Krivine’s Theorem and the Indices of a Banach Lattice

Abstract

In this paper we shall present an exposition of a fundamental result due to J.L. Krivine about the local structure of a Banach lattice. In [3] Krivine proved that £ p (1 ≤ p ≤ ∞) is finitely lattice representable in any infinite dimensional Banach lattice. At the end of the introduction of [3] it is then stated that a value of p for which this holds is given by, what we will call below, the upper index of the Banach lattice. He states that this follows from the methods of his paper and of the paper [5] of Maurey and Pisier. One can ask whether the theorem also holds for p equal to the lower index of the Banach lattice. At first glance this is not obvious from [3], since many theorems in [3] have as a hypothesis that the upper index of the Banach lattice is finite. This can e.g. also be seen from the book [6] of H.U. Schwarz, where only the result for the upper index is stated, while both indices are discussed. One purpose of this paper is clarify this point and to present an exposition of all the ingredients of a proof of Krivine’s theorem for both the upper and lower index of a Banach lattice. We first gather some definitions and state some properties of the indices of a Banach lattice. For a discussion of these indices we refer to the book of Zaanen[7].