Characterizations Of Some Mappings On Operator Algebras

The aim of this dissertation is to study some mappings on operator algebras. These mappings include derivations, Jordan derivations, higher derivations, Jordan higher deriva-tions, Lie derivations, Lie higher derivations, Lie triple derivations, centralizers and map-pings associated with Hochschild 2-cocycles. The operator algebras we discuss cover nest algebras, J-subspace lattice algebras, CSL algebras, completely distributive subspace lattice algebras, triangular algebras and generalized matrix algebras.This dissertation splits into seven chapters. In Chapter 1, we introduce the back-ground of this study, review the developments and achievements until now, and give the preliminary concepts we need in this paper.In Chapter 2, we characterize Jordan derivable mappings in terms of Peirce decom-position and determine Jordan all-derivable points for some general bimodules. Then we generalize these results to the case of Jordan higher derivable mappings. Meanwhile, we find the connection between (Jordan) all-derivable points and (Jordan) higher all-derivable points, i.e., an element in an algebra A over a field of characteristic zero is a (Jordan) all-derivable point if and only if it is a (Jordan) higher all-derivable point. Finally we show that every Jordan higher derivation on the norm-closed subalgebra A of B(X) such that V{x:x(?)f∈A}=X and∧{ker(f):x(?)f∈A}= (0) is bounded. An immediate but noteworthy application of our main results shows that for a nest N on a Banach space X with the associated nest algebra algN, if there exists a non-trivial idem-potent P∈algN such that ran(P)∈N, then every C∈algN is a Jordan all-derivable point of L(algN, B(X)) and a Jordan higher all-derivable point of algN.In Chapter 3, we study Lie higher derivations on triangular algebras and show that under certain conditions every Lie higher derivation on a triangular algebra is proper. We also characterize Lie triple derivations on triangular algebras and give necessary and sufficient conditions such that every Lie triple derivation is proper.Let A be a unital algebra over a number field F. A linear mappingδfrom A into itself is called a generalized (m, n,l)-Jordan centralizer if it satisfies (m+n+l)S(A2)-mδ(A)A-nAδ(A)-lAδ(I)A∈FI for every A∈A, where m≥0, n≥0,1≥0 are fixed integers with m+n+l≠0. In Chapter 4, we study generalized (m, n,l)-Jordan centralizers on generalized matrix algebras and some reflexive algebras algL, where L is a CSL or satisfies∨{L:L∈J(L)}= X orΛ{L_:L∈J(L)}=(0), and prove that each generalized (m, n, l)-Jordan centralizer of these algebras is a centralizer when m+l≥1 and n+l≥1.Let A be a unital algebra,δbe a linear mapping from A into itself and m, n be fixed integers with m≠0. We callδan (m, n)-derivable mapping at C, if mδ(AB)+ nδ(BA)= mδ(A)B+mAδ(B)+nδ(B)A+nBδ(A) for all A,B∈A with AB=C. The above definition includes the cases of derivable mappings, Jordan derivable mappings and Lie derivable mappings at some point. In Chapter 5, we characterize (m, n)-derivable mappings on generalized matrix algebras and show that ifδis an (m, n)-derivable mapping at 0 (resp.IA(?)0,I) from U into itself, thenδis a derivation, a Jordan derivation or a Lie derivation according to different choices of m and n. We also show that ifδis a norm-continuous (m, n)-derivable mapping at 0 on CSL algebras with m+n≠0 andδ(I)=0, thenδis a derivation.In Chapter 6, we discuss generalized (Jordan) derivations associated with Hochschild 2-cocycles on upper triangular matrices and prove that every generalized Jordan deriva-tion associated with Hochschild 2-cocycles from the algebra of upper triangular matrices to its bimodule is the sum of a generalized derivation and an antiderivation.In Chapter 7, we give a summarization of the whole text, and pose some questions remaining unsolved.