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Uniformly Closed Algebras Generated by Boolean Algebras of Projections in Locally Convex Spaces

Published online by Cambridge University Press:  20 November 2018

Werner Ricker
Werner Ricker*
Affiliation:
Universität Tübingen, T¨ubingen, Federal Republic of Germany
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The theory of operator algebras in Banach spaces generated by Boolean algebras of projections is by now well known. It is systematically exposed in the penetrating studies of W. Bade, [1], [2] and [6, Chapter XVII]. Many of these results, a priori independent on normability of the underlying space, have recently been extended to the setting of locally convex spaces; see [3], [4], [5], [11] and [15], for example.

However, one of Bade's fundamental results, stating that the closed algebra generated by a complete Boolean algebra in the uniform operator topology is the same as the closed algebra that it generates in the weak operator topology, has remained remarkably resistant in attempts to extend it to locally convex spaces. Recently however, a class of Boolean algebras in non-normable spaces, called boundedly σ-complete Boolean algebras, was exhibited in which the analogue of Bade's result is valid, [14; Theorem 5.3].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 5 , 01 October 1987 , pp. 1123 - 1146
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Bade, W. G., On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 (1955), 345–359.Google Scholar
2. Bade, W. G., A multiplicity theory for Boolean algebras of projections in Banach spaces., Trans. Amer. Math. Soc. 92 (1959), 508–530.Google Scholar
3. Dodds, P. G. and de Pagter, B., Orthomorphisms and Boolean algebras of projections., Math. Z. 187 (1984), 361–381.Google Scholar
4. Dodds, P. G. and Ricker, W., Spectral measures and the Bade reflexivity theorem, J. Funct. Anal. 61 (1985), 136–163.Google Scholar
5. Dodds, P. G., de Pagter, B. and Ricker, W., Reflexivity and order properties of scalar-type spectral operators in locally convex spaces, Trans. Amer. Math. Soc. 293 (1986), 355–380.Google Scholar
6. Dunford, N. and Schwartz, J. T., Linear operators, part III: Spectral operators, (Wiley-Interscience, New York, 1971).Google Scholar
7. Kluvanek, I., The range of a vector-valued measure, Math. Systems Theory 7 (1973), 44–54.Google Scholar
8. Kluvanek, I., Conical measures and vector measures, Ann. Inst. Fourier (Grenoble) 27 (1977), 83–105.Google Scholar
9. Kluvanek, I. and Knowles, G., Vector measures and control systems, (North Holland, Amsterdam, 1976).Google Scholar
10. Lewis, D. R., Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157–165.Google Scholar
11. Ricker, W., On Boolean algebras of projections and scalar-type spectral operators, Proc. Amer. Math. Soc. 87 (1983), 73–77.Google Scholar
12. Ricker, W., Closed spectral measures in Fréchet spaces., Internat. J. Math. & Math. Sci. 7 (1984), 15–21.Google Scholar
13. Ricker, W., A spectral mapping theorem for scalar-type spectral operators in locally convex spaces, Integral Equations Operator Theory 8 (1985), 276–288.Google Scholar
14. Ricker, W., Spectral measures, houndedly σ-complete Boolean algebras and applications to operator theory, Trans. Amer. Math. Soc, to appear in 304 (1987).Google Scholar
15. Walsh, B., Structure of spectral measures on locally convex spaces., Trans. Amer. Math. Soc. 720 (1965), 295–326.Google Scholar