This is an announcement for the paper "Operators on Banach spaces of Bourgain-Delbaen type" by Matthew Tarbard.

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.

Archive classification: math.FA

Remarks: Oxford University DPhil Thesis

Submitted from: matthew.tarbard@sjc.ox.ac.uk

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1309.7469

or

http://arXiv.org/abs/1309.7469