Sequential closedness of Boolean algebras of projections in Banach spaces
Complete and σ-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 σ-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 σ-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).
- Department of Mathematics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
- Department of Applied Mathematical Analysis, Faculty EEMCS, Delft University of Technology, Mekelweg 4, 2628CD Delft, The Netherlands
- Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, D-85071 Eichstätt, Germany