Remarks on James's distortion theorems

Abstract

If a Banach space X contains a complemented subspace isomorphic to Co (respectively, I 1 ), then X contains complemented almost isometric copies of Co (respectively, (}). If a Banach space X is such that X* contains a subspace isomorphic to L^O, 1] (respectively, t°°), then X" contains almost isometric copies of L l [0,1] (respectively, £°°).In [3], James proved that if a Banach space contains a subspace which is isomorphic to Co (respectively, 1}), then it contains almost isometric copies of Co (respectively, I 1 ). In this short note we shall prove complemented versions of these results and show that a dual Banach space containing a subspace isomorphic to L 1^, 1] (respectively, £°°) must contain almost isometric copies of L^O, 1] (respectively, t°°). In particular, the L^O, 1] result is in sharp contrast to a result of Lindenstrauss and Pelczyriski PROOF: Let Y be a complemented subspace of X which is isomorphic to c 0 . Let P be a bounded linear projection from X onto Y.

Banach spaces with the Daugavet property

“…The proof even shows the stronger result that X contains asymptotically isometric copies of 1 in the sense of [10].…”

A note on asymptotically isometric copies of $l^1$ and $c_0$

“…In [4] it is shown that an isomorphic l 1 -copy does not necessarily contain asymptotically isometric l 1 -copies although by James' classical distortion theorem it always contains almost isomorphic In the present note we modify a construction of Godefroy in order to show that every nonreflexive subspace of any L-embedded Banach space contains an asymptotic l 1 -copy and thus, in particular, fails the fixed point property. Analogous results hold for c 0 and M-embedded spaces.…”

“…In [4] it is shown that an isomorphic l 1 -copy does not necessarily contain asymptotically isometric l 1 -copies although by James' classical distortion theorem it always contains almost isomorphic l 1 -copies. (For definitions see below.)…”

“…Following [4,5,6] we say that (x n ) spans l 1 asymptotically isometrically (or just that (x n ) spans l 1 asymptotically) if there is a sequence (δ n ) in [0, 1[ tending to 0 such that…”

“…(Note that in the definition of asymptotic c 0 's and l 1 's in [4] the sequence (δ n ) is supposed to be decreasing but that this difference is not essential.) Finally we say that a Banach space is isomorphic (respectively almost isometric, respectively asymptotically isometric) to l 1 (to c 0 ) if it has a basis with the corresponding property.…”