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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm167-1-4,
author = {D. H. Fremlin and B. de Pagter and W. J. Ricker},
title = {Sequential closedness of Boolean algebras of projections in Banach spaces},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {45-62},
zbl = {1073.46018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm167-1-4}
}
D. H. Fremlin; B. de Pagter; W. J. Ricker. Sequential closedness of Boolean algebras of projections in Banach spaces. Studia Mathematica, Tome 166 (2005) pp. 45-62. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm167-1-4/