PHYSICAL SCIENCES LIBRARY

The present work is a first step towards a systematic constructive development of the theory of operator algebras over a Hilbert space H. Among the topics investigated in the thesis are locally convex topologies, the extension and characterisation of ultraweakly continuous linear functionals on B(H), and conditions that ensure the (constructive) existence of the adjoint of a bounded linear operator on H. We also study the relationship between a linear subset of B(H) and the dual of its predual, and the comparison of projections in a von Neumann algebra. The two appendices to the thesis deal, respectively, with weak continuity properties and the locatedness of the range of an operator. Ackow ledgements I would like to express my deep gratitude to Professor Douglas Bridges, my super-visor, for his continuous advice, support and encouragement. Special thanks to Professor Crist ian Calude for his scientific cooperation and guidance over the last 10 years. A warm thank you to Associate Professor Hajime Ishihara, Dr. Peter Schuster, Associate Professor Doru ~tefanescu, and Professor loan Tomescu for their help. I would also like to thank my colleagues Robin Havea and Mihaela Baroni for their kind and efficient support, Bas Spitters for his suggestions and Julian Visch for his assistance. Thanks to Viv Bridges, Jane Strachan and Glenys Williams for their friendship and their beeing there for me. The financial support offered by the Royal Marsden Foundation of New Zealand has made possible this project and is acknowledged with thanks. Many thanks to the University of Waikato and the University of Canterbury. Last, but not least, a word of thanks to my parents, Petru and Cornelia Vata, for their love and faith in me