ON TREE CHARACTERIZATIONS OF G δ −EMBEDDINGS AND SOME BANACH SPACES S. DUTTA AND V. P. FONF Abstract. We show that a one-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 SE 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 SE * 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 SE 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 in general replaced with the assumption that every normalized w * -null (w-null in (b)) sequence has a subsequence which is a boundedly complete basic sequence. 1. Introduction In [2] Bourgain and Rosenthal introduced the following notion of G δ - embedding. A bounded linear one-to-one operator T : E → Y from a Banach space E into a Banach space Y is called a G δ -embedding if the image T (D) of every norm closed bounded and separable subset D ⊆ E is 2000 Mathematics Subject Classification. Primary 46B03. Research of S. Dutta was supported in part by the Institute for Advanced Studies in Mathematics at Ben-Gurion University of the Negev. Research of V. P. Fonf was supported in part by Israel Science Foundation, Grant No. 139/03. Key words and phrases. G δ -embeddings, boundedly complete basic sequences, trees, regulators, , skipped blocking decompositions, point of continuity property. 1