δ and f are uniquely determined.

Category Algebras and States on Categories

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups.

Lie centralizers at the zero products on generalized matrix algebras

Let [Formula: see text] be a 2-torsion free unital generalized matrix algebra, and [Formula: see text] be a linear map satisfying [Formula: see text] In this paper, we study the structure of [Formula: see text] and under some mild conditions on [Formula: see text] we present the necessary and sufficient conditions for [Formula: see text] to be in terms of centralizers. We then provide the characterizations of Lie centralizers on [Formula: see text] and our results generalize some of the previous results. We also refer to some applications of our results for triangular algebras and some operator algebras.

On Additivity of Jordan Higher Mappings on Generalized Matrix Algebras

In this article, the additivity of higher multiplicative mappings, i.e., Jordan mappings, on generalized matrix algebras are studied. Also, the definition of Jordan higher triple product homomorphism is introduced and its additivity on generalized matrix algebras is studied.

Lie (Jordan) centralizers on generalized matrix algebras

Let be a commutative ring with unit element, be -algebras, be an -bimodule, and be a -bimodule. The -algebra is a generalized matrix algebra defined by the Morita context In this article, we study Lie (Jordan) centralizer on generalized matrix algebras and obtain the necessary and sufficient conditions for a Lie centralizer map to be proper. Further, we prove that every Jordan centralizer is a centralizer on generalized matrix algebras under certain assumptions.