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....
We create a new family of Banach spaces, the James-Schreier spaces, by amalgamating two important classical Banach spaces: James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. We then investigate the properties of these James-Schreier spaces, paying particular attention to how key properties of their 'ancestors' (that is, the James space and the Schreier space) are expressed in them. Our main results include...
A properly infinite C*-algebra has no non-zero traces. We construct properly infinite Banach *-algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly "close" to the class of C*-algebras, in the sense that they can be hermitian or *-semisimple.
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