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This thesis consists of two separate and distinct parts. Part One is concerned with the problem of characterizing of homomorphisms and derivations on the algebra L1(w). Chapter 1.1. is on general properties of L1(w). In this chapter we prove that every continuous endomorphism of L1(w) has an extension to a continuous endomorphism of M(w). In Chapter 1.2 we characterize isomorphisms from one semi-simple algebra L1(wl) onto another semi-simple algebra L'(W2). In this chapter we also study the endomorphisms of L1 (R+). In Chapter 1.3 we characterize the isometric isomorphisms of a radical L1 (w). We also find a necessary and sufficient condition for two radical algebras L1(w1) and L1(w2) to be isometrically isomorphic. Chapter 1.4 is on derivations of L1(w). In this chapter we characterize derivations on a radical L1 (w) and we find necessary and sufficient conditions on w for the existence of non-zero derivations. Part Two is on isometric representations of the algebras M(G). The main results of this part are in Chapter 2.2. In this chapter we prove that there is an isometric isomorphism from M(G) into BB(H) and the algebra L1 (G) is not isometrically isomorphic with an algebra of operators on a Hilbert space.

The fundamental link between prespectral measures and Banach function spaces is to be found in a theorem of T.A. Gillespie which relates cyclic spaces isomorphically to certain Banach function spaces. We obtain here an extension of this result to the wider class of precyclic spaces. We then consider the properties of weak sequential completeness and reflexivity in Banach function spaces: necessary and sufficient conditions are obtained which in turn, via the aforementioned isomorphisms, both extend and simplify analogously formulated existing results for cyclic spaces. Finally the concept of a homomorphism between pairs of Banach function spaces is examined. The class of such mappings is determined and a complete description obtained in the form of a (unique) disjoint sum of two mappings, one of which is always an isomorphism and the other of which is arbitrary in a certain sense, or null. It is shown moreover that the isomorphic component itself is composed of two other isomorphisms in a manner analogous to the geometrical composition of a rotation and a dilatation.

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Action structures have been proposed as an algebraic framework for models of concurrent behaviour. In this thesis, refinements of action structures are developed, providing an abstract treatment of the structural aspect of processes, as well as a setting in which to study their dynamics. Concrete models of concurrent computation such as Petri nets and the π-calculus have been cast as action structures in a uniform manner, giving rise to a concrete class of action structures, called <I>action calculi</I>. As a result, action calculi are here adopted as the point of departure towards an abstract algebraic treatment of process construction and concurrent computation. The refinement of action structures to <I>control structures</I> gives a semantic space for action calculi; and includes a semantic account of names, based around a semantic counterpart to the syntactic notion of free names called <I>surface</I>. Two variants of action calculi are explored in analogous fashion. Present in these variants are some intuitively appealing aspects, such as greater expressivity of dataflow; a semantic treatment of name hiding or restrictions; and, in one of the variants, garbage collection of restricted but unused names and characterisations of surface in terms of restriction. While the treatment of process constructors reveals rich structural issues, the algebraic framework given by control structures provides considerable support for studying the dynamical aspects of processes. In particular, it allows a comparison of diverse action calculi upon their dynamic properties; illustrated here is a method achieving this. The method involves an examination of action calculi dynamics through the images of the calculi on a common <I>static</I> model called a <I>classifier</I>.

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 S-Y.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 Marshall-Chang 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.

Hardy’s inequality is a basic inequality in the theory of the Hardy spaces. In this thesis, we outline the development of the Hardy spaces from their complex analytic roots to their real variable interpretation of the 1960’s and 1970’s and we then prove a generalisation of the classical Hardy’s inequality in the context of Rⁿ under polynomial changes of variables. Central to our proof of this generalisation is a weighted restriction theorem for polynomial curves, which is global in a certain sense.  We also give a simpler proof which only works in 2 dimensions.

This thesis is concerned with the study of splitting for bounded and completely bounded Hochschild cohomology of von Neumann algebras. Having as a starting point the notions of a split and a split exact complex, which are standard in homological algebra, we define five types of splitting for the (completely) bounded Hochschild cohomology group of <i>A,</i> with coefficients in <i>X, Hⁿ_c(b) (A,X). </i>In general we could say that the study of splitting is the study of the invertibility of the coboundary map ∂<i>ⁿ</i>. We show that all types of splitting are closely connected to geometric properties of the space of n-boundaries <i>Bⁿ_*(A, X)</i> and of the space of n-cocycles <i>Zⁿ_*(A,X)</i> and discuss the relation between the different types of splitting. Then we define module actions on spaces of maps from, into and between <i>A</i>-modules. Given a <i>A</i>-module <i>X</i> and a space <i>Y</i> we make <i>L¹_*</i> (<i>Y, X</i>) into an <i>A</i>-module containing <i>X</i>. The modules <i>L¹_*</i>(<i>Y, X</i>) inherit duality and normality from the module <i>X</i>; the completely bounded case is particularly interesting since we have to define a matricial norm structure on the tensor product of two matricially normed spaces <i>U</i> and <i>V</i> such that the tracial dual of <i>U</i><i>V</i> is completely isometrically isomorphic to the space of completely bounded maps from <i>U</i> into the tracial dual of <i>V</i>. On the other hand we define a module structure on <i>L¹_*</i> (<i>X, Y</i>) which generalises the notion of the dual <i>A</i>-module of <i>X</i>. The completely bounded case is again non-trivial because we must consider a new matricial norm structure on <i>L¹_*</i>(<i>X, Y</i>) generalising the matricial norm structure of the tracial dual. Duality and normality of <i>L¹</i>_* (<i>X, Y</i>) are also discussed. We continue by studying the relation between splitting and the modules <i>L¹_*</i> (<i>Y, X</i>).

This thesis is concerned with some problems in three areas of Banach algebra theory. These are dealt with separately in Chapters 2, 3 and 4. Chapter 2 is concerned with certain automatic continuity problems for homomorphisms and derivations on Banach algebras. The main result is that if there exists a discontinuous homomorphism from a Banach algebra onto a semi-prime Banach algebra, or a discontinuous derivation on a semi-prime Banach algebra, then there exists a topologically simple radical Banach algebra. The main result of Chapter 3 is that there are no Jordan derivations which are not also associative derivations on any semi-prime algebra over a field not of characteristic 2. It follows from this that every Jordan derivation on a semi-simple Banach algebra is a derivation, and therefore continuous. The background to Chapter 4 is a theorem which states that if A is a C'-algebra with identity, acted on by a group G of isometric automorphisms in such a way that A is G-abelian, then the mot of G-invariant states of A is a simplex. This was proved by Lanford and Ruelle in connection with the C*-algebra approach to statistical mechanics. Methods are developed to provide an alternative proof of this result and to investigate the possibility of similar results holding in special cases when A is not a C*-algebra.

A Euclidean <I>SU</I>(2) monopole consists of a connection and Higgs field on an <I>SU</I>(2) bundle over π<SUP>3</SUP>, satisfying certain partial differential equations. Monopoles may equivalently be described in terms of holomorphic vector bundles on twistor space, algebraic curves in twistor space, rational maps, or solutions to Nahm's equations (a set of ODEs for matrix-valued functions), all satisfying some further conditions. Research by Atiyah, Donaldson, Hitchin, Nahm and others has provided a beautiful and relatively complete picture of these different viewpoints and the links between them. Monopoles have also been studied on hyperbolic space π<SUP>3</SUP>, although the corresponding picture in this case is less well understood. One difficulty is that the conditions which must be imposed in order for all the various correspondences to be valid have not yet been completely determined. A partial answer is given in Chapter 2, where it is proved that any hyperbolic monopole arising from a spectral curve satisfies a certain natural boundary condition. The proof uses the algebraic geometry of the spectral curve and is similar to Hurtubise's proof of the analogous result in the Euclidean case. A large part of this thesis concentrates on the "Braam-Austin" description of hyperbolic monopoles. This is the hyperbolic version of Nahm's description of Euclidean monopoles; a monopole corresponds to a pair of discrete matrix-valued functions satisfying some difference equations. Euclidean monopoles appear as limits of hyperbolic monopoles as the curvature of π<SUP>3</SUP> tends to zero. This "Euclidean limit" is described geometrically and is studied in terms of Braam-Austin data. Explicit conditions are given for such a sequence to have a subsequence converging to a Euclidean monopole. The result depends on a conjecture (§ 4.5) about properties of Braam-Austin monopole solutions.