Closed ideals in the Banach algebra of operators on a Banach space
Volume 67 / 2005
Banach Center Publications 67 (2005), 245-264
MSC: Primary 47L10, 46H10; Secondary 47L20, 46B03.
DOI: 10.4064/bc67-0-20
Abstract
In general, little is known about the lattice of closed ideals in the Banach algebra ${\scr B}(E)$ 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 ${\scr B}(F)$, 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.
