Sequential closedness of Boolean algebras of projections in Banach spaces

D. H. Fremlin; B. de Pagter; W. J. Ricker

Studia Mathematica (2005)

  • Volume: 167, Issue: 1, page 45-62
  • ISSN: 0039-3223

Abstract

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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).

How to cite

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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 -

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