THE l-INDEX OF TSIRELSON TYPE SPACES

Let E be a separable Banach space not containing a copy of l. The complexity of the l(n)’s inside E may be measured by Bourgain’s l-index [3] or by locating so called lα-spreading models [8]. It is easy to see that the existence of l 1 α-spreading models implies a large l-index. In general, the implication is not reversible [7, Remark 6.6(i)]. However, suppose that T is the standard Tsirelson space constructed by Figiel and Johnson [5] (the dual of the original Tsirelson space [10]). It is known that there is a constant K such that every normalized block basic sequence in T is K-equivalent to a subsequence of the unit vector basis of T (see e.g., [4]). Using this observation, one can show that the existence of l-block trees in T with large indices leads to the existence of large lα-spreading models. The result can be used to calculate the l-index of T . In this paper, we show that a similar method can be applied to certain general Tsirelson type spaces. In particular, it is shown that if ω1 > α = ω α1 ·m1 + · · ·+ ωn ·mn in Cantor normal form and αn is not a limit ordinal, then there exists a Banach space whose l-index is ω. This gives a partial answer to Question 1 in [7]. If M is an infinite subset of N, denote the set of all finite, respectively infinite subsets of M by [M ], respectively [M ]. A subset F of [N] is hereditary if G ∈ F whenever G ⊆ F ∈ F . F is spreading if whenever F = {n1, . . . , nk} ∈ F with n1 < · · · < nk and m1 < · · · < mk satisfies mi ≥ ni for 1 ≤ i ≤ k then {m1, . . . ,mk} ∈ F . F is compact if it is compact in the product topology in 2. A set F of finite subsets of N is called regular if it has all three properties. If E and F are finite subsets of N, we write E < F , respectively E ≤ F , to mean maxE < minF , respectively maxE ≤ minF (max ∅ = 0 and min ∅ = ∞). We abbreviate {n} < E and {n} ≤ E to n < E and n ≤ E respectively. Given F ⊆ [N], a sequence of finite subsets {E1, . . . , En} of N is said to be F -admissible if E1 < · · · < En and {minE1, . . . ,minEn} ∈ F . If M and N are regular subsets of [N], we let