Operators on Banach Spaces of Bourgain{Delbaen Type Matthew Tarbard St John's College University of Oxford arXiv:1309.7469v1 [math.FA] 28 Sep 2013 A thesis submitted for the degree of Doctor of Philosophy Michaelmas 2012 To my family, for all their love and support. Acknowledgements Firstly, I would like to thank my supervisor, Professor Richard Haydon, whose invaluable suggestions, help and support made the work in this thesis possible. I especially thank the hard work of my examiners, Professor Charles Batty and Dr Matthew Daws, who provided a very intellectually stimulating viva. Their careful reading of the the- sis also led to the correction of several errors. I have been fortunate enough to present some of the results in this thesis at various semi- nars and conferences. These opportunities, and the people I have met through these events, helped shape my thoughts and ideas, which led to some of the results in this thesis. I would therefore like to thank Dr Bunyamin Sari for inviting me to the BIRS Banach Space Theory Workshop 2012, where I learnt a great deal about Bourgain-Delbaen spaces and other interesting problems. I would also like to thank Professor William B. Johnson, who brought to my attention at the BIRS workshop a corollary of my result that I had not originally spotted. I would also like to thank Professor Charles Batty, Dr Matthew Daws and Dr Niels Laustsen for inviting me to speak at seminars at the Universities of Oxford, Leeds and Lancaster respectively. Special thanks also go to Dr Richard Earl, who has provided me with fantastic teaching opportunities at Worcester College throughout my DPhil, as well as introducing me to a number of great people, notably, Dr Brian King and Dr Martin Galpin. I am thankful to Richard, Brian and Martin for introducing me to bridge, and for the considerable generosity they have shown me (particularly at the Worcester College Bar)! I am of course very grateful to all my friends and family, who have supported me through- out my DPhil. I am particularly thankful to Victoria Mason, who has provided endless encouragement and support whilst writing this thesis, and contributed significantly to the non-academic aspect of my life. Special thanks also go to Stephen Belding, David Hewings, Tanya Gupta and Carly Leighton for always being around to socialise with and making Oxford a more interesting place to live. Finally, I'd like to acknowledge that my doctoral studies were funded by the EPSRC (En- gineering and Physical Sciences Research Council). Abstract The research in this thesis was initially motivated by an outstanding problem posed by Argyros and Haydon. They used a generalised version of the Bourgain-Delbaen construction to construct a Banach space XAH for which the only bounded linear operators on XAH are compact perturbations of (scalar multiples of) the identity; we say that a space with this property has very few operators. The space XAH possesses a number of additional interesting properties, most notably, it has `1 dual. Since `1 possesses the Schur property, weakly compact and norm compact operators on XAH coincide. Combined with the other properties of the Argyros-Haydon space, it is tempting to conjecture that such a space must necessarily have very few operators. Curiously however, the proof that XAH has very few operators made no use of the Schur property of `1. We therefore arrive at the following question (originally posed in [2]): must a HI, L1, `1 predual with few operators (every operator is a strictly singular perturbation of λI) necessarily have very few operators? 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 δ > 0 and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance δ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have `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 `1 Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with `1 dual, it behaves somewhat differently to the first class of spaces. Finally, we briefly consider shift-invariant `1 preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples. Contents 1 Introduction 1 1.1 Historical background . 1 1.2 Overview of the thesis . 3 1.3 Notation and Elementary Definitions . 5 1.4 Preliminary Results . 6 1.4.1 Basic sequence techniques . 6 1.4.2 Separation Theorems . 9 1.4.3 Hereditary Indecomposability . 10 1.4.4 Elementary Results from Operator Theory . 12 1.4.5 Complexification . 16 1.4.6 Strictly Singular Operators . 17 2 The Bourgain-Delbaen Construction 28 2.1 Introduction . 28 2.2 The generalised Bougain-Delbaen Construction . 29 2.3 The Bourgain-Delbaen Construction . 37 2.4 Connecting the two constructions . 39 2.5 The operator algebras for the original Bourgain-Delbaen spaces . 48 2.5.1 Construction of the basic operator . 49 2.5.2 Non-separability of L(X) ........................ 54 3 Spaces with few but not very few operators 60 3.1 The Main Theorem . 60 3.2 Corollaries of the Main Theorem . 63 3.2.1 On the structure of the closed ideals in L(Xk). 63 3.2.2 The Calkin algebra L(Xk)=K(Xk). 64 3.2.3 Commutators in Banach Algebras . 65 3.2.4 Invariant subspaces. 66 3.3 The Basic Construction . 66 i 3.4 Rapidly Increasing Sequences and the operator S : Xk ! Xk . 78 3.5 Operators on the Space Xk ........................... 88 3.6 Strict Singularity of S : Xk ! Xk ........................ 95 3.7 The HI Property . 98 4 A Banach space with `1 Calkin algebra 100 4.1 The Main Theorem . 100 4.2 The Construction . 102 4.3 L(X1)=K(X1) is isometric to `1(N0) . 119 5 Shift invariant `1 preduals 121 5.1 Introduction . 121 5.2 Connection to the BD construction . 127 5.3 Concluding Remarks . 133 References 134 ii Chapter 1 Introduction 1.1 Historical background Knowledge about the types of bounded linear operator that exist from a Banach space into itself can reveal much about the structure of the underlying Banach space. In particular, it is possible to infer a great deal about the structure of the space X when its operator algebra, L(X), is `small'. The first substantial results in this direction are those of Gowers and Maurey, presented in [16] and [15]. As a motivational example, we consider the space Xgm constructed by Gowers and Maurey in [16]. Here it was shown that all (bounded linear) operators defined on a subspace Y of Xgm (and mapping into Xgm) are strictly singular perturbations of the inclusion operator iY : Y ! Xgm. More precisely, every such operator is expressible in the form λiY + S, where λ is a scalar and S : Y ! Xgm is a strictly singular operator. We shall give a precise definition of a strictly singular operator in Section 1.4.6. For now, it suffices to think of the strictly singular operators as those which are, in some sense, small. Indeed, it is a well known result of Fredholm theory that strictly singular perturbations of Fredholm operators are still Fredholm, with the same index. The representation of operators on subspaces of Xgm just discussed allows us to infer some remarkable structural properties of the space Xgm. We obtain the following: 1. Xgm is not decomposable, that is, it cannot be written as a (topological) direct sum of two of its infinite dimensional subspaces. This is because a non-trivial projection is not expressible as a strictly singular perturbation of the identity. 2. In fact, we conclude by the same argument that Xgm is hereditarily indecomposable, that is to say, no closed infinite dimensional subspace of Xgm is decomposable. It follows that no subspace of Xgm has an unconditional basis, i.e. Xgm has no un- 1 conditional basic sequence. Indeed, if (ei)i=1 were an unconditional basis for some 1 1 1 subspace Y ⊆ Xgm, then we could decompose Y as Y = [e2i]i=1 ⊕ [e2i−1]i=1. (We remark, and contrast this to, the well known fact that every Banach space contains a basic sequence.) 3. Xgm is not isomorphic to any of its proper subspaces. Indeed, it follows from the operator representation and elementary results from Fredholm theory that every op- erator from Xgm to itself is either strictly singular (and thus not an isomorphism), or Fredholm with index 0. In the latter case, if Xgm were isomorphic to a proper subspace Y , then there would be an isomorphic embedding T : Xgm ! Xgm that maps onto Y (we simply take an isomorphism from Xgm ! Y and then compose with the inclusion operator sending Y into Xgm). In particular T is not strictly singular and so must be Fredholm with index 0. However, this is clearly not the case as T is injective but not onto Xgm. One could of course consider the relationship between a Banach space and its operator algebra from a different perspective to that just described. Instead of assuming we have some well behaved properties of the operator algebra and asking what consequence this has for the structure of the underlying Banach space, we may choose to impose some kind of structural conditions on a Banach space and see what affect this has on the associated operator algebra.