- Predual Which Is Not Isometric to a Quotient of C ( Α )

About twenty years ago Johnson and Zippin showed that every separable L1(μ)-predual was isometric to a quotient of C(∆), where ∆ is the Cantor set. In this note we will show that the natural analogue of the theorem for l1-preduals does not hold. We will show that there are many l1–preduals which are not isometric to a quotient of any C(K)-space with K a countable compact metric space. We also prove some general results about the relationship between l1-preduals and quotients of C(K)-spaces with K a countable compact metric space. About 20 years ago Johnson and Zippin [J-Z] proved the following theorem. Theorem. Suppose that X is a separable L1(μ)-predual, then there is subspace Y of C(∆), where ∆ is the Cantor set, such that X is isometric to C(∆)/Y . Because the space X might be C(∆), C(∆) is the smallest L1(μ)-predual that one could use for such a result. If we consider the class of l1-preduals then it is conceivable that a smaller space might be sufficient although the space would need to depend on some measurement of the size of the l1-predual. A natural class of spaces to consider is the spaces C(α) where α is a countable ordinal. (This is the same as the class of C(K)-spaces with K a countable compact Hausdorff space by a classical result of Mazurkiewicz and Sierpiński, [M-S].) Thus one can consider the following question. Question. If X is an l1–predual is there a countable ordinal α such that X is isometric (isomorphic) to a quotient of C(α)? We will show that the isometric question has a negative answer and prove some technical results which are useful for deciding whether an l1–predual is isomorphic to a quotient of C(α). The isomorphic question remains open and at the end of the paper we discuss some variants of the isomorphic problem. Note also that we consider only X for which X is isometric to l1 because even the Johnson–Zippin result is false for isomorphic l1-preduals, [B-D]. Throughout this paper l1-predual will mean a Banach space with dual isometric to l1. If α is an ordinal C(α) will denote the space of continuous functions on the ordinals less than or equal to α with the order topology. If A is a subset of a Banach space, [A] is the norm closed linear span of A. Notation and standard This paper is in final form and no version of it will be submitted for publication elsewhere. 1991 Mathematics Subject Classification. Primary 46E10, Secondary 46B04.. Research supported in part by National Science Foundation grant DMS-8902327 Typeset by AMS-TEX 1