Mappings On Operator Algebras And Quantum Logics

This thesis is divided into two parts.In the first part we focus on mappings on operator algebras both in Banach spaces and in Hilbert spaces. We give some conditions under which mappings are derivations on operator algebras in Chapter 2 and 3. Generalized Jordan derivations and generalized Jordan triple derivations on prime rings and standard operator algebras are introduced and studied in Chapter 4. In Chapter 5 we introduce the concepts of topological reflexivity in several topologies, namely, the discrete, the norm, the strong and the weak operator topology. We prove that the spaces of (α, β)-derivations on certain operator algebras are topologi-cally reflexive in the weak operator topology. The automorphisms and (α,β)-derivations of reflexive algebras in Banach spaces are also characterized in this chapter. In Chapter 6, we pay our attention to additive rank one nilpotence preservers on B(H). As application of this result, we characterise many additive perservers on B(H). In Chapter 7 we study the two-dimensional case of Jordan triple mappings on standard operator algebras in Banach spaces.In the second part of this thesis we maily study the mappings on quantum logics and some related subjects. With the help of D-homomorphisms and D-antihomomorphisms of D-posets we study the relation between ideals and filters in D-posets in Chapter 8. We also consider the sets of states on D-posets and the relation between sets of states and the poset structure of D-posets. The definitions of supports, local filters and local ideals are also introduced and their interrelation is considered. In the last chapter we study the ideals, filters, supports, local ideals and local filters in pseudoeffect algebras.