Jordan Modules And Jordan Ideals Of Completely Distributive Lattice Algebras

In this thesis, we study when Jordan modules of completely distributive lattice algebras are actually modules. The main result of this thesis consists of three parts.In Chapter 3, we find that when a completely distributive lattice satisfies a certain condition, the reflexive Jordan modules of the lattice algebra are actually modules. We also state that if the span of rank one operators in the lattice algebra is weakly dense (especially, such as CDC algebras, nest algebras) in the algebra, then all weakly closed Jordan modules are actually modules. In the end, we point out if all reflexive Jordan modules of a subspace lattice algebra are modules, then a specific structure will not exist in the lattice.In chapter 4, we consider the weakly closed Jordan modules of nest algebras on Banach spaces. The result reads as follows: if every subspace in a nest is topologically complementable, then each weakly closed Jordan module of the associated nest algebra is a module.In chapter 5, we prove that all reflexive Jordan ideals of a completely distributive lattice algebra are actually ideals. We also point out that if the span of rank one operators in the lattice algebra is weakly dense, then all weakly closed Jordan ideals of the algebra are ideals.