Concerning Bourgain’s ℓ1-Index of a Banach space

Abstract

A well known argument of James yields that if a Banach space X contains ℓ n 1 's uniformly, then X contains ℓ n 1 's almost isometrically. In the first half of the paper we extend this idea to the ordinal ℓ1-indices of Bourgain. In the second half we use our results to calculate the ℓ1-index of certain Banach spaces. Furthermore we show that the ℓ1-index of a separable Banach space not containing ℓ1 must be of the form ω α for some countable ordinal α.

Note on distortion and Bourgain l1-index

“…The case of Remark 2.3. By Lemma 6.5 (and Remark 6.6(iii)) of [10] the universal constant C (arbitrarily close to 1) in the assumption of Corollary 2.2 is automatic for α = ω γ , with γ a limit ordinal.…”

“…Since I b (Z) > ω α , Lemma 5.8 of [10] implies that I b (Z n , K) ≥ ω α for some K ≥ 1 and any n ∈ N.…”

“…Recall the close relation ( [10]) between I b (X) and I(X), the original Bourgain 1 -index defined as the block index but by trees of not necessarily block sequences: for I(X) ≥ ω ω we have I b (X) = I(X), and if I(X) = ω n+1 for some n ∈ N, then I b (X) = ω n+1 or ω n .…”

“…The p -strongly asymptotic spaces were introduced and studied in [6]. The Bourgain 1 -index and 1 -block index of various spaces in relation to existence of higher order spreading models, distortability and quasiminimality were investigated in [10,12,13,14].…”

“…1 is finitely (almost isometrically) represented on block sequences in X. (ii) For any α < ω 1 by Theorem 5.19 of [10], and Proposition 1.6, any block subspace Y of the Tsirelson type space…”