Title:

Continuity of derivations and uniform algebras on odd spheres

The thesis is composed of two separate and distinct parts. Part one is concerned with the problem of determining when certain linear mappings are necessarily continuous with particular attention being given to derivations. Chapter 1 consists of a discussion of the separating space of a linear mapping. Chapter 2 contains a description of the Banach algebra LI[0,1] and some of its properties. In Chapter 3 we consider derivations on L1[0,1], proving in Theorem 3.1 that they are necessarily continuous. In Chapter 4 we extend this result to module derivations and in Theorem 4.2 we give sufficient conditions on a Banach algebra B such that every module derivation from B is continuous. When B is separable and commutative we can improve Theorem 4.2 and then it is easily seen that one of the sufficient conditions is best possible. In Chapter 5 we give sufficient conditions on a Banach algebra B such that certain higher derivations from any Banach algebra onto B are automaticaly continuous. Part two is concerned with the recent result of D.E. Marshall and SY.A. Chang that every closed subalgebra of L7(T) (where T is the unit circle) containing H (T) is a Douglas algebra. Using their techniques we give a proof of this result and discuss generalisations of these ideas and related concepts to higher dimensions. Chapter 6 consists of a discussion of Douglas algebras, functions of vanishing mean oscillation (VMO), Carleson measures and other topics. In Chapter 7 we generalise the space of VMO and provide a characterisation of the new space in terms of Carleson measures. Using these ideas we prove the MarshallChang theorem in Chapters 8 and 9. Chapter 10 discusses the subject of Douglas algebras in higher dimensions. Chapter 11 gives a description of a particular class of Hankel operators on L2(S) (where S is the unit sphere in Cn). In Chapter 12 we characterise the Toeplitz operators amongst operators on H2(S) in terms of an operator equation. In Chapters 10, 11 and 12 we pose several open questions.
