Mappings On Reflexive Algebra

This paper introduced a subspace lattice L on Banach space X and a subspace lattice algebra algL associated to the subspace lattice L. Let B be a unital algebra, mapping T:algLâ†'B is linear mapping, satisfy one of the conditions as follows:(Z1)ab=0(?)T(a)T(b)=0,(Z2)ab=ba=0(?)T(a)T(b)=0,(Z3)aob=(?)(a)°T(b)=0. If the subspace lattice L which is on Hilbert space is completely distributive commutative subspace lattice or P-subspace lattice, we prove that under certain conditions, mapping T is an isomorphism or Jordan isomorphism multiplied by a central invertible element. Finally we introduced a von Neumann algebra M with a faithful normal semi-finite trace,which includes measurable operator algebra S(M), locally measurable operator algebra LS(M) and Ï„-measurable operator algebra S(M,Ï„). Moreover we talk about derivations and local derivations on these algebras, especially every derivation on von Neumann algebra of type I has uniquely decomposition.