Characterizations Of Mappings On Operator Algebras

This dissertation aims to study some kinds of mappings on operator algebras. It consists of six chapters.In Chapter 1, we introduce some preliminary concepts and properties about map-pings such as derivations, Jordan derivations and Lie derivations and reflexive algebras such as nest algebras, CSL algebras, completely distributive subspace lattice algebras, JSL algebras and double triangle subspace lattice algebras.In Chapter 2, we characterize derivations and Jordan derivations on some reflexive algebras. For a subspace lattice (?) on a Banach space X, we say that (?) is a P-subspace lattice if V{L∈(?):L_?L}= X or∧{L_:L∈(?),L_?L}=(0). In Sections 2 and 3, we show that ifδis a linear mapping from a P-subspace lattice algebra Alg(?) into B(X), then the following three conditions are equivalent:(1)δis derivable at zero point, i.e.,δ(AB)=δ(A)B+Aδ(B) whenever AB=0; (2)δis Jordan derivable at zero point, i.e.,δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) whenever AB+BA=0; (3)δis a generalized derivation andδ(I)∈(Alg(?))'. We also investigate left (right) multipliers, isomorphisms and local generalized derivations on P-subspace lattice algebras. These results can apply to reflexive algebras such as JSL algebras, discrete nest algebras and subspace lattice algebras with X_≠X or (0)+≠(0). In Section 4, we prove that every linear mappingδJordan derivable at zero point on a strongly double triangle subspace lattice algebra AlgD has the formδ(A)=τ(A)+λA((?)A∈AlgD), whereτis a derivation and A is some scalar. In Section 5, we have that a bounded linear mappingδon a CDCSL algebra Alg(?) is Jordan derivable at zero point if and only ifδis a generalized derivation andδ(I)∈(Alg(?))'.In Chapter 3, we investigate Lie derivations on reflexive algebras. Let A be a nest subalgebra of a factor von Neumann algebra or a CSL algebra whose invariant subspace lattice is generated by finitely many commuting independent nests. In Section 2, we show that ifδis a linear mapping from A into A such thatδ([A, B])=[δ(A),B]+[A,8(B)] whenever AB=0 or AB=P (where P is some non-trivial projection in A), thenδcan be decomposed into a sum of a derivation and a center-valued mapping. Let C be a commutative unital ring and let M be a unital C-bimodule. In Section 3, we prove that every Lie triple derivation from Tn(C) into a 2-torsion free unital Tn(C)-bimodule Tn(M) is a sum of a derivation and a linear mapping having its range in the center of Tn(M).In Chapter 4, we consider derivations and Jordan derivations on triangular algebras. In Section 2, we show that if N is a nontrivial nest on a complex separable Hilbert space, G is an element in AlgN andδ:AlgN→AlgN is a linear mapping such thatδ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) whenever AB=G, thenδis a derivation. In Section 3, we prove that if a triangular algebra Tri(A, M,B) satisfies some conditions, then every linear mappingδ:Tri(A,M,B)→Tri(A,M,B) derivable at G=Diag(W,0) (where W in invertible in A) is a derivation.In Chapter 5, we introduce generalized left multipliers and generalized Jordan left multipliers. Letδbe an additive mapping from a ring R into an R-bimodule M and letβ:R×R→M be a biadditive mapping such thatβ(x,yz)-β(x,y)z-β(xy,z)=0 for all x,y∈R. We call (δ,β) is a generalized left multiplier (resp. generalized Jordan left multiplier) ifδ(xy)=δ(x)y+β(x, y) (resp.δ(x2)=δ(x)x+β(x,x)) for all x,y∈R. In Section 2, we show that every generalized Jordan left multiplier is a generalized left multiplier on some rings. As a corollary, we prove that every generalized Jordan derivation is a generalized derivation on some rings and algebras.In Chapter 6, we summarize the paper and give some open questions.