Maps Completely Preserving Jordan *- Zero-products And Other Invariants On Operator Spaces

The preserver problems is to characterize the maps keeping some features unchanged on operator spaces in order to infer the concrete forms of the above maps.Generally,the invariants involve function,relation,transformation or subset etc.The mapping also has a variety of forms such as linear,multiplicative and additive.Concepts such as Jordan *-product between operators are crucial in Basic Mathematics,a large number of experts have discussed different*-product between operators under the framework of preserver problems.Subsequently,a series of outstanding achievement are acquired.With the continuous deepening of research,people have begun to explore the issues of completely preserver problems.Affected and inspired by the current results,we choose the Jordan zero-product,Jordan 1-*-zero-product,Jordan ?-*-zero-product and indefinite Jordan 1-?-zero-product as invariants,and study maps on standard operator algebras,factor von Neumann algebras and indefinite inner product spaces of infinite dimensional complex Hilbert spaces respectively in this paper.Thereupon,we get the characterizations of surjection completely preserving these invariants.The following are our main results:1.By characterizing the bijections preserving orthogonality of projection operators in both directions on the infinite dimensional complex Hilbert spaces,we obtain the concrete form of surjections completely preserving Jordan1-*-zero-product between *-standard operator algebras.Our results show that such maps are a scalar multiple of isomorphisms or conjugate isomorphisms.2.By characterizing the bijections preserving orthogonality of idempotents in both directions on the infinite dimensional complete indefinite inner product spaces,we obtain the concrete form of surjective maps completely preserving indefinite Jordan 1-?-zero-product between ?-standard operator algebras.Our results show that such maps are nonzero scalar multiple of isomorphisms or conjugate isomorphisms.3.Every surjection completely preserving Jordan zero-product on infinitedimensional complex Hilbert spaces is a nonzero scalar multiple of either a linear isomorphism or a conjugate linear isomorphism.4.Every surjection completely preserving Jordan 1-*-zero-product on infinite dimensional complex Hilbert spaces is a nonzero scalar multiple of either a linear *-isomorphism or a conjugate linear *-isomorphism.5.Every additive surjection completely preserving Jordan?-*-zero-product on infinite dimensional complex Hilbert spaces is a nonzero scalar multiple of either a linear *-isomorphism or a conjugate linear *-isomorphism.