On dualL ^{1}spaces and injective bidual Banach spaces
 Richard Haydon
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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 ℒ_{1}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 ω_{1}, but which is such thatl ^{1}(ω_{1}) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω_{1}. 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.
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 Hagler, J., Stegall, C. (1973) On Banach spaces whose duals contain complemented subspaces isomorphic to ℰ[0,1]*. J. Functional Analysis 13: pp. 233251 CrossRef
 Haydon, R. (1977) On Banach spaces which contain l1(τ) and types of measures on compact spaces. Israel J. Math. 28: pp. 313324 CrossRef
 Lindenstrauss, J., Tzafriri, L. (1973) Classical Banach Spaces. SpringerVerlag, BerlinHeidelbergNew York
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 Title
 On dualL ^{1}spaces and injective bidual Banach spaces
 Journal

Israel Journal of Mathematics
Volume 31, Issue 2 , pp 142152
 Cover Date
 19780601
 DOI
 10.1007/BF02760545
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
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 Authors

 Richard Haydon ^{(1)}
 Author Affiliations

 1. Brasenose College, Oxford, England