Szlenk and w∗-dentability indices of the Banach spaces C([0,

Philip A. H. Brooker

¹

Abstract

a b s t r a c tLet α be an infinite ordinal and γ the unique ordinal satisfying ω ω γ ≤ α < ω ω γ +1 . We show that the Banach space C ([0, α]) of all continuous scalar-valued functions on the compact ordinal interval [0, α] has Szlenk index equal to ω γ +1 and w * -dentability index equal to ω 1+γ +1 .

Three and a half asymptotic properties

“…It was argued in [6] that C 0 (ω ω ξ ), the space of all continuous, scalar-valued functions f on ω ω ξ + = [0, ω ω ξ ] such that f (ω ω ξ ) = 0, is ξ-AUF. Brooker [2] showed that Sz(C 0 (ω ω ξ +)) = ω ξ+1 , so D ξ = T ξ,∞ , and therefore D ξ does not contain any of the other classes mentioned in Theorems 6.2 or 6.3.…”

ξ-asymptotically uniformly smooth, ξ-asymptotically uniformly convex, and (β) operators

“…This shows that α ξ (K) = 0. But since Sz(C([1, ω ω ζ ])) = ω ζ+1 [4], α ξ (co w * (K)) > 0. This example shows that α ξ is not convexifiable.…”

Szlenk and w⁎-dentability indices of C(K)

“…Thus our results recover the known facts that if ω ω ζ ξ < ω ω ζ+1 , Sz(C([0, ξ])) = ω ζ+1 . This was shown by Samuel [15] in the case that ξ is countable, by Lancien and Hájek when ξ < ω 1 ω, and by Brooker [3] in the general case. We also recover the values of Dz([0, ξ]) and Sz(L p (C([0, ξ]))), which was shown for countable ξ in [8], and in the general case by Brooker [3].…”