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

Authors:	W J Ricker

Institution:	School of Mathematics, University of New South Wales, Sydney, New South Wales, 2052 Australia

Abstract:	A classical result of W. Bade states that if $\mathcal {M}$ is any $\sigma -$ complete Boolean algebra of projections in an arbitrary Banach space then, for every $x_0\in X,$ there exists an element (called a Bade functional for with respect to $\mathcal {M})$ in the dual space , with the following two properties: (i) $M\mapsto \langle Mx_0,x'\rangle$ is non-negative on $\mathcal {M}$ and, (ii) whenever $M\in \mathcal {M}$ satisfies $\langle Mx_0,x'\rangle =0.$ It is shown that a Fréchet space has this property if and only if it does not contain an isomorphic copy of the sequence space $\omega = \mathbb C^\mathbb N.$

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