Commuting And Skew-commuting Maps On Operator Algebras

Additive or linear commuting maps and skew-commuting maps are important maps in operator theory and operator algebras,and their structures and properties have been studied by many authors.In this thesis,we mainly study the structures of nonlinear commuting maps,skew-commuting maps and related maps on operator algebras.Let A be any associative ring or algebra and ?:A?A a map.? is called commuting if[?(a),b]=[a,?(b)]=a?(b)-?(b)a for all a,b?A;is called anti-commuting if[?(a),b]=-[a,?(b)]for all a,b?A;is called skew-commuting if {?(a),b}=-{a,?(b)}=-(a?(b)+?(b)a)for all a,b?A;is called anti-skew-commuting if {?(a),b}={a,?(b)} for all a,b?A.(?)(*)Assume that A contains a nontrivial idempotent e1 and the unit element 1,and for any a E A,it satisfies the following conditions:where e2 = 1-e1.In the thesis,we fist give the structures of the nonlinear(skew-)commuting maps and anti-(skew-)commuting maps on A satisfying(*),and then these results are applied to prime rings and some operator algebras.The following results are obtained:1.Let A be a unital 2-torsion free prime ring with a nontrivial idempotent e1.Then a nonlinear map ?:A? A is commuting if and only if ?(a)=za+f(a)for all a?A,where z?Z(A)and f:A?Z(A)is a map;is anti-commuting if and only if ?(a)?Z(A)for all a?A;is skew-commuting if and only if ??0;is anti-skew-commuting if and only if ?(a)=za for all a E A,where z?Z(A).2.Let M be a von Neumann algebra.Then a nonlinear map ?:M?M is commuting if and only if ?(A)=ZA+f(A)for all A E M,where Z?Z(M)and f:M?Z(M)is a map;is skew-commuting if and only if ??0.In addition,if M has no central summands of type ?1,then ? is anti-commuting if and only if ?(A)?Z(M)for all A?M;is anti-skew-commuting map if and only if there exists some Z?Z(M)such that ?(A)=ZA for all A?M.3.Let N be a nest on a Banach space X and AlgN the associated nest algebra.Assume that 0 is a nonlinear map on AlgN.If there exists a non-trivial element in N that is complemented in X,then ? is commuting if and only if ?(A)=?A+f(A)I for all A?AlgN,where ??F and f:AlgN?F is a functional;is skew-commuting if and only if ??0.If ? is anti-commuting or anti-skew-commuting,then ? has no good structure.