Factorizations of natural embeddings of l p n into L r . II

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 u 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 lk 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 lk with k exponential in n. 1991 by Pacific Journal of Mathematics.

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