In general, little is known about the lattice of closed ideals in the Banach algebra ℬ(E) of all bounded, linear operators on a Banach space E. We list the (few) Banach spaces for which this lattice is completely understood, and we give a survey of partial results for a number of other Banach spaces. We then investigate the lattice of closed ideals in ℬ(F), where F is one of Figiel's reflexive Banach spaces not isomorphic to their Cartesian squares. Our main result is that this lattice is uncountable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-20,
author = {Niels Jakob Laustsen and Richard J. Loy},
title = {Closed ideals in the Banach algebra of operators on a Banach space},
journal = {Banach Center Publications},
volume = {68},
year = {2005},
pages = {245-264},
zbl = {1083.47062},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-20}
}
Niels Jakob Laustsen; Richard J. Loy. Closed ideals in the Banach algebra of operators on a Banach space. Banach Center Publications, Tome 68 (2005) pp. 245-264. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-20/