Equivalent Characterization Of Derivations On Algebras With Involution

In this thesis,we mainly study some equivalent characterization of derivations on algebras with an involution*.It is divided into four chapters.The main contents of each chapter are as follows:In Chapter 1,the introduction mainly introduces the research background,research status and the related concepts.In Chapter 2,we characterize the nonlinear bi-skew Jordan type derivations on*-algebras satisfying certain conditions.Let A be a unital*-algebra which contains the unit I and a nontrivial projection P.In this chapter,we show that,if A satisfies(?)XAP=0 implies X=0 and(?)XA(I-P)=0 implies X=0,then a map Φ:A→A such thatΦ(A1(?)A2(?)…(?)An)=(?)A1(?)…(?)Ak-1(?)Φ(Ak)(?)Ak+1(?)…(?)An for all A1,A2,…,An ∈ A if and only if Φ is an additive*-derivation.In Chapter 3,we characterize the nonlinear mixed Jordan triple*-derivations on*-algebras satisfying certain conditions.Let A be a unital*-algebra which contains the unit I and a nontrivial projection P.In this chapter,we show that,if A satisfies(?)XAP=0 implies X=0 and(?)XA(I-P)=0 implies X=0,then a map Φ:A→A such thatΦ([A·B,C]*)=[Φ(A)·B,C]*+[A·Φ(B),C]*+[A·B,Φ(C)]*for all A,B,C ∈ A if and only if Φ is an additive*-derivation.In Chapter 4,we characterize the second nonlinear mixed Jordan triple*derivations on factor von Neumann algebras.Specifically,let A be a factor von Neumann algebra with dim(A)≥2,then a map Φ:A→A satisfiesΦ([A,B]*·C)=[Φ(A),B]*·C+[A,Φ(B)]*·C+[A,B]*·Φ(C)for all A,B,C ∈A if and only if Φ is an additive*-derivation.