An ordinal version of some applications of the classical interpolation theorem

Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space E1 with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space E2 with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we prove that the Szlenk indices of E1 and E2 can be controlled by the Szlenk index of E, where the Szlenk index is an ordinal index associated with a separable Banach space which provides a transfinite measure of the separability of the dual space. Introduction. Let E be a Banach space with a separable dual. Zippin’s theorem ([Z]) shows that E embeds in a Banach space E1 with a shrinking basis, and in [D-F-J-P] it is shown that E is a quotient of a Banach space E2 with a shrinking basis. These two results use the interpolation scheme of [D-F-J-P]. Close to the index introduced by W. Szlenk in [S], the Szlenk index of E, denoted by Sz(E), is defined by slicing the dual unit ball of E with w∗-open sets. Here, we show that we can control the Szlenk indices of E1 and E2 by the Szlenk index of E. More precisely, there exist universal maps φ1 : ω1 → ω1 and φ2 : ω1 → ω1 such that if Sz(E) ≤ α < ω1 then we can choose E1 and E2 with Sz(E1) ≤ φ1(α) and Sz(E2) ≤ φ2(α) (Theorems 3.1 and 4.2). We do not know φ1 and φ2 more precisely, in particular we do not know if φ1 or φ2 can be the identity map. We use tools from descriptive set theory (see [K-L]) and some results from [B1] (see also [B2]). This study is closely related to the Borel regularity of the interpolation scheme of [D-F-J-P]. The first section is devoted to notations and recalls, and the second one to preliminary lemmas. In the third section, we prove that φ1 exists, following [G-M-S] in the proof of Zippin’s theorem. As a corollary, we obtain 1991 Mathematics Subject Classification: Primary 46B20.