Operators on Banach Spaces of Bourgain-Delbaen Type

Abstract

We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $\delta >0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $\delta$ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have ${\ell }_1$ dual. The first class of spaces are HI and possess few, but not very few operators. We thus have a negative solution to the Argyros-Haydon question. We remark that all these spaces have finite dimensional Calkin algebra, and we investigate the corollaries of this result. We also construct a space with ${\ell }_1$ Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with ${\ell }_1$ dual, it behaves somewhat differently to the first class of spaces. Finally, we briefly consider shift-invariant ${\ell }_1$ preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples.

Oxford University DPhil Thesis

Cited by in corpus (7)

• Separable Spaces of Continuous Functions as Calkin Algebras
• Unital Banach algebras not isomorphic to Calkin algebras of separable Banach spaces
• Unconditionally saturated Banach space with the scalar-plus-compact property
• A Banach space whose algebra of operators is Dedekind-finite but it does not have stable rank one
• When are full representations of algebras of operators on Banach spaces automatically faithful?
• Fréchet algebras with a dominating Hilbert algebra norm
• A Banach space with an infinite dimensional reflexive quotient algebra L(X)/SS(X)