On dualL 1-spaces and injective bidual Banach spaces

Abstract

In a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l 1(τ) embeds in a Banach spaceX if and only ifL 1([0, 1]τ) embeds inX *. An extension (to a rather wider class of cardinals) of the basic lemma of that paper is here applied so as to yield an affirmative answer to a question posed by Rosenthal concerning dual ℒ1-spaces. It is shown that ifZ * is a dual Banach space, isomorphic to a complemented subspace of anL 1-space, and κ is the density character ofZ *, thenl 1(κ) embeds inZ *. A corollary of this result is that every injective bidual Banach space is isomorphic tol ∞(κ) for some κ. The second part of this article is devoted to an example, constructed using the continuum hypothesis, of a compact spaceS which carries a homogeneous measure of type ω1, but which is such thatl 1(ω1) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω1. The dual space ℰ(S)* is isometric to $$(L{}^1[0,1]^{\omega _1 } ) \oplus \left( {(\sum\limits_{\omega _1 } {{}^ \oplus L{}^1[0,1] \oplus l^1 (\omega _1 )} } \right)_1 ,$$ , and is a member of a new isomorphism class of dualL 1-spaces.