On injective Banach spaces and the spaces $C\left( S \right)$

Abstract

A Banach space is injective (resp. a (Pi space) if every isomorphic (resp. isometric) imbedding of it in an arbitrary Banach space Y is the range of a bounded (resp. norm-one) linear projection defined on Y. In §1 we study linear topological properties of injective Banach spaces and the spaces C(S) themselves; in §2 we study their conjugate spaces. (Throughout, S denotes an arbitrary compact Hausdorff space.) For example, by applying a result of Gaifman [3], we obtain in §1 that there exists a <?i space which is not isomorphic to any conjugate Banach space. We also obtain there that S satisfies the countable chain condition (the C.C.C.) if and only if every weakly compact subset of C(S) is separable. (S is said to satisfy the C.C.C. if every uncountable family of open subsets of S contains two distinct sets with nonempty intersection.) In §2 we classify up to isomorphism (linear homeomorphism) all the conjugate spaces (£*, B**, 5***, etc.) of the (Pi spaces B =L°°(/x) for some finite measure /x, or B = /°°(r) for some infinite set T. (The isomorphic classification of the spaces L°°(jx) for finite measures /z is given in [8].) We also determine in §2 the injective quotients of the above spaces B, and show that every injective Banach space of dimension the continuum, has its dual isomorphic to (Z)*. (Dimension of a Banach space Y (denoted dim Y) equals the minimum of the cardinalities of subsets of Y with dense linear span.) We include some of the proofs; full details of these and other results will appear in [7].