Some Maps Of Completely Distributive Csl Algebra

In this thesis, we mainly study some important maps of completely distributive and commutative subspace lattice algebras, which are centralizers, Jordan centralizers, module isomorphisms, generalized derivations and generalized Jordan derivations. Also, we investigate the additive maps derivable at some points on the reflexive algebra on a Banach space, whose invariant subspace lattice has the nontrivial smallest element, or have the nontrivial greatest element. The thesis consists of five chapters.In Chapter 1, we introduce some terminology and notation, and summarize the background. Also, we state the main results of this thesis.Chapter 2 studies centralizers and Jordan centralizers on CDC algebras. We prove that every centralizer is quasi-spatial implemented, and every Jordan centralizer is a centralizer. At the end of this chapter, it is shown that every local centralizer on CDC algebras is a centralizer.In Chapter 3, the general form of module isomorphisms on CDC algebras preserving the rank one operators is described. It turns out that such a module isomorphism is trivial.In Chapter 4, we prove that every additive Jordan derivation on CDC algebras is an additive derivation. Then, from this conclusion it follows that every additive generalized Jordan derivation on CDC algebras is an additive generalized derivation.In Chapter 5, we consider the reflexive algebra whose invariant subspace lattice has the nontrivial smallest element, or has the nontrivial greatest element. Firstly, we state every additive map which is derivative at zero point is a sum of an inner derivation and a numeral multiplicative operator. Secondly, we prove that every generalized derivable map at zero is generalized derivable. Thirdly, we prove that every linear derivable map at unit operator on this algebra is a derivation.