FACTORIZATIONS OF NATURAL EMBEDDINGS OF LP(N) INTO LR .2.

This is a continuation of the paper by Figiel, Johnson and Schechtman with a similar title. Several results from there are strengthened, in particular: 1. If T is a "natural" embedding of l2n into L1 then, for any well-bounded factorization of T through an L1 space in the form T = uv with v of norm one, u well-preserves a copy of l1k with k exponential in n. 2. Any norm one operator from a C(K) space which well-preserves a copy of l2n also well-preserves a copy of l infinity k with k exponential in n. As an application of these and other results we show the existence, for any n, of an n-dimensional space which well-embeds into a space with an unconditional basis only if the latter contains a copy of l infinity k with k exponential in n .