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Properties 
and 
, 
Embeddings in Banach Spaces with 1-Unconditional Basis and 
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
If 
the set of quasi-midpoints of 
and 
in 
is given by
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
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
where 
.
It is easy to see that if 
is equilateral in 
, and 
are as above, then
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
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
Thus 
. Since by hypothesis 
, we obtain that 
if 
and 
. Next we will show that 
.
To this effect let 
. Define 
and 
. Then for every 
By the unconditionality of 
and since by (4) we have that 
,
Therefore
Now let 
be such that 
. Such an element exists since 
. Recalling that 
, we obtain that
Using again that 
, we get
or equivalently
On the other hand, since by (2) 
, we have
Similarly
By (3.8) and (3.9), and (3.5) we obtain
Hence
Finally, from (3.7) and (3.11) we have 
and
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:
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:
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
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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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