Problems in functional analysis

This thesis is concerned with two, rather different, areas of analysis. A major part is devoted to problems in distributive lattice theory. Motivated by the Priestley representation theorem for a distributive lattice L we obtain information about the lattices of ideals, filters and congruence relations of L. Using purely lattice-theoretic means we give extension theorems for homomorphisms in the class of pseudocomplemented distributive lattices. These are analogous to the classical Hahn-Banach theorem. Finally, Lρ-spaces are constructed from matrix measures and their properties investigated. Chapter 1 marshals material used in subsequent chapters. In §1.2 the Priestley representation theorem is stated which allows a distributive lattice with zero and identity to be identified with the clopen decreasing subsets of a certain ordered topological space; the relevant theory of ordered spaces is summarized. The class of distributive lattices with pseudocomplementation is defined in §1.3 together with the equational subclasses. The Priestley duality for a pseudocomplemented lattice L is given as well as for the centre and dense set of L.