The strong closure of $\sigma$-complete Boolean algebras of projections

Abstract.

A result of T. A. Gillespie implies that the strong operator closure of any abstractly $\sigma$-complete Boolean algebra of projections in a Banach space X which does not contain a copy of c ₀ is Bade complete. It is shown that the same conclusion is valid for another (extensive) class of Banach spaces X, namely those which are weakly compactly generated. As a consequence, it follows that a Boolean algebra of projections in a separable Banach space is abstractly $\sigma$-complete iff it is abstractly complete. It is also shown that a Banach space X has the property that the strong closure of every abstractly complete Boolean algebra of projections in X is Bade complete iff X does not contain a copy of $\ell ^\infty \!$.