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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: W. J. Ricker
Journal: Proc. Amer. Math. Soc. 125 (1997), 2401-2407
MSC (1991): Primary 47B15, 46G10, 47C05
MathSciNet review: 1415365
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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 $X$ then, for every $x_0\in X,$ there exists an element $x'$ (called a Bade functional for $x_0$ with respect to $\mathcal {M})$ in the dual space $X'$, with the following two properties: (i) $M\mapsto \langle Mx_0,x'\rangle $ is non-negative on $\mathcal {M}$ and, (ii) $Mx_0=0$ whenever $M\in \mathcal {M}$ satisfies $\langle Mx_0,x'\rangle =0.$ It is shown that a Fréchet space $X$ 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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Additional Information

W. J. Ricker
Affiliation: School of Mathematics, University of New South Wales, Sydney, New South Wales, 2052 Australia

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04028-8
PII: S 0002-9939(97)04028-8
Received by editor(s): March 4, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society