Sequential closedness of Boolean algebras of projections in Banach spaces

D. H. Fremlin ¹   ; B. de Pagter ²   ; W. J. Ricker ³

¹ Department of Mathematics University of Essex Wivenhoe Park Colchester CO4 3SQ, United Kingdom
² Department of Applied Mathematical Analysis Faculty EEMCS Delft University of Technology Mekelweg 4 2628CD Delft, The Netherlands
³ Mathematisch-Geographische Fakultät Katholische Universität Eichstätt-Ingolstadt D-85071 Eichstätt, Germany

Studia Mathematica, Tome 167 (2005) no. 1, pp. 45-62

Résumé

Complete and $\sigma $-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950's. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for $\sigma $-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria which characterize when a $\sigma $-complete Boolean algebra of projections is sequentially closed. These criteria are used to show that both possibilities occur: there exist examples which are sequentially closed and others which are not (even in Hilbert space).

Détail

DOI : 10.4064/sm167-1-4

Keywords: complete sigma complete boolean algebras projections acting banach space introduced bade basic every complete boolean algebra projections necessarily closed set strong operator topology here address analogous question sigma complete boolean algebras always sequentially closed set strong operator topology atomic answer shown affirmative general develop criteria which characterize sigma complete boolean algebra projections sequentially closed these criteria possibilities occur there exist examples which sequentially closed others which even hilbert space