Related Maps Preserving K-Jordan Product On Operator Algebras

Preserver problem on operator algebras is one of the most active cross research fields for operator theory and operator algebras.It has achieved a series of beautiful and pro-found results.This paper mainly discussed the maps about the k-Jordan product on operator algebras.Let k?1 be any positive integer.Define the k-Jordan product of a1,a2,…,dk+1 for an associative ring bypk(a1,a2,…,ak,ak+1)=P1(Pk-1(a1,a2,…,ak),ak+1),where p0(a1)=a1,P1(a1,a2)={a1,a2}=a1a2+a2a1 is the usual Jordan product.Particularly,when a2=a3=…=ak=ak+1,the k-Jordan product of a1,a2 is denoted by {a1,a2}k={{a1,a2}k-1,a2)1.Assume that R is a unital ring containing an nontrivial idempotent element e1,the characteristic is not 2,and satisfies e1ae1·e1Re2={0}=e2Re1·e1ae1=>e1ae1=0,e1Re2·e2ae2={0}=e2ae2·e2Re1=>e2ae2=O,where e2=1-e1.Assume that f:R?R is a map.(1)If f is surjective,then f satisfies {f(a),f(e)}k={a,e}k for all a ?R and e?{e1,1-e1,1} if and only if f(a)=f(1)a holds for all a ?R,with f(1)is in the center of R withf(1)k+1=1.As applications,such maps on some important operator algebras,such as triangular algebras,nest algebras,upper triangular block matrix algebras,prime algebras and von Neumann algebras,are characterized completely.(2)If f is bijective and satisfies f(pk(a1,…,ak+1))=pk(f(a1),…,f(ak+1))for all a1,…,ak+1?R then f is additive.Based on this,2-Jordan multiplicative isomorphism on standard operator algebras are characterized.(3)Let X and Y be two Banach spaces over the real or complex filed F with dimen-sions greater than 1,and let A and B be standard operator algebras on X and Y,respec-tively.Assume that ?:A?S is a unital additive surjective map and ?(FP)(?)F?(P)for every rank one idempotent operator P ? A.IF {?(A),?(B)}k=0 for any A,B?A with {A,B}k=0,then either ?(F)=0 holds for all F?F(X),or one of the following is true:(i)?(A)=TAT-1 for all A?A,for T:X?Y is an invertible bounded linear or conjugate linear operator;(ii)?(A)=TA*T-1 for all A ? A,for T:X*?Y is an invertible bounded linear or conjugate linear operator.