Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces
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- by W. J. Ricker PDF
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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 $\omega = \mathbb {C}^{\mathbb {N}}.$References
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Additional Information
- W. J. Ricker
- Affiliation: School of Mathematics, University of New South Wales, Sydney, New South Wales, 2052 Australia
- Received by editor(s): March 4, 1996
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2401-2407
- MSC (1991): Primary 47B15, 46G10, 47C05
- DOI: https://doi.org/10.1090/S0002-9939-97-04028-8
- MathSciNet review: 1415365