Research Of Problems Related To Quasi-product Preserving Maps On B(X)

We can define a new operation by the addition and multiplication in a ring R, (?)x, y∈R,xoy=x+y-xy, we call it the quasi-product. In the ring quasi-product is a very important operation, such as the research of Jacobson radical and lie quasi-nilpotent. Quasi-product forms a semigroup structure, which becomes a semigroup of operators in operator algebra. The main purpose of this paper is to describe the characteristics of quasi-product isomorphic on a class of operator algebras.We characterize the quasi-product isomorphism on B(X) in the first. We found that a finite rank operator in B(X) can be written as a quasi-product of finite rank one idempotent operators. Thus the problem of describe the characteristics of maps preserving quasi-product can be reduced to find the characteristics of maps on rank one idempotent operator spaces preserving quasi-product. Using the known conclusions of maps preserving commutativity on rank one idempotent operators, we get the main conclusions of this chapter:let X be a complex Banach space, dim X> 2, and φ is quasi-isomorphism on B(X), then the flowing conclusions mast be true.(1) if dim X=∞, then there exists a bounded invertible linear or conjugate linear operator T on X, which makes(2) if 2< dim X<∞, dim X=n, let B(X)=Mn, there is a ring isomorphism Ï„ on C and an invertible matrix TMn, which makesWe also study the features of maps on B(H) preserving quasi-unitary operators. Let H be a complex Hilbert space, and φ:B(H)â†'B(H) is a additive surjective map. If φ be a bilateral preserving quasi-unitary operator, then the following con-ditions are equivalent,(1) φ(P) bilateral preserving idempotent operators,(2) φ(P) bilateral preserving projection operator,(3) φ(P) bilateral preserving the orthogonality of idempotent operators.(4) φ(P) bilateral preserving the partial order of idempotent operators.