Acta Applicandae Mathematicae 27: 111-121, 1992. (~) 1992 Kluwer Academic Publishers. Printed in the Netherlands. 111 Krivine's Theorem and the Indices of a Banach Lattice ANTON R. SCHEP Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, U.S.A (Received: 27 April 1992) Abstract. In this paper we shall present an exposition of a fundamental result due to J.L. Krivine about the local smacture of a Banach lattice, In [3] Krivine proved that gp (1 < p < oo) 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]. Mathematics Subject Classifications (1991): 46A40, 46B42 Key words: Krivine's theorem, Banach lattice I. Introduction DEFINITION 1.1 Let 1 <__ p <__ oo. A Banach lattice E has the strong g.p- decomposition property (or satisfies a lower p-estimate) if there exists a constant M such that for all disjoint elements z~,..., Xn in E we have 1 Ilz llp _< M i=1 for p < c~ and max Ilz/ll _- M l<i<n i=1 in case ~ = co.