Subspace Lattice Algebras Research Articles

Multiplicative generalized Lie n-derivations on completely distributive commutative subspace lattice algebras

Multiplicative generalized Lie n-derivations on completely distributive commutative subspace lattice algebras

Linear Mappings Characterized by Action on Zero Products or Unit Products

Let $$\mathcal {A}$$ be a unital algebra and $$\mathcal {M}$$ be a unital $$\mathcal {A}$$ -bimodule. We characterize the linear mappings $$\delta $$ and $$\tau $$ from $$\mathcal {A}$$ into $$\mathcal {M}$$ , satisfying $$\delta (A)B+A\tau (B)=0$$ for every $$A,B \in \mathcal {A}$$ with $$AB=0$$ when $$\mathcal {A}$$ contains a separating ideal $$\mathcal {T}$$ of $$\mathcal {M}$$ , which is in the algebra generated by all idempotents in $$\mathcal {A}$$ . We apply the result to $$\mathcal {P}$$ -subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that $$\mathcal {A}$$ is a unital Banach algebra and $$\mathcal {M}$$ is a unital Banach $$\mathcal {A}$$ -bimodule, we give a complete description of linear mappings $$\delta $$ and $$\tau $$ from $$\mathcal A$$ into $$\mathcal M$$ , satisfying $$\delta (A)B+A\tau (B)=0$$ for every $$A,B\in \mathcal {A}$$ with $$AB=I$$ .

2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS

AbstractLet${\mathcal{A}}$be a semisimple Banach algebra with minimal left ideals and$\text{soc}({\mathcal{A}})$be the socle of${\mathcal{A}}$. We prove that if$\text{soc}({\mathcal{A}})$is an essential ideal of${\mathcal{A}}$, then every 2-local derivation on${\mathcal{A}}$is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and${\mathcal{J}}$-subspace lattice algebras, is a derivation.

Local Maps of JSL Algebras

A JSL algebra is a reflexive algebra with the $${\mathcal {J}}$$ -subspace lattice and a standard subalgebra of it is a subalgebra of it containing all finite rank operators in it. We prove that every local Lie derivation of a JSL algebra is a Lie derivation, every 2-local derivation of a standard subalgebra of a JSL algebra is a derivation and every surjective 2-local isomorphism between standard subalgebras of JSL algebras is an isomorphism. These results can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.

A characterization of generalized derivations of JSL algebras

Let Alg ℒ be a J -subspace lattice algebra on a Banach space X and M be an operator in Alg ℒ. We prove that if δ: Alg ℒ → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B) for all A,B ∈ Alg ℒ with AMB = 0, then δ is a generalized derivation. This result can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.

Characterizations of Jordan left derivations on some algebras

A linear mapping δ from an algebra A into a left A-module M is called a Jordan left derivation if δ(A2)=2Aδ(A) for every A∈A. We prove that if an algebra A and a left A-module M satisfy one of the following conditions—(1) A is a C∗-algebra and M is a Banach left A-module; (2) A=AlgL with ∩{L−:L∈JL}=(0) and M=B(X); and (3) A is a commutative subspace lattice algebra of a von Neumann algebra B and M=B(H)—then every Jordan left derivation from A into M is zero. δ is called left derivable at G∈A if δ(AB)=Aδ(B)+Bδ(A) for each A,B∈A with AB=G. We show that if A is a factor von Neumann algebra, G is a left separating point of A or a nonzero self-adjoint element in A, and δ is left derivable at G, then δ≡0.

Full-derivable points of $\mathcal {J}$-subspace lattice algebras

Let $\mathcal{L}$ be a $\mathcal{J}$-subspace lattice on a complex Banach space $X$ and $\mbox{\rm Alg\,}{\mathcal L}$ the associated $\mathcal{J}$-subspace lattice algebra. We say that an operator $Z\in {\rm Alg\,}{\mathcal L}$ is a full-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a derivation and is a full-generalized-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself generalized derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)-A\delta(I)B=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a generalized derivation. In this paper, we prove that if $Z\in\mbox{\rm Alg\,}{\mathcal L}$ is an injective operator or an operator with dense range, then $Z$ is a full-derivable point as well as a full-generalized-derivable point of ${\rm Alg\,}{\mathcal L}$.

Lie Triple Derivations on𝒥-Subspace Lattice Algebras

We describe the structure of Lie triple derivations on𝒥-subspace lattice algebras. The results can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras, respectively.

Characterization of (generalized) Lie derivations on $${\mathcal{J}}$$ -subspace lattice algebras by local action

Let \({\mathcal{L}}\) be a \({\mathcal{J}}\)-subspace lattice on a Banach space X over the real or complex field \({\mathbb{F}}\) with dim X ≥ 2 and Alg\({\mathcal{L}}\) be the associated \({\mathcal{J}}\)-subspace lattice algebra. For any scalar \({\xi \in \mathbb{F}}\), there is a characterization of any linear map L : Alg \({\mathcal{L} \rightarrow {\rm Alg} {\mathcal{L}}}\) satisfying \({L([A,B]_\xi) = [L(A),B]_\xi + [A,L(B)]_\xi}\) for any \({A, B \in{\rm Alg} {\mathcal{L}}}\) with AB = 0 (rep. \({[A,B]_ \xi = AB - \xi BA = 0}\)) given. Based on these results, a complete characterization of (generalized) ξ-Lie derivations for all possible ξ on Alg\({\mathcal{L}}\) is obtained.

On derivable mappings

A linear mapping δ from an algebra A into an A -bimodule M is called derivable at c ∈ A if δ ( a ) b + a δ ( b ) = δ ( c ) for all a , b ∈ A with a b = c . For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C ∈ A has a right inverse in B ( X ) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B ( X ) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J -subspace lattice algebras, and norm-closed unital standard algebras of B ( X ) . As an application, every Jordan derivation from such an algebra into B ( X ) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B ( X ) can be characterized by boundedness and derivability at any fixed C ∈ A , provided C has a right inverse in B ( X ) . We also show that if A is a canonical subalgebra of an AF C ∗ -algebra B and M is a unital Banach A -bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation.

Jordan Derivations of Reflexive Algebras

Let \({\mathcal L}\) be a subspace lattice on a Banach space X and suppose that \({\vee\{L\in\mathcal L: L_- (0)\}=(0)}\) . Then each Jordan derivation from Alg\({\mathcal L}\) into B(X) is a derivation. This result can apply to completely distributive subspace lattice algebras, \({\mathcal J}\) -subspace lattice algebras and reflexive algebras with the non-trivial largest or smallest invariant subspace.

Additive maps derivable at some points on [formula omitted]-subspace lattice algebras

Let L be a J-subspace lattice on a real or complex Banach space dim X with X>2 and AlgL be the associated J-subspace lattice algebra. Let δ:AlgL→AlgL be an additive map. It is shown that, if δ is derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B) whenever AB=0, then δ(A)=τ(A)+λA, ∀A, where τ is an additive derivation and λ is a scalar; if δ is generalized derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B)-Aδ(I)B whenever AB=0, then δ is a generalized derivation. It is also shown that, if X is complex, then every linear map derivable at unit operator on AlgL is a derivation.

Jordan elementary operators on [formula omitted]-subspace lattice algebras

In this paper, we give a characterization of Jordan elementary operators of standard subalgebras of J-subspace lattice algebras. In particular, the general form of elementary operators and reverse elementary operators is described. These results can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.

Lie derivations of $\mathcal J$-subspace lattice algebras

We describe the structure of Lie derivations of $\mathcal J$-subspace lattice algebras. The results can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras, respectively.

Additivity of Elementary Maps on Rings#

Let ℛ and ℛ′ be rings. Under some assumptions on ℛ, we study the additivity of maps M: ℛ → ℛ′ and M*: ℛ′ → ℛ that are surjective and satisfy M(x M*(y)z) = M(x)yM(z) and M*(yM(x)u) = M*(y)x M*(u) for x, z ∈ ℛ and y, u ∈ ℛ′. In particular, if ℛ is a prime ring containing a non-trivial idempotent, or a standard operator algebra, or a nest algebra on a Hilbert space, or an atomic Boolean subspace lattice algebra on a Banach space, then we can conclude that both M and M* are additive.

Elementary operators on 𝒥-subspace lattice algebras

The abstract concept of an elementary operator was recently introduced and studied by other authors. In this paper, we describe the general form of elementary operators between standard subalgebras of 𝒥-subspace lattice algebras. The result can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.

Invertibility preserving linear maps on [formula omitted]-subspace lattice algebras

It is shown that every unital surjective linear map between J -subspace lattice algebras which preserves invertibility in both directions is a Jordan isomorphism. This result can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.

Jordan isomorphisms of [formula omitted]-subspace lattice algebras

Let L be a J -subspace lattice. A subalgebra of Alg L is called a standard subalgebra if it contains all finite rank operators in Alg L . In this paper, a characterization of Jordan isomorphisms between standard subalgebras of J -subspace lattice algebras is given. This result can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras, respectively.

Decomposability of Finite-Rank Operators in Commutative Subspace Lattice Algebras

In this paper, we explore conditions for a finite-rank operator in a commutative subspace lattice (CSL) algebra to be decomposable (that is, it can be written as the sum of rank one operators in that algebra). We introduce correlation coefficients for rank two operators, the property (F), and correlation matrices for finite-rank operators, based on which we prove that a rank two operator is decomposable if and only if it has only finitely many correlation coefficients, and if a finite-rank operator has the property (F) (has only finitely many correlation matrices) then it is decomposable.

Derivations and isomorphisms of certain reflexive operator algebras

In this paper, we will prove that every derivation of completely distributive subspace lattice (CDS) algebras on Banach space is automatically continuous. This is new even in the Hilbert space case. As an application of this result, we obtain that every additive derivation of nest algebras on Banach spaces is inne. We will also prove that every isomorphism between nest algebras on Banach space is automatically continuous, and in addition, is spatial.