Weakly compact operators and the Dunford-Pettis property on uniform spaces

Let (X, U) be a Hausdorff uniform space and Cb(X) the space of all bounded continuous real-valued functions on X . . The subspace of Cb(X ), consisting of the all uniformly continuous functions with respect to U, is denote by Cub(X ).In this paper we give a characterization of weakly compact operators and ,Q~ continuous defined from Cub(X) into a Banach space E,where (~u is the finest locally convex topology agreeing with the pointwise topology on each uniformly equicontinuous and bounded subsets of .Cub(X). We also show that has the Strict Dunford-Pettis Property and the Dunford-Pettis Property, both under special conditions.


INTRODUCTION AND NOTATIONS
All uniform spaces (X, U) are assumed to be Hausdorff uniform spaces. Basic references for the measures theory on topological spaces are in Varadarajan [7] . We will denote by Cb(X ) the space of all real-valued bounded continuous functions defined on X, and Cub(X ), the subs pace of Cb(X), consists of those functions which are uniformly continuous. ?~l will denote the collection of all uniformly equicontinuous and bounded (U.E.B.) subsets of Cub(X). ,Qu will denote the finest locally convex topology agreeing with the pointwise topology on each H E ~l. A uniform measure on X is defined to be a bounded linear functional on Cub(X) which is pointwise continuous on each H E ~'l (see ~1~ , ~Z~ , ~4~ ) and the space of all uniform measures will be denoted by Mu(X ). It is well known that the dual of is Mu(X). Let M(X ) be the dual of (Cub(X), ~~.~~) , where the ~~.~~ denotes the supremum norm. We denote by H-top the locally convex topology on M (X ) of uniform convergence on the U.E.B. sets. Let Md(X) be the subspace of M(X ) generated by the Dirac measures. It was proved in [4] that the H-top closure of Md(X ) is the space Mu (X ).
We denote by ~3~ the locally convex topology on Cb(X ) agreeing with the compactopen topology on each norm-bounded subset of Cb(X). Sentilles [6] proved that the ' dual of is the space Mt(X) of all tight measures on X . One of the Sentilles's results that we will use here is the following: "a subset A of Mt(X) is uniformly tight if, and only if, A is 03B2t-equicontinuous. It is also known that Cub(X) is 03B2t-dense on Cb(X) (see [I]).
The proof of the following technical lemma is a simple verification and it will be omitted.. Lemma 1. The canonical mapping ~ : : defined by f --> f is an isomorphism, where X denotes the completion of (X , Ll ) and f is the unique uniform extension of f to A~. .

WEAKLY COMPACT OPERATORS
Let E be a Banach space. In this section we will study E-valued linear normcontinuous operators on Cub(X), in particular, weakly compact operators. Definition 1. Let T be a E-valued linear continuous operator on . We shall say that T is a tight additive operator if its restriction to the unit ball of Cub(X) is continuous for the compact-open convergence topology.
Note that T is tight additive if, and only if, T is 03B2t-continuous and so, by the density of Cub(X ) in Cb(X) via the topology T has a unique continuous extension to Cb(X).
From now on we will assume that T : Cub(X) -+ E is a weakly compact operator, that is, T transforms the unit ball of Cub (X) into a relatively weakly compact subset of E. By the denseness, = = Mt(X), the space of all tight measures (see [6]), and by the weakly compactness of T, C E. On the other hand, T has a unique 03B2t-continuous extension T to Cb(X) and, by the latter, Then, T is also a weakly compact operator.
The following theorem will give a characterization of tight operators and the proof is based on the well known result which says that /3u and the norm topology have the same bounded sets (see [1]). . Theorem 2. Suppose that U is metrizable and (X, u) is complete. Then, the following statements are equivalent : In the next theorem we shall use the following notations: If (X, U) is a uniform A space and d is uniformly continuous pseudometrics (u.c.p.) on X, then Xd denotes the completion of the metric space which comes from X, d and the corresponding projection, ~rd.
Theorem 3. Let T be a weakly compact E-valued operator defined on Cub(X).

DUNFORD-PETTIS AND STRICT DUNFORD-PETTIS PROPERTY
In this section we will analyze the Strict Dunford-Pettis and the Dunford-Pettis Property of the locally convex space We begin with the definition of these properties which were given by Grothendieck in his well known paper "Sur les applications lineares faiblement compact d'espace du type C(K)",Canad. J. Math. 5(1974), 183-201. Definition 2. We shaU say that a Hausdorff locally convex space E has the Dunford-Pettis Property (resp. Strict Dunford-Pettis Property) if for any Banach space F and every linear continuous and weakly compact operator T : E -~ F, T (C) is relatively compact (resp. {T (xn)~ is Cauchy) in F for any absolutely convex weakly compact subset C (resp. weak Cauchy sequence {xn~) in E. Proof.
First we shall assume that (X , LI ) is a complete metrizable uniform space. Let T be a weakly compact and ,Q~ -continuous linear operator defined from Cub(X) into an arbitrary Banach space F. By Th. 2.2, T is a tight operator and then T admits a unique extension T to Cb(X ) which is ,Dtcontinuous.
We shall first prove that has the Strict Dunford-Pettis Property. Let {fn}n~N be a 03C3 (Cub(X), Mu(X))) 2014 Cauchy sequence in Cub(X). Since Mu(X) = Mt(X) and {fn}n~N is in Cb(X), we have that this sequence is 03C3 (Cb(X), Mt(X ))) -Cauchy. Now, by [3], we know that has the Strict Dunford-Pettis Property, therefore = T fn}n~N is convergent in F. nEN If U is metrizable, then the conclusion follows from Lemma 1.1, since A is ~3,~ -continuous and weakly compact operator. Now, to prove that has the Dunford-Pettis Property, we again assume that (X, U) is complete metrizable uniform space and follows the similar argument given above.
One of the open problems that we still face, is whether or not has the Strict Dunford-Pettis Property. We already proved that the answer is yes if U is metrizable. In the next theorem we will assume that has the Strict Dunford-Pettis Property and we will prove that it has the Dunford-Pettis Property under the condition that X is u-compact.
possesses the Strict Dunford-Pettis Property and X is 03C32014compact, then has the Dunford-Pettis Property. Proof.
Let {Kn}n~N be an increasing sequence of compact subsets of X such that U Kn is dense on X. We will denote by Ln the closed absolutely convex hull of n=1 Kn in Mu(X) (X is a uniform subspace of Mu (X ) ) . Since Kn is a compact subset of Mu(X) in the H-top and {Mu(X), xtop) is complete, we have that Ln is a compact subset of Moreover, {Ln}n~N is an increasing sequence.
We claim that U Ln is H-top dense in Mu(X). In fact, take p e Mu(X) and a n=1 balanced neighborhood V of p. Since Md(X) is H-top dense in Mu(X), V contains P P some element v = 0 3 A 3 0 3 B 1 i 0 3 B 4 x i of Md(X), with x1, x2, ..., xP E X. Suppose that 0 a =03A3 i=~ ~_| 03B1i| 1 (if a = 0, V ~ ( Ln) ~ 0 and we are done) and take neighborhoods Wi P of 03B4xi, i = 1, 2, .., p, such that 03A3 03B1iWi C V. Since Wi n X is a neighborhood of 03B4xi i=1 P P in X, we get 03B4yi E Kni such that E W= n X. Thus, 03B1i03B4yi ~ 03B1iWi C V and the above argument to V, we get V Fn U L~) 7~ 0.
From this, we have that (Mu(X), H -top) is a 03C32014compact space, which implies that r is also a 03C32014compact space. The conclusion of the theorem follows from [3] ~ Th. 1