Open Access

Properties and , Embeddings in Banach Spaces with 1-Unconditional Basis and

Fixed Point Theory and Applications20092010:342691

DOI: 10.1155/2010/342691

Received: 24 September 2009

Accepted: 3 November 2009

Published: 11 November 2009

Abstract

We will use García-Falset and Lloréns Fuster's paper on the AMC-property to prove that a Banach space that embeds in a subspace of a Banach space with a 1-unconditional basis has the property AMC and thus the weak fixed point property. We will apply this to some results by Cowell and Kalton to prove that every reflexive real Banach space with the property and its dual have the and that a real Banach space such that is sequentially compact and has has the .

1. Introduction

In 1988 Sims [1] introduced the notion of weak orthogonality and asked whether spaces with WORTH have the weak fixed point property . Since then several partial answers have been given. For instance, in 1993 García-Falset [2] proved that if is uniformly nonsquare and has WORTH then it has the , although Mazcuñán Navarro in her doctoral dissertation [3] showed that uniform nonsquareness is enough. In this work she also showed that WORTH plus 2-UNC implies the In both of these cases the space turns out to be reflexive. In 1994 Sims [4] himself proved that WORTH plus -inquadrate in every direction for some implies the and in 2003 Dalby [5] showed that if has and is -inquadrate in every direction for some , then has the .

Recently in 2008 Cowell and Kalton [6] studied properties and in a Banach space , where coincides with WORTH if is separable and in coincides with in if is a separable Banach space. Among other things they proved that a real Banach space with embeds almost isometrically in a space with a shrinking 1-unconditional basis and observed that and are equivalent if is reflexive.

We proved, using property AMC shown by García-Falset and Lloréns Fuster [7] to imply the , that spaces that embed in a space with a 1-unconditional basis have the Combining this with Cowell and Kalton's results we were able to show that a reflexive real Banach space with WORTH and its dual both have , giving a partial answer to Sims' question. We also showed that a separable space such that has and is sequentially compact has the .

2. Notations and Definitions

Let be a real Banach space and a closed nonempty bounded convex subset of .

Definition 2.1.

If , we define
(2.1)
If the set of quasi-midpoints of and in is given by
(2.2)

Definition 2.2.

is the quotient space endowed with the norm where is the equivalence class of in , which we also will denote by . For we will also denote by the equivalence class in . If is as above, let . If is a Banach space and for we define , by
(2.3)

If for we denote by .

It is known that is also closed bounded and convex in and that .

Definition 2.3.

Let be the set of strictly increasing sequences of natural numbers and a nonempty bounded convex subset of a Banach space . A sequence in is called equilateral in , if for every such that for every , the following equality holds in . If and , then
(2.4)

where .

It is easy to see that if is equilateral in , and are as above, then
(2.5)

Now we define the property which interests us in this paper, it was given by García-Falset and Lloréns Fuster in 1990 [7].

Definition 2.4.

A bounded closed convex subset of a Banach space with has the AMC property, if for every weakly null sequence which is equilateral in , there exist , , with for every , such that the set
(2.6)

is nonempty and . is said to have AMC if every weakly compact nonempty subset of with has the AMC property.

3. Embeddings into Spaces with 1-Unconditional Basis and the wFPP

Lin in [8] showed that if has an unconditional basis with unconditional constant , then has the . García-Falset and Lloréns Fuster proved that in fact under these conditions has the AMC property which in turn implies the . We will follow the proof of this closely to establish the next theorem.

Theorem 3.1.

Let be a Banach space and suppose that there exists a Banach space with a -unconditional basis and a subspace of such that where . Then has AMC and thus the .

Proof.

Let be an isomorphism with and . Let be a nonempty weakly compact convex subset of with and . We will show that has the AMC property.

Let be a weakly null equilateral sequence in and let . Then is weakly compact and is weakly null in . Hence there exists a sequence and projections with respect to the basis in with

(a) where ,

(b) for all ,

(c) .

Let be given by . Then clearly for every . Let be given by and and let be . Recall that we will write instead of . By (a), (b), and (c) and since is equilateral we have that

(1) , and ,

(2) , ,

(3) , ,

(4)for all , .

Therefore, since is -unconditional
(3.1)

Thus . Since by hypothesis , we obtain that if and . Next we will show that .

To this effect let . Define and . Then for every
(3.2)
By the unconditionality of and since by (4) we have that ,
(3.3)
Therefore
(3.4)
Now let be such that . Such an element exists since . Recalling that , we obtain that
(3.5)
Using again that , we get
(3.6)
or equivalently
(3.7)
On the other hand, since by (2) , we have
(3.8)
Similarly
(3.9)
By (3.8) and (3.9), and (3.5) we obtain
(3.10)
Hence
(3.11)
Finally, from (3.7) and (3.11) we have and
(3.12)

Therefore, if we conclude that and thus has the AMC property.

Remark 3.2.

It is evident that if the space has a unconditional basis, if is small enough, the above result remains true for some .

4. Some Consequences

There has always been the conjecture that a space with property WORTH has the . We show here that this is correct as long as is reflexive. We also show that property in implies the in Banach spaces so that is sequentially compact and that WORTH together with WABS implies the as well. All these results are consequences of some theorems by Cowell and Kalton [6]. First we need to recall some definitions.

Definition 4.1.

A Banach space has the WORTH property if for every weakly null sequence and every , the following equality holds:
(4.1)

This definition was given by Sims in [1]. The next definition was stated by Dalby [5].

Definition 4.2.

A Banach space has the property if for every weak null sequence and every , the following equality holds:
(4.2)

If is separable and has , this coincides with the property defined in [6].

Definition 4.3.

A Banach space has the Weak Alternating Banach-Saks (WABS) property if every bounded sequence in has a convex block sequence such that
(4.3)

Cowell and Kalton in [6] proved the following three results.

Theorem 4.4.

If is a separable real Banach space, then has the property if and only if for any there is a Banach space with a shrinking -unconditional basis and a subspace of such that .

Dalby [5] observed that property in a space implies property WORTH in and it follows that if is reflexive, then both properties are equivalent. From this and another theorem we are not going to mention here, Cowell and Kalton obtained the next theorem.

Theorem 4.5.

If is a separable real reflexive space, then has property WORTH if and only if for any there is a reflexive Banach space with a -unconditional basis and a subspace of such that .

The third result we are going to use is as follows.

Theorem 4.6.

If is a separable real Banach space, then has both the properties and WABS if and only if for any there is a Banach space with a shrinking -unconditional basis and a subspace of such that .

From this and our previous work it follows directly the following:

Theorem 4.7.

If is a real separable space such that either

(I) has property ,

(II) is reflexive and has property WORTH, or

(III) has both the properties WORTH and WABS,

then has the property AMC and thus the .

It is known that reflexivity implies WABS, and thus (II) implies (III), but we want to include (II) in order to deduce the next corollary. Properties and WABS are inherited by subspaces, and if has property and is sequentially compact, then the dual of any subspace of also has this property. Hence we have the following result.

Corollary 4.8.

Let be a real Banach space.

(1)If is reflexive and has property , then and both have the .

(2)If has properties WORTH and WABS, then has the .

(3)If has and is sequentially compact, then has the .

Proof.

If is a Banach space that satisfies (1), (2), or (3), every separable subspace has the and hence, since the is separably determined, has the . If is separable and reflexive and has property , then it has property as well and this implies by definition that also has property . Therefore both have the and hence the result follows for nonseparable reflexive spaces.

Declarations

Acknowledgment

This work is partially supported by SEP-CONACYT Grant 102380. It is dedicated to W. A. Kirk.

Authors’ Affiliations

(1)
Centro de Investigación en Matemáticas (CIMAT)

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Copyright

© H. Fetter and B. Gamboa de Buen. 2010

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