Representation and factorization theorems for almost-Lp-spaces

We extend the notions of p-convexity and p-concavity for Banach ideals of measurable functions following an asymptotic procedure. We prove a representation theorem for the spaces satisfying both properties as the one that works for the classical case: each almost p-convex and almost p-concave space is order isomorphic to an almost-Lp-space. The class of almost-Lp-spaces contains, in particular, direct sums of (infinitely many) Lp-spaces with different norms, that are not in general p-convex – nor p-concave –. We also analyze in this context the extension of the Maurey–Rosenthal factorization theorem that works for p-concave operators acting in p-convex spaces. In this way we provide factorization results that allow to deal with more general factorization spaces than Lp-spaces.