This book presents a systematic investigation of the theory of those commutative, unital subalgebras (of bounded linear operators acting in a Banach space) which are closed for some given topology and are generated by a uniformly bounded Boolean algebra of projections. One of the main aims is to employ the methods of vector measures and integration as a unifying theme throughout. This yields proofs of several classical results which are quite different to the classical ones. This book is directed to both those wishing to learn this topic for the first time and to current experts in the field.
Includes bibliographical references (p. -123) and index.
Preface.- Introduction.- Vector measures and Banach spaces.- Abstract Boolean algebras and Stone spaces.- Boolean algebras of projections and uniformly closed operator algebras.- Ranges of spectral measures and Boolean algebras of projections.- Integral representation of the strongly closed algebra generated by a Boolean algebra of projections.- Bade functionals: an application to scalar-type spectral operators.- The reflexivity theorem and bicommutant algebras.- Bibliography.- Appendix.- List of symbols.- Subject index.
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