Criteria for Closedness of Spectral Measures and Completeness of Boolean Algebras of Projections

The intimate connection between spectral measures and s-complete Boolean algebras of projections in Banach spaces was intensively investigated by W. Bade in the 1950s, w‘‘Linear Operators III: Spectral Operators,’’ Wiley-Interscience, New York, 1971; Chap. XVIIx. It is well known that the most satisfactory situation occurs when the Boolean algebra is actually complete rather than just s-complete. This makes it desirable to have available criteria not just in Banach spaces but also in the nonnormable setting. which can be used to determine the completeness of Boolean algebras of projections. Such criteria, which should be effective and applicable in practice, as general as possible and apply to extensive classes of spaces, are presented in this article. A series of examples shows that these criteria are close to optimal.