Left Derivations In Banach Algebras And Prime Rings

This thesis surveys the achievements concerning left deriv-eations on Banach algebras and (semi-) prime rings. On the one hand, we sketch the development of left derivations on Banach algebras. The main results focus on the ranges of left derivations and Jordan left derivations. On the other hand, we summarize the development of left derivations on (semi-) prime rings. The main conclusions describe left derivations and related maps on (semi-) prime rings.Let f:R→R be an additive mapping of a ring R. We call f a left derivation of R if f(xy)= xf(y)+yf(x) holds for all x,y∈R. Particularly, left derivations on Banach algebras are linear left derivations. Namely,f is also linear.The first chaper introduces the development of left derivations on Banach algebras. The main results focus on the ranges of left derivations on Banach algebras. The development experiences from commutative Banach algebras to non-commutative Banach algebras, and also from continuous left derivations to discontinuous left derivations. The ranges of Jordan left derivations on Banach algebras were discussed after determining the ranges of left derivations on Banach algebras. And then the ranges of Jordan left derivations on Banach algebras under some mild assumption were decided.Firstly, we outline the research process of the ranges of left derivations on Banach algebras.Singer and Wermer discussed the ranges of continuous linear left derivations on a commutative Banach algebra in 1955 and obtainedTheorem 1.1 (1) The image of a continuous linear left derivation on a commutative Banach algebra is contained in its Jacobson radical;(2) A continuous linear left derivation on a commutative semisimple Banach algebra must be zero.Then Thomas discussed the ranges of linear left derivations on a commutative Banach algebra in 1988 and obtainedTheorem 1.2 (1) The image of a linear left derivation on a commutative Banach algebra is contained in its Jacobson radical;(2) A linear left derivation on a commutative semisimple Banach algebra must be zero.In 1990, Bresar and Vukman presented the range of a conti-nuous linear left derivation on a commutative Banach algebra, and obtained the following theorem.Theorem 1.3 (1) The image of a continuous linear left derivation on a Banach algebra is contained in its Jacobson radical;(2) A continuous linear left derivation on a semisimple Banach algebra must be zero.In 1990, Jung discussed the range of a linear left derivation on a Banach algebra through introducing "spectrally bounded".Theorem 1.4 (1) The image of a linear left derivation on a Banach algebra is contained in its Jacobson radical if and only if it is spectrally bounded.(2) A spectrally bounded linear left derivation on a semisimple Banach algebra must be zero.Then we introduce some conclusions about the ranges of Jordan left derivations on Banach algebras under some mild assumptions.In 1999, Jung presented the ranges of spectrally bounded left derivations on Banach algebras and obtainedTheorem 1.5 Let d be a spectrally bounded Jordan left derivation on a Banach algebra A. If [d(x),x]∈rad(A) for all x∈A, then d(A) (?) rad(A).In 2002, Park and Jung discussed the ranges of Jordan left derivations on Banach algebras and obtainedTheorem 1.6 Let d be a Jordan left derivation of a Banach algebra A such that d2(x)= 0 for all x∈A.Then d maps A into rad(A).In 2008, Vukman presented the ranges of Jordan left derivations on a semisimple Banach algebra and obtainedTheorem 1.7 Let A be a semisimple Banach algebra and let D:A→A be a linear left Jordan derivation. In this case D= 0.And then Vukman put forward a conjecture:let A be a Banach algebra, and let D:A→A be a linear left Jordan derivation, then D maps A to its Jacobson radical. In the second chapter, we mainly talked over the development of left derivations on (semi-) prime rings. Fistly, we describe left derivations on (semi-) prime rings; Secondly, we discuss Jordan left derivations on (semi-) prime rings; Thirdly, we introduce the structure of the maps with (Jordan) left derivation properies on some subsets; At last, we give some conclusions on related concepts coming from left derivations in (semi-) prime rings.The research of left derivations on (semip-) prime rings began with the results by Bresar and Vukman in 1990.In section one, we describe important conclusions on the structure of the left derivations on (semi-) prime rings, given by Bresar and Vukman in 1990.Theorem 2.1 (1) A left derivation on a prime ring must be zero.(2) A left derivation on a semiprime ring must be a derivation whose image is contained in the center of the ring.In section two, we describe the left Jordan derivations on (semi-) prime rings.In 1990, Bresar and Vukman talked over the structure of Jordan left derivations on prime rings with characteristic different from 2 and 3.Theorem 2.2 Let R be a prime ring of characteristic different from 2 and 3. If R admits a nonzero Jordan left derivation D:R→R, then R is commutative.In 1996, Jun and Kim gave the structure of Jordan left derivations on a 2-torsion free prime ring.Theorem 2.3 Let R be a ring and X be a 2-torsionfree left R-module. Suppose that aRx= 0 with a∈R,x∈X implies that either a= 0 or x= 0. If there exists a nonzero Jordan let derivation D:R→X then R is commutative.In 2008, Vukman gave the ranges of Jordan left derivations on a 2-torsion free semiprime ring.Therorem 2.4 Let R be a 2-torsion free semiprime ring and let D:R→R be a left Jordan derivation. In this case D is a derivation which maps R into Z(R).In section three, we introduce some results on the maps with the properties as left Jordan derivation on special Lie ideal of prime rings.In 2000, Ashraf and Rehman gave the following theorem.Theorem 2.5 Let R be a 2-torsion free prime ring and let U be a Lie ideal of R such that u2∈U, for all u∈U. If d:R→R is an additive mapping such that d(u2)= 2ud(u) for all u∈U, then d(uv)= ud(v)+vd(u) for all u,v∈U.In 2001, Rehman and Ali improved theorem 2.5 as follows.Theorem 2.6 Let R be a 2-torsion free prime ring and let U be a Lie ideal of R such that u2∈U, for all u∈U. If d:R→R is an additive mapping such that d(u2)= 2ud(u) for all u∈U, then either U(?)Z(R) or d(U)= (0).In section four, we describe some related concepts coming from left derivations.In 2001, Ali and Ashraf gave the concepts of generalized left derivations (generalized Jordan left derivations, respectively) as following.Definition 2.3 An additive mapping g:R→R is called a generalized left derivation (generalized Jordan left derivation) if there exists a (Jordan) left derivationδ:R→R such that g(xy)= xg(y)+yδ(x), for all x,y∈R (g(x2)= xg(x)+xδ(x),for all x∈R).In 2005, Ashraf gave the concept of left (θ,φ)-derivations (resp. Jordan left (θ,φ)-derivations) as followsDefinition 2.4 Suppose thatθ,φare endomorphisms of R. An additive mapping y:R→R is called a left(θ,φ)-derivation (resp. Jordan left (θ,φ)-derivation) ifγ(xy)=θ(x)γ(y)+φ(y)γ(x), for all x,y∈R (resp. y(x2)=θ(x)γ(x)+φ(x)γ(x), for all x∈R).In 2004, Zaidi, Ali and Ashraf researched the map which has the properties of Jordan left (θ,θ)-derivation on some special Jordan ideals of prime rings as followsTheorem 2.7 Let R be a 2-torsionfree prime ring and let J be a Jordan ideal and a subring of R. Ifθis an automorphism of R andδ:R→R is an additive mapping satisfyingδ(u2)= 2θ(u)δ(u), for all u∈J, then either J(?)Z(R) orδ(J)= (0).In 2005, Ashraf researched the map which have the properties of left (0,0)-derivation on some special Lie ideal and obtainedTheorem 2.8 Let R be a 2-torsion free prime ring and let U be a Lie ideal of R such that u2∈U, for all u∈U. Suppose thatθis an automorphism of R. If d: R→R is an additive mapping satisfying d(u2)= 20(u)d(u) for all u∈U, then either d(U)= (0) or U(?)Z(R).Theorem 2.9 Let R be a 2-torsion free prime ring and U a Lie ideal of R such that u2∈U, for all u∈U. Suppose thatθis an automorphism of R. If d: R→R is an additive mapping satisfying d(u2)= 2θ(u)d(u) for all u∈U, then d(uv)=θ{u)d(v)+0(v)d(u) for all u,v∈U.In 2008, Ashraf and Ali discussed the structure of a generalized Jordan left derivation in a 2-torsionfree prime ring as followsTheorem 2.10 Let R be a 2-torsionfree prime ring. Let G:R→R be a generalized Jordan left derivation with associated Jordan left derivationδ:R→R. Then every generalized Jordan left derivation is a generalized left derivation on R.Theorem 2.11 Let R be a 2-torsionfree prime ring. If R admits a generalized left derivation with associated Jordan left derivation 8, then either 8= 0 or R is commutative.