Maps Completely Preserving *-Zero Products In The *-Standard Operator Algebras

As a method to effectively reveal the natures of operator algebras,the preserver problems have been used to solve the classification problems of operator algebras,and have attracted the attention of mathematics researchers at home and abroad.Mainly through the discussion of preserving a characteristic invariants(function,transformation,determinant,rank,etc.),they achieve the characterizations of problems of preserving the invariant mappings.But the maturity of any theory is gradually developed,and there are certain rules for preserving the developments of problems.At the beginning of the research,the mapping was mainly a linear one that preserved invariants on matrix algebras;in the middle period,the mapping was mainly a linear one that preserved invariants on operator algebras(C~*algebras,von Neumann algebras,and set algebras,etc.);later in the study,mathematical researchers removed the linearity to study additive or more general surjections,which produced good theoretical results.In addition,The relationship contained in its invariants is closely related to quantum mechanics,cryptography,and variational equations,which will inspire us to continue to study and preserving problems.With the continuous deepening and innovation of research,the ideas of completely preserving problems have been proposed.More and more outstanding scholars have begun to use this ideas and methods to study the mappings that completely preserve certain characteristic invariants and get good descriptions.In this paper we takesJordan-?~*-zero product,AB-BA~T=0andA~*B+B~*A=0 as the characteristic invariants and studies the mappings that completely preserve the invariants on*-standard operator algebras,standard operator algebras and*-standard operator algebras.The main results of this article are as follows:1.We characterize general unital surjections completely preservingJordan-?~*-zero product between the*-standard operator algebras of Hilbert spaces on the complex field?,and obtain that the mappings are a constant multiple of isomorphisms or conjugate isomorphisms.The orthogonality of the bilateral preserving projection operators is used to obtain the specific structural forms.2.We research the general surjections completely preserving the products AB-BA~T=0 between standard operator algebras in Hilbert spaces on the complex field?,and the results show that the mappings are ring isomorphisms.3.The general surjections that completely preserve the productsA~*B+B~*A=0between*-standard operator algebras are discussed,and the specific structure of the mappings on the set of projection operators is obtained.