, Volume 167, Issue 1, pp 27-48
Date: 03 Oct 2008

On tree characterizations of G δ-embeddings and some Banach spaces

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Abstract

We show that a one-to-one bounded linear operator T from a separable Banach space E to a Banach space X is a G δ-embedding if and only if every T-null tree in S E has a branch which is a boundedly complete basic sequence. We then consider the notions of regulators and skipped blocking decompositions of Banach spaces and show, in a fairly general set up, that the existence of a regulator is equivalent to that of special skipped blocking decomposition. As applications, the following results are obtained.

(a) A separable Banach space E has separable dual if and only if every w*-null tree in S E * has a branch which is a boundedly complete basic sequence.

(b) A Banach space E with separable dual has the point of continuity property if and only if every w-null tree in S E has a branch which is a boundedly complete basic sequence.

We also give examples to show that the tree hypothesis in both the cases above cannot be replaced in general with the assumption that every normalized w*-null (w-null in (b)) sequence has a subsequence which is a boundedly complete basic sequence.

The research of S. Dutta was supported in part by the Institute for Advanced Studies in Mathematics at Ben-Gurion University of the Negev.
The research of V. P. Fonf was supported in part by Israel Science Foundation, Grant No. 139/03.