Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-97-04028-8
MathSciNet review: 1415365
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. Bade, W., On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345-360. MR 17:513d
  • 2. Bessaga, C. and Pelzynski A.A., On a class of $B_0-$spaces, Bull. Acad. Polon. Sci. 5 (1957), 375-377. MR 19:562b
  • 3. Diestel, J. and Uhl. J.J. Jnr, Vector measures, Math. Surveys No.15, Amer. Math. Soc., Providence, 1977. MR 56:12216
  • 4. Dodds, P. G., de Pagter, B. and Ricker, W.J., Reflexivity and order properties of scalar-type spectral operators in locally convex spaces, Trans. Amer. Math. Soc. 293 (1986), 355-380. MR 87d:47046
  • 5. Fernández, A. and Naranjo, F., Rybakov's theorem for vector measures in Fréchet spaces, Indag. Math. (New Series), to appear.
  • 6. Jarchow, H., Locally convex spaces, Stuttgart, Teubner, 1981. MR 83h:46008
  • 7. Köthe, G., Topological vector spaces $I$, Springer-Verlag, Heidelberg, 1969.
  • 8. Rybakov, V.I., Theorem of Bartle, Dunford and Schwartz on vector-valued measures, Mat. Zametik, 7 (1970), 247-254 (= Math. Notes7, 147-151). MR 41:5591
  • 9. Walsh, B., Structure of spectral measures on locally convex spaces, Trans. Amer. Math. Soc. 120 (1965), 295-326. MR 33:4690

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47B15, 46G10, 47C05

Retrieve articles in all journals with MSC (1991): 47B15, 46G10, 47C05


Additional Information

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

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

American Mathematical Society