Boundedness Criteria for Boolean Algebras of Projections

Let E and F be two commuting bounded Boolean algebras of projections on a Banach spaceX. Various geometric conditions onXare established which ensure that the Boolean algebra of projections generated by E and F is also bounded. In particular, this is the case ifXis a Banach lattice or a subspace of ap-concave Banach lattice, where 1⩽p<∞. Several consequences of this boundedness property are also discussed.