Maps Completely Preserving Skew Jordan Zero-products And Maps Completely Preserving Skew Commutativity

The skew Jordan zero-products and the skew commutativity between the operators are very important concepts in mathematical theory,and they have an important role in the differential equations,quantum mechanics and cryptography.Hence,many scholars have studied the skew commutativity between operators under the framework of preserver problem.The preserver problems is to study the maps keeping some invariant properties(some certain properties,subsets or transformations,etc.)on operator algebras and operator spaces,and to characterize the concrete forms of the above maps.Subsequently,the algebraic or geometric properties of operator algebras or operator spaces are shown by the structure forms of mappings.The study of preserver problems on different operator spaces and operator algebras has become a very active research topic on functional analysis and operator algebra theory,at the same time,some extensively and beautiful results are got.In recent years,completely preserver problems on operator algebras or operator spaces have been studied.Furthermore,many scholars have begun to discuss it deeply.In this paper,we choose the skew Jordan zero-products and skew commutativity as invariants,and study maps on standard operator algebras,standard operator algebras on indefinite inner product spaces and factor von Neumann algebras,respectively.Therefore,we get the characterizations of maps preserving these invariant.The following are our main results:1.On infinite dimensional complex Hilbert spaces,we study the maps completely preserving skew Jordan zero-products or skew commutativity on*-standard operator algebras.We show that such maps must be a scalar multiple of isomorphisms or conjugate isomorphisms.2.On indefinite inner product spaces,we study the maps completely preserving indefinite skew Jordan zero-products or indefinite skewcommutativity.We show that such maps must be a scalar multiple of isomorphisms or conjugate isomorphisms.3.On infinite dimensional complex Hilbert spaces,we characterize the maps completely preserving skew Jordan zero-products or skew commutativity on factor von Neumann algebras and obtain the concrete forms of the maps.