### Remarks on James's distortion theorems

Patrick N. Dowling, Narcisse Randrianantoanina, Barry Turett
1998 Bulletin of the Australian Mathematical Society
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
more » ... 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 [4] , who show that L x [0,1] is arbitrarily distortable, and so /^[0,1] can be equivalently renormed so as not to contain almost isometric copies of i x [0,1] (with its usual norm). As for the E°°r esult, it is known that if a Banach space contains a subspace isomorphic to i 00 then it must contain almost isometric copies of t°°. This result was proved by Partington in [6] . Unaware of Partington's result, Hudzik and Mastylo [2] reproved this result in the setting of function spaces. THE RESULTS Two Banach spaces X and Y are said to be A-isometric (with A ^ 1), if there exists a linear isomorphism T : X ->• Y so that ||T|| \\T~1\\ ^ A. A Banach space X is said to contain almost isometric copies of the Banach space Y if, for each e > 0, there exists a subspace Z of X so that Y and Z are (1 + e)-isometric. PROPOSITION 1. If X is a Banach space which contains a complemented subspace isomorphic to CQ , then X contains complemented almost isometric copies of CQ . 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.