LINEAR OPERATORS BETWEEN PARTIALLY ORDERED BANACH SPACES AND SOME RELATED TOPICS By

If X is a partially pre-ordered Banach space, with closed positive wedge, there are various questions of interest about this wedge. Is it generating? Is it normal? what ordering does it induce on X? It is possible to deduce some properties of wedges from other. properties of related wedges. we consider Banach spaces of linear operators between two such spaces, and define 11: › 0 iff Tx.,;() whenever x we investigate properties of the positive wedge of these spaces. Firstly we look at spaces of all bounded linear operators. If the range is the reals, then a great deal is known, and a summary of these results, together with the definitions is given in Chapter I. In Chapter II we look at the general case. we find conditions for the wedge to be normal, and certain cases in which it is generating. we also investigate when the wedge is one of some special forms that are of interest, and finally look at the order-structure of the space. The opportunity is taken to state some results for projective tensor products. Chapter III deals with the same problems for spaces of compact operators, generally with range a simplex space. we deal with certain categories of partially ordered Banach spaces in Chapter IV. we determine injective elements in these categories, and also projective elements in certain categories of compact convex sets. Finally we look at the structure of the spaces A(K) in Chapter V, obtaining a characterisation of the set of facially continuous functions as the union of all the subalgebras of A(K). We also generalise the Banach-.Stone theorem, and a result on direct sum decompositions of these spaces. Acknowledgements I wish to express my gratitude to my supervisor, Dr. F. Jellett for his advice and encouragement during the preparation of this thesis. My thanks are also due to Chelsea College for the Research Studentship of which I have been in receipt during part of the preparation of this thesis. CONT.ENTS Chapter I- Partially ordered Banach spaces and their duality. 1.1. Definitions and some special spaces----------------- 1. 1.2. Normality and positive generation 14