Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

Abstract

A classical result of W. Bade states that if M is any a-complete Boolean algebra of projections in an arbitrary Banach space X then, for every xo G X, there exists an element x' (called a Bade functional for xo with respect to M) in the dual space X', with the following two properties: (i) M F-* (Mxo, x') is non-negative on M and, (ii) Mxo = 0 whenever M E M satisfies (Mxo, x') = 0. It is shown that a Frechet space X has this property if and only if it does not contain an isomorphic copy of the sequence space w = CN. A Boolean algebra M of selfadjoint projections in a Hilbert space H has the property that for every x0 C H the inner product (Exo, x0), with E C M, is nonnegative and vanishes only if Exo = 0. A satisfactory extension to the Banach space setting of this useful property of the inner product in Hilbert spaces is the following remarkable result of W. Bade, [1, Theorem 3.1]. Theorem 1. Let M be a a-complete Boolean algebra of projections in a Banach space X. Then, for each xo C X, there exists a continuous linear functional x' E X' (called a Bade functional for x0 with respect to M) with the properties (i) (Mxo, x') > 0, for all M C MA, and (ii) if (Mxo, x') = 0 for some M c M, then Mx0 = 0. Theorem 1 fails to be true in the non-normable setting, even in Frechet (locally convex) spaces. Indeed, the Boolean algebra M generated by the co-ordinate projections in the Frechet space w of all complex sequences, equipped with the topology of co-ordinatewise convergence, is u-complete but the element x0 = (1, 1, . . . ) cannot have any Bade functional with respect to M since the continuous dual space w' consists of all complex sequences which have only finitely many non-zero terms. Let M be a a-complete Boolean algebra of projections in a locally convex Hausdorff space X. Then M is said to have Property-(B) if, for every x0 E X, there exists x' E X' satisfying (i) and (ii) of Theorem 1. The space X is said to have the Bade property if every a-complete Boolean algebra of projections in X has Property-(B). Bade's classical theorem above asserts that every Banach space has the Bade property. The above example shows that Frechet spaces in general do Received by the editors March 4, 1996. 1991 Mathematics Subject Classification. Primary 47B15, 46G10, 47C05. (?)1997 American Mathematical Society