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Properties
and
,
Embeddings in Banach Spaces with 1-Unconditional Basis and ![](//media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_IEq4_HTML.gif)
Fixed Point Theory and Applications volume 2010, Article number: 342691 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ1_HTML.gif)
If the set of quasi-midpoints of
and
in
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ4_HTML.gif)
where .
It is easy to see that if is equilateral in
, and
are as above, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ7_HTML.gif)
Thus . Since by hypothesis
, we obtain that
if
and
. Next we will show that
.
To this effect let . Define
and
. Then for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ8_HTML.gif)
By the unconditionality of and since by (4) we have that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ9_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ10_HTML.gif)
Now let be such that
. Such an element exists since
. Recalling that
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ11_HTML.gif)
Using again that , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ12_HTML.gif)
or equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ13_HTML.gif)
On the other hand, since by (2) , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ14_HTML.gif)
Similarly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ15_HTML.gif)
By (3.8) and (3.9), and (3.5) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ16_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ17_HTML.gif)
Finally, from (3.7) and (3.11) we have and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ18_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ19_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F342691/MediaObjects/13663_2009_Article_1262_Equ21_HTML.gif)
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.
References
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García-Falset J: The fixed point property in Banach spaces whose characteristic of uniform convexity is less than 2. Journal of the Australian Mathematical Society. Series A 1993,54(2):169–173. 10.1017/S1446788700037095
Mazcuñán Navarro EM: Geometría de los espacios de Banach en teoría métrica del punto fijo, Tesis doctoral. Universitat de Valencia, Valencia, Spain; 2003.
Sims B: A class of spaces with weak normal structure. Bulletin of the Australian Mathematical Society 1994,49(3):523–528. 10.1017/S0004972700016634
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Acknowledgment
This work is partially supported by SEP-CONACYT Grant 102380. It is dedicated to W. A. Kirk.
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Fetter, H., Gamboa de Buen, B. Properties and
,
Embeddings in Banach Spaces with 1-Unconditional Basis and
.
Fixed Point Theory Appl 2010, 342691 (2009). https://doi.org/10.1155/2010/342691
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DOI: https://doi.org/10.1155/2010/342691