Commuting Boolean algebras of projections
Charles A. McCarthy · Pacific Journal of Mathematics · 1961
Introduction* One of the more important problems in the theory of spectral operators is to decide when the sum and product of two bounded commuting spectral operators is again spectral.J. Wermer [7] has shown that the sum and product of two bounded commuting spectral operators on Hubert space is again spectral.N. Dunford [4, Theorem 19] and S. R. Foguel [5, Theorem 7] have shown that if the Boolean algebra of projections generated by the resolutions of the identity of two bounded commuting spectral operators on a weakly complete Banach space is bounded, then the sum and product of these operators are spectral.We therefore wish to determine conditions that insure the boundedness of the Boolean algebra of projections generated by two bounded commuting algebras of projections on a Banach space.We shall show that it suffices that one of the original algebras be strongly complete, countably decomposable, and contains no projection of infinite multiplicity.The example of S. Kakutani [6] shows that the Boolean algebra of projections generated by two commuting, strongly complete, algebras of bound 1, but both of infinite multiplicity on a non weakly complete space, need not be bounded.By slightly reworking his example, we shall show that the order of magnitude of our estimates is sharp, even for spaces of finite dimension.By taking a suitable direct sum of these examples, we obtain a separable reflexive Banach space on which we have two commuting, strongly complete, Boolean algebras of projections, both of bound 1, neither having a projection of infinite uniform multiplicity, but such that the algebra of projections they generate is unbounded.On this same Banach space we also show that the sum and product of two bounded commuting spectral operators need not be spectral.This paper is divided into four sections: the first is devoted to the proof of a combinatorial inequality, the second contains our main theorem on the boundedness of projections, the third section consists of examples.The last section is an appendix to section two.l A combinatorial inequality • The required inequality is the