Geometry of banach spaces with diameter two properties and octahedral norms

Resumen

The aim of this dissertation is to analyse and solve several problems in the theory of diameter two properties, octahedrality, and to take advantage of the strong connection between the diameter two properties and the octahedrality of the norm to study new examples of Banach spaces whose norm is octahedral. The content of the thesis is organised in three chapters: -Chapter 1: It is devoted to provide basic background about the diameter two properties, connections with different properties of the geometry of Banach spaces, classical examples of Banach spaces with the diameter two properties and different stability results by taking classical l_p-sums. -Chapter 2: It is devoted to analysing different problems related to the diameter two properties and to present the strong connections between the diameter two properties and the octahedral norms. This section is divided in five sections: In Section 2.1 we deal with the question whether the slice-D2P and the D2P are or not equivalent properties. As a consequence of our study, we deduce that a Banach space which contains an isomorphic copy of c0 can be equivalently renormed to have the slice-D2P but its unit ball contains non-empty relatively weakly open subsets of arbitrarily small diameter. The content of this section is based on the following paper: -J. Becerra Guerrero, G. López-Pérez and A. Rueda Zoca, Big slices versus big relatively weakly open subsets in Banach spaces, J. Math. Anal. App. 428, 2 (2015), 855-865. In Section 2.2 we analyse the possibility that the D2P and the SD2P, which are known to be different properties, are actually different in the extreme way that the slice-D2P and the D2P are. As a consequence, we prove that every Banach space containing an isomorphic copy of c0 can be equivalently renormed to have the D2P but whose unit ball contains convex combinations of slices of arbitrarily small diameter, which shows that the D2P and the SD2P are too different in the extreme way that slice-D2P and D2P are. The content of this section is based on the following paper: -J. Becerra Guerrero, G. López-Pérez and A. Rueda Zoca, Extreme differences between weakly open subsets and convex combination of slices in Banach spaces, Adv. Math. 269 (2015), 56-70. In Section 2.3 we consider almost square Banach spaces, a geometric property of Banach spaces that implies the SD2P and that is introduced in a paper of T. Abrahamsen-J.Langemets-V.Lima. We show that a Banach space X admits an equivalent renorming to be almost square if, and only if, X contains an isomorphic copy of c0, which solves an open problem posed by T. Abrahamsen-J.Langemets-V.Lima. The content of this section is based on the second section of the following paper: -J. Becerra Guerrero, G. López-Pérez and A. Rueda Zoca, Some results on almost square Banach spaces, J. Math. Anal. Appl. 438, 2 (2016), 1030-1040. Section 2.4 is devoted to proving that the norm of a Banach space X is octahedral if, and only if, X* has the w*-SD2P which solves an open problem posed by R. Deville in 1988. The main application of this equivalence in Chapter 2 is to prove that every Banach space which contains an isomorphic copy of l_1 satisfies that, for every r>0, there exists an equivalent renorming on X such that every convex combination of slices of the unit ball of X has diameter, at least, 2-r. This can be seen as a kind of partial answer to the open problem posed by G. Godefroy in 1989 of whether every Banach space containing an isomorphic copy of l_1 admits an equivalent renorming so that the bidual norm is octahedral. The content of this section is based on the following paper: -J. Becerra Guerrero, G. López-Pérez and Abraham Rueda Zoca, Octahedral norms and convex combination of slices in Banach spaces, J. Func. Anal. 266, 4 (2014), 2424-2436. We end the Chapter 2 with Section 2.5, where we exhibit further research, remarks and open questions related to the content of Chapter 2. Chapter 3: It is devoted to analysing the octahedrality of the norm of two different kind of Banach spaces. This chapter is divided in four sections: In Section 3.1 we study the octahedrality of the operator norm in spaces of operators. We get that, given two Banach spaces X and Y , then the operator norm on every subspace H of L(X,Y) containing the finite rank operators is octahedral as soon as the norms of X* and Y are octahedral. We also prove that the assumption of octahedrality on just one factor is not sufficient. As a consequence, all the results of the section imply that the SD2P is preserved from both factors by taking projective tensor product but not from just one of them, which gives a complete answer to a question posed by T. Abrahamsen-V. Lima-O. Nygaard in 2013 in the case of the projective tensor product and the SD2P. The results are based on the paper: -J. Becerra Guerrero, G. López-Pérez and A. Rueda Zoca, Octahedral norms in spaces of operators, J. Math. Anal. Appl. 427, 1 (2015), 171-184, and on the third section of the paper -J. Langemets, V. Lima and A. Rueda Zoca, Octahedral norms in tensor products of Banach spaces, Q. J. Math. 68, 4 (2017), 1247-1260. In Section 3.2 we consider the vector-valued Lipschitz-free spaces and prove that the norm of a Lipschitz-free space F(M,X) is octahedral whenever M is unbounded or it is bounded but it is not uniformly discrete under the additional assumption of extensions of X-valued Lipschitz functions. We also construct vector-valued Lipschitz-free spaces F(M,X) where not only its norm fails to be octahedral but also its unit ball contains points of Fréchet differentiability. The content of the section is based on the paper -J. Becerra Guerrero, G. López-Pérez and A. Rueda Zoca, Octahedrality in Lipschitz-free Banach spaces, Proc. R. Soc. Edinb. Sect. A Math. 148A, 3 (2018), 447-460. In Section 3.3 we focus on octahedrality of the norm of real Lipschtz free spaces, where we introduce a geometric property of metric spaces, the long trapezoid property, which characterises the octahedrality of Lipschitz-free spaces in the sense that a metric space M has the long trapezoid property if, and only if, the norm of F(M) is octahedral. By making use of this characterisation we prove, for instance, that the norm of F(M) is octahedral if M is an infinite subset of l_1. The content of the section is based on the paper: -A. Prochákza and A. Rueda Zoca, A characterisation of octahedrality in Lipschitz free spaces, Ann. Inst. Fourier (Grenoble) 62, 2 (2018), 569-588. We end the chapter with Section 3.4, where we exhibit further research, remarks and open questions related to the content of Chapter 3.