Richard Haydon

A non-reflexive Grothendieck space that does not containl∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)*, every weak* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.

Some more characterizations of Banach spaces containing l 1

In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l 1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations: a Banach space B contains no subspace isomorphic to l 1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points .

On dualL 1-spaces and injective bidual Banach spaces

AbstractIn a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l1(τ) embeds in a Banach spaceX if and only ifL1([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 anL1-space, and κ is the density character ofZ*, thenl1(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 thatl1(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

Baire trees, bad norms and the Namioka property

(L{}^1[0,1]^{\omega _1 } ) \oplus \left( {(\sum\limits_{\omega _1 } {{}^ \oplus L{}^1[0,1] \oplus l^1 (\omega _1 )} } \right)_1 ,

Injective Banach Lattices.

, and is a member of a new isomorphism class of dualL1-spaces.

Non-Separable Banach Spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyńskis about Banach spaces containingL1. Let τ be a regular cardinal satisfying the hypothesis that κω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]τ. 2) A Banach spaceX has a subspace isomorphic tol1(τ) if and only ifX∗ has a subspace isomorphic toL1({0, 1}τ). An example is given to show that a more recent result of Rosenthals about Banach spaces containingl1 does not have an obvious transfinite analogue. A second example (answering a question of Rosenthals) shows that there is a Banach spaceX which contains no copy ofl1 (ω1), while the unit ball ofX∗ is not weakly∗ sequentially compact.

On a measure-theoretic problem of Arveson

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.

Countable unions of compact spaces with the Namioka property

LetT be an (into linear) isometry on a (real or complex) Lorentz function spaceLw,p,1≤p<∞. We show that iff andg have disjoint support, thenT f andT g also have disjoint support. Using this result, we give a characterization of the isometries ofLw,p.