Sequential closedness of Boolean algebras of projections in Banach spaces

D. H. Fremlin, B. de Pagter, and W. J. Ricker. "Sequential closedness of Boolean algebras of projections in Banach spaces." Studia Mathematica 167.1 (2005): 45-62. <http://eudml.org/doc/285116>.

@article{D2005,
abstract = {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).},
author = {D. H. Fremlin, B. de Pagter, W. J. Ricker},
journal = {Studia Mathematica},
keywords = {Boolean algebras of projections; sequential closedness; Bade functionals; band projections; spaces of measures},
language = {eng},
number = {1},
pages = {45-62},
title = {Sequential closedness of Boolean algebras of projections in Banach spaces},
url = {http://eudml.org/doc/285116},
volume = {167},
year = {2005},
}

TY - JOUR
AU - D. H. Fremlin
AU - B. de Pagter
AU - W. J. Ricker
TI - Sequential closedness of Boolean algebras of projections in Banach spaces
JO - Studia Mathematica
PY - 2005
VL - 167
IS - 1
SP - 45
EP - 62
AB - 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).
LA - eng
KW - Boolean algebras of projections; sequential closedness; Bade functionals; band projections; spaces of measures
UR - http://eudml.org/doc/285116
ER -