Published online by Cambridge University Press: 20 January 2009
The property of weak sequential completeness plays a special role in the theory of Boolean algebras of projections and spectral measures on Banach spaces. For instance, if X is a weakly sequentially complete Banach space, then
(i) every strongly closed bounded Boolean algebra of projections on X is complete (3, XVII.3.8, p. 2201); from which it follows easily that
(ii) every spectral measure on X of arbitary class (Σ, Γ), where Σ is a σ-algebra of sets and Γ is a total subset of the dual space of X, is strongly countably additive; and hence that
(iii) every prespectral operator on X is spectral.
(See also (1, Theorem 6.11, p. 165) for (iii).)