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.

1. Department of Mathematics
   University of Copenhagen
   Universitetsparken 5
   DK-2100 Copenhagen \O
   Denmark
   
2. Mathematical Sciences Institute
   Australian National University
   Canberra ACT 0200, Australia