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We develop a discrete framework for the interpolation of Banach spaces, which contains e.g. the well-known real and complex interpolation methods, but also more exotic methods like the $pm$-method, the Radamacher interpolation method and the $ell^p$-interpolation method, as concrete examples. Our method is based on a sequential structure imposed on a Banach space, which allows us to deduce properties of interpolation methods from properties of sequential structures. Our framework has both a formulation modelled after the real and the complex interpolation methods. This allows us to extend results previously known for either the real or the complex interpolation method to all interpolation methods that fit into our framework. As applications of this observation we prove abstract Stein interpolation and the interpolation of intersections for all methods that fit into our framework.
We prove that every Banach space, not necessarily separable, can be isometrically embedded into a $mathcal L_{infty}$-space in a way that the corresponding quotient has the Radon-Nikodym and the Schur properties. As a consequence, we obtain $mathcal
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $ell_1$. We present many examples and several properties of this class. We give some applicatio
Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imp
For every $alpha<omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $omega^{alphaomega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $omega^{alphaomega}$. In order to