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∞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∞k with k exponential in
n.
