Characterizing Some Mappings On Operator Algebras

In this paper, we give a characterization of the linear mappings on operator al-gebras. The mappings that we study include left derivations, Jordan left derivations, (m,n)-Jordan derivations, generalized derivations and generalized Jordan derivations. The algebras that we study include C*-algebras, von Neumann algebras, nest algebras, completely distributive subspace lattice algebras, J-subspace lattice algebras, P-subspace lattice algebras, commutative subspace lattice algebras and generalized matrix algebras. This paper splits into seven chapters.In Chapter one, we introduce the background of this study, review the developments and achievements until now, and give some preliminary concepts we need in this paper.In Chapter two, we prove that if an algebra A and a left A-module M satisfy one of the following conditions, then every Jordan left derivation from A into M identically equals zero. (1) A is a C*-algebra and M is a Banach left A-module; (2) A=AlgL with ∩{L_:L∈Jc]= (0) and M= B(X), where L is a subspace lattice on a Banach space X; (3) A=B∩AlgL and M=B(H), where B is a von Neumann algebra on a Hilbert space H and L(?)B and is a commutative subspace lattice on H.In Chapter three, we prove that if A is a unital algebra over the complex field C and M is a unital A-bimodule with a left (right) separating set generated algebraically by all idempotents in A, then every (m, n)-Jordan derivation from A into M is identical is a|mn(m-n)(m+n)|-torsion-free generalized matrix algebra and M is a faithful unital (A,B)-bimodule, then every (m, n)-Jordan derivation from U into itself is equal to zero wheneverIn Chapter four, we prove that a unital algebra is zero Jordan product determined if it is generated algebraically by its idempotents. Hence we affirmatively answer two questions posed by M. Bresar in 2009 and M. Kosan et al. in 2014, respectively. We also investigate whether a unital algebra with a nontrivial idempotent is zero Jordan product determined, and give necessary and sufficient conditions such that every triangular algebra is zero Jordan product determined. As applications, we characterize the local properties of Jordan left derivations, (m, n)-Jordan derivations, Jordan derivations, Lie derivations, Jordan homomorphisms and Lie homomorphisms on zero Jordan product determined algebras.In Chapter five, we give a characterization of generalized derivations and generalized Jordan derivations through zero products or zero Jordan products. Let A be a unital algebra, M be a unital.A-bimodule and δ be a linear mapping from A into M. Firstly, we prove that if A contains an ideal J generated algebraically by idempotents such that {M∈M:JMK= 0, for each J,K∈J}={0}, and δ satisfies AB= BC= 0 implies Aδ(B)C=0 for each A, B, C∈A, then δ is a generalized derivation. In particular, if δ is a local derivation from A into M, then δ is a derivation. We also show that if A contains an ideal J generated algebraically by idempotents such that {M∈M:JM= MJ=0, for each J∈J}={0}, and δ is derivable at zero point, that is AB=0 implies Aδ(B)+δ(A)B=0 for each A,B∈A, then δ is a generalized derivation. Obviously, if M contains a separating set J generated algebraically by all idempotents in A, then J satisfies the above two conditions. Finally we prove that if A contains an ideal J generated algebraically by idempotents such that {M∈M:JMJ= 0, for each J∈J}={0}, and δ satisfies AB= BA= 0 implies A(?)δ(B)+δ(A)(?)B=0 for each A,B∈A (in particular,δ is a derivable mapping or a Jordan derivable mapping at zero point), then δ is a generalized Jordan derivation.In Chapter six, we prove that if A is a unital algebra over the complex field C, M is a unital left A-module contains a right separating set generated algebraically by idempotents in A, and δ is a linear mapping from A into M such that AB= BA= 0 implies A8(B)+Bδ(A)= 0 for each A, B∈A, then δ(A)=Aδ(I) for every A∈A. We also show that if A is a factor von Neumann algebra, then every mapping left derivable at a right separating element or a nonzero self-adjoint element is equal to zero.In Chapter seven, we give a summarization of the whole paper, and pose some questions remaining unsolved.