PROXIMITY TO L(1) AND DISTORTION IN ASYMPTOTIC L(1) SPACES

For an asymptotic l(1) space X with a basis (x(i)) certain asymptotic l(1) constants, delta(alpha)(X) are defined for alpha < omega(1). delt a(alpha)(X) measures the equivalence between all normalized block base s (y(i))(i=1)(k) of (x(i)) which are S-alpha-admissible with respect t o (x(i)) (S-alpha is the alpha th-Schreier class of sets) and the unit vector basis of l(1)(k). This leads to the concept of the delta spect rum of X, Delta(X), which reflects the behavior of stabilized limits o f delta(alpha)(X). The analogues of these constants under all renormin gs of X are also defined and studied. We investigate Delta(X) both in general and for spaces of bounded distortion. We also prove several re sults on distorting the classical Tsirelson's space T and its relative s. (C) 1997 Academic Press.