Maps Completely Preserving Commutativity Or Jordan Zero-products On Operator Algebras

The commutativity and Jordan zero-products between the operators are widely used in the fields of mathematics quantum mechanics and so on.Therefore,in recent years,many scholars have a strong interest in the preserver problems on operator space.The preserver problems are to characterize the maps keeping some properties invariant on the operator space,and to obtain the concrete forms of the above maps.The study of preserver problems on different operator spaces or operator algebras has become a hot research object on functional analysis and operator algebra theory.With the deepening of the research,completely preserver problems have been deeply and extensively studied.In this paper,we discuss the surjections completely preserving commutativity on factor von Neumann algebras,and the surjections completely preserving commutativity or Jordan-zero products on nest algebras and J-subspace lattice algebras,respectively.The following are our main results:1.Between factor von Neumann algebras on infinite dimensional complex Hilbert spaces,every surjective map completely preserving commutativity is a nonzero scalar multiple of either a linear isomorphism or a conjugate linear isomorphism.2.Between nest algebras on infinite dimensional complex Banach spaces,every surjective map completely preserving commutativity or Jordan-zero product is a nonzero scalar multiple of either an isomorphism or a conjugate isomorphism.3.Between J-subspace lattice algebras on infinite dimensional complex Banach spaces,the restriction of ? to K completely preserving commutativity or Jordan zero-products is a nonzero scalar multiple of a linear isomorphism for every K ? J((?)).