Ordered Banach Spaces
Even though there are many excellent monographs on Banach spaces, the topic of ordered Banach spaces is not very often included. And even if it is, the authors usually deal only with the more particular case of Banach lattices. The aim of this book is to fill (at least partially) this gap and to provide an up-to-date study of the structure of convex cones contained in Banach spaces and some of the related operators. The book originates from three different sources. The first is conical measures (introduced by G. Choquet and used for developing integral representation theory). The second is the well-established theory of Banach lattices and the third is the Krivine theorem on finite representability of finite dimensional ℓp spaces in Banach lattices. By blending them together, the author obtains an original and instructive contribution to classical functional analysis.
The book starts by presenting various forms of the Hahn-Banach theorem and proving theorems of M. G. Krein, V. L. Klee and T. Ando on cones in Banach spaces. The second chapter is devoted to a presentation of B. Maurey’s theorem on factorization of operators through Lp spaces. An important ingredient here and later on is the Rosenthal lemma. To study convex cones, chapter 3 introduces conical measures and their basic properties. The next part contains the first main results of the book on p-summing operators and their factorization. The author then proceeds with an investigation of representability of finite ℓp spaces in normal cones of Banach spaces. He also shows a relationship between type and cotype of a Banach space with the index introduced in the previous chapter. The author then studies positive operators starting in C(K) spaces and he continues the investigation of p-summing operators on convex cones. The Pietsch inequality and a composition theorem are proved. The last chapter describes the situation of tensor products and positive maps. Five appendices and several open problems complete the book making the presentation rather self-contained.