In the week from June 12th to June 17th, 1989, the functional analysis group of the Mathematics Department, Johannes-Kepier-Universitat at Linz played host to an international conference on the geometry of Banach spaces which was sponsored by the International Mathematical Union. This conference united 119 participants from 27 countries, including most of the leading specialists in the field, for a week of intensive study and discussion on recent progress. The scientific programme consisted of 23 plenary lectures which were held by a number of experts invited by the members of the organizing committee. In addition, there were a series of half-hour talks held in parallel sessions.

It is the pleasant duty of the editors of the proceedings of the above conference to express their gratitude to the following persons and organisations whose support, financial or otherwise, played an important role in its successful realisation:

The members of the organizing committee

for their help in arranging of the details of the scientific programme.

The following organizations for their financial assistance:

1. – International Mathematical Union

2. – Österreic his dies Bundesministeriu m für Wlssensc haft und Forschung

3. – Linzer Hochschulfonds

4. – Land Oberösterreich

5. – Handelskammer Oberösterreich

6. – Verband der Versicherungsunternehmen Österreichs,

7. – Wiener Städtische Versicherung.

Abstract. Let m be any cardinal. The main result characterizes ℓ1 (m) as the only Banach lattice whose positive cone is metrizable in the weak topology. Two related theorems on ℓ1-sums of Banach spaces are also proved.

Introduction

The three theorems of the paper concern ℓ1-sums of Banach spaces. The first, which states that the Banach space ℓ∞(m) contains the ℓ1-sum of 2m copies of itself, is essentially a translation into Banach space terms of a theorem of Pondiczery on the product topology [10]. The case m = ℵ0 of this result is reminiscent of the general theorem [9] that if X is any separable Banach space containing ℓ1, then X* contains M/(0, 1), the space of finite Borel measures on [0, 1]: the connection resides in the observation that M[0, l] is linearly isomorphic to the ℓ1-sum of c copies of L1 (0, 1). This observation is used to prove Theorem 2, which says that if X is as above then X* contains 2C mutually non-isomorphic closed linear subspaces.

Recall that the unit ball of a Banach space X is metrizable in the weak topology ifX* is separable, and recall too the consequence of the Baire category theorem that if X is infinite-dimensional then the weak topology on X and the weak-star topology on X* are not metrizable. However, there are still some interesting unbounded subsets of Banach spaces that are metrizable in the weak or the weak-star topology. Recall, for example, that the positive cone of the dual of C(K) (that is, the space of continuous functions on a compact metric space K), consisting of the non-negative finite Borel measures on K, is metrizable in the weak-star topology.

Introduction, definitions and discussion of results.

Although the example given by Enflo in 1973 [5] settled the approximation problem and the basis problem for Banach spaces, a number of closely related problems have continued to arouse interest. If X is a separable Banach space, there are a number of natural properties intermediate between X having the approximation property and having a basis.

Let us first make some definitions. Suppose X is a separable Banach space. Then X has the approximation property (AP) if there is a net of finite-rank operators Tα so that Tαx → x for x ∈ X, uniformly on compact sets. is said to have the bounded approximation property (BAP) if this net can be replaced by a sequence Tn; alternatively X has (BAP) if there is a sequence of finite-rank operators, Tn, such that sup Tn∥ > ∞ and Tnx → x for x ∈ X. A sequence Tn with these properties will be called an approximating sequence. If X has an approximating sequence Tn with limn → ∞ ∥Tn∥ = 1 then X has the metric approximation property (MAP).

An important principle [15] that we will use frequently is that if Tn is any approximating sequence for X then there is an approximating sequence Sn satisfying SmSn = Sn whenever m > n and such that for some subsequence Tkn of Tn then limn → ∞∥Tkn − Sn∥ = 0. (See Lemma 2.4 of [15]).

Introduction

Let X,Y always denote infinite Polish spaces and M(X) stands for the Banach space of Radon measures on X with the variation norm. While the isometric type of sublattices of L1(X,μ) for some μ ε M(X) is completely understood (they are either l1,L1[0,1], or l1⊕ L1[0,1], see e.g. [S] III Prop. 11.2) the structure of sublattices of M(X) is much more complicated. Interesting examples of such sublattices are the Henkin-measures on the unit sphere of Cn for n > 1 (see [Rn], Chap. IX), Rajchman measures on the unit circle (see [Ke], Chap. IX), invariant mesures of a family of measurable tranformations of X (see [Ph], Chap. X) and, more generally, the invariant measures of a H-sufficient statistic in the sense of Dynkin (see [Dy], [Ma]). The first two examples are actually bands in M(X) and there is a very nice characterization of such bands in terms of the compact subsets of X that they annihilate due to Mokobodski ([Ke], Chap. IX. 1). The last two examples are usually true sublattices of M(X) isometric to M(0,1).

In this note we characterize sublattices L of the latter kind, (i.e. L is isometric to M(0, 1))in terms of the existence of strongly affine projections, the w*-Radon-Nikodymproperty, martingale compactness, a choquet-type integral representation theorem and finally in terms of the embedding of their unit sphere into M(X) (see section 2 for precise statements).