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

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Fixed Point Theory and Applications Volume 2010, Article ID 342691,7pages doi:10.1155/2010/342691

Research Article

Properties WORTH and WORTHH

∗

,

1 δ

Embeddings in Banach Spaces with

1-Unconditional Basis and wFPP

Helga Fetter and Berta Gamboa de Buen

Centro de Investigaci´on en Matem´aticas (CIMAT), Apdo. Postal 402, 36000 Guanajuato, GTO, Mexico

Correspondence should be addressed to Helga Fetter,[email protected]

Received 24 September 2009; Accepted 3 November 2009

Academic Editor: Mohamed A. Khamsi

Copyrightq2010 H. Fetter and B. Gamboa de Buen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We will use Garc´ıa-Falset and Llor´ens Fuster’s paper on the AMC-property to prove that a Banach spaceXthat1δembeds in a subspaceXδof a Banach spaceYwith 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 WORTH and its dual have the FPP and that a real Banach spaceXsuch thatBX∗isw∗sequentially compact and

X∗has WORTH∗has the wFPP.

1. Introduction

In 1988 Sims1introduced the notion of weak orthogonalityWORTHand asked whether spaces with WORTH have the weak fixed point propertywFPP. Since then several partial answers have been given. For instance, in 1993 Garc´ıa-Falset2proved that ifXis uniformly nonsquare and has WORTH then it has the wFPP, although Mazcu ˜n´an 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 wFPP.In both of these cases the spaceX turns out to be reflexive. In 1994 Sims4himself proved that WORTH plusε0-inquadrate in

every direction for someε0<2 implies the wFPP and in 2003 Dalby5showed that ifX∗has

WORTH∗and isε0-inquadrate in every direction for someε0<2, thenXhas the wFPP.

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

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the wFPP. 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 FPP, giving a partial answer to Sims’ question. We also showed that a separable spaceXsuch thatX∗ has WORTH∗and BX∗isw∗sequentially compact has the wFPP.

2. Notations and Definitions

LetX, · Xbe a real Banach space andKa closed nonempty bounded convex subset ofX.

Definition 2.1. Ifx∈X, we define

Rx, K sup{x−zX:z∈K}. 2.1

Ifx, y∈Kthe set of quasi-midpoints ofxandyinKis given by

Mx, y

z∈K: max

z−xX,z−yX ≤

1

2x−yX

. 2.2

Definition 2.2. X is the quotient space l∞X/c0X endowed with the norm z

lim sup_nznXwherezis the equivalence class ofzninl∞X, which we also will denote by

zn. Forx∈Xwe will also denote byxthe equivalence classx, x, x, . . .inX. IfKis as

above, letK {zn∈X:zn ∈K, n1,2, . . .}. IfY is a Banach space andTn :X → Y for

n1,2, . . . ,we defineTn T,T:X → Yby

Tzn Tnzn. 2.3

IfTnT forn1,2, . . .we denoteTbyT.

It is known that K is also closed bounded and convex in X and that T lim sup_nTn.

Definition 2.3. Let SNbe the set of strictly increasing sequences of natural numbers and Ka nonempty bounded convex subset of a Banach spaceX. A sequencexninKis called

equilateral inK, if for everyβ, γ ∈ SNsuch thatβn/γnfor everyn∈N, the following equality holds inX. Ifxβ xβnandxγ xγn, then

xβxγxβ−xγDxn diamK, 2.4

whereDxn lim sup_nlim sup_mxn−xmX.

It is easy to see that ifxnis equilateral inK, andβ, γ∈ SNare as above, then

xβ−xγlim_n xβn−xγnX lim_n xβnXlim_n xγnX. 2.5

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Definition 2.4. A bounded closed convex subsetK of a Banach spaceX, · Xwith 0 ∈ K

has the AMC property, if for every weakly null sequencexnwhich is equilateral inK, there

existρ∈0,1,x∈K,β, γ ∈ SNwithβn/γnfor everyn∈N, such that the set

Mρ

xn, β, γ

Mxβ,xγ

∩ zn:dzn, K≤ρdiamK

2.6

is nonempty andRx, Mρxn, β, γ < diamK.X is said to have AMC if every weakly

compact nonempty subsetKofXwith 0∈Khas the AMC property.

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

Lin in8showed that ifXhas an unconditional basisenwith unconditional constantλ <

331/2₋₃_{/2, then}_X_{has the wFPP. Garc´}_ı_{a-Falset and Llor´ens Fuster proved that in fact under}

these conditionsXhas the AMC property which in turn implies the wFPP. We will follow the proof of this closely to establish the next theorem.

Theorem 3.1. Let X, · X be a Banach space and suppose that there exists a Banach space

Y, · Ywith a1-unconditional basisenand a subspaceXδ ofY such thatdX, Xδ < 1δ whereδ <√13−3/2. ThenXhas AMC and thus the wFPP.

Proof. LetS : X → Xδ be an isomorphism withS ≤1δandS−1 ≤1. LetK ⊂ Xbe a

nonempty weakly compact convex subset ofX with 0∈Kand diamK 1. We will show thatKhas the AMC property.

Letxnbe a weakly null equilateral sequence inK and letKδ SK. ThenKδ is

weakly compact andSxnis weakly null inY. Hence there exists a sequenceβ ∈ SNand

projections with respect to the basiseninY with

aPn:Y → spemn, emn1, . . . , ernwheremn≤rn< mn1, blimnPnyY 0 for ally∈Y,

climnSxβn−PnSxβn_Y 0.

Letγ ∈ SNbe given byγn βn1. Then clearly βn/γnfor everyn ∈ N. Let P , Q : Y → Ybe given by P Pn and Q Pn1and let S : X → Ybe

S S, S, . . .. Recall that we will writeSinstead ofS. By a,b, andcand sincexnis

equilateral we have that

1Sxβ ≤1δ,Sxγ ≤1δandSxβ−Sxγ ≤1δ,

2P S xβSxβ,QS xγSxγ,

3QS xβ0,P S xγ0,

4for ally∈Y,PyQy0.

Therefore, sinceenis 1-unconditional

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ThusxβxγS−1Sxβxγ ≤1δ. Since by hypothesisxβ−xγ ≤1, we obtain thatxβ

xγ/2∈Mρxn, β, γifδ <1 andρ 1δ/2. Next we will show thatR0, Mρxn, β, γ<

1.

To this eﬀect letw wn∈ Mρxn, β, γ. Defineu PQSw andr I−P

QSw. Then for every x∈K

2u ur−Sx u−rSx. 3.2

By the unconditionality ofenand since by4we have thatP Sx QSx 0,

u−rSxPQSwSx−I−PQSw−Sx

PQSwSx I−PQSw−Sxur−Sx.

3.3

Therefore

2u ≤ 2ur−Sx2Sw−Sx. 3.4

Now letv ∈ K be such thatw −v ≤ ρ. Such an element exists sincew ∈ Mρxn, β, γ.

Recalling thatρ 1δ/2, we obtain that

2u ≤ 2Sw −Sv ≤1δ2. 3.5

Using again thatw ∈Mρxn, β, γ, we get

2w ≤ wxβxγ 2

w−xβ

2

w−xγ

2

≤wxβxγ 2 1 2 3.6 or equivalently

wxβxγ

2

≥2w −

1

2. 3.7

On the other hand, since by2P Sx βSxβ, we have

Sw Sxβ−2P S wSwP S xβ−2P S w

I−PSwPSxβ−Sw

I−PSw−PSxβ−SwSw−P S xβSw−Sxβ

≤1δ w−xβ≤

1δ 2 .

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Similarly

SwSxγ−2QS w≤

1δ

2 . 3.9

By3.8and3.9, and3.5we obtain

2SwSxβSxγ 2

≤1δ 2

PQSw 1δ 2u

≤1δ 1δ2.

3.10

Hence

wxβxδ

2

≤

1δ2δ

2 . 3.11

Finally, from3.7and3.11we have 2w − 1/2≤1δ2δ/2 and

w ≤ δ23δ3

4 . 3.12

Therefore, ifδ < √13−3/2 we conclude thatR0, Mρxn, β, γ < 1 and thusX has the

AMC property.

Remark 3.2. It is evident that if the spaceY has a1λ unconditional basis, ifλ is small enough, the above result remains true for someδ.

4. Some Consequences

There has always been the conjecture that a space with property WORTH has the wFPP. We show here that this is correct as long asXis reflexive. We also show that property WORTH∗ inX∗ implies the wFPP in Banach spacesXso thatBX∗isw∗sequentially compact and that

WORTH together with WABS implies the wFPP as well. All these results are consequences of some theorems by Cowell and Kalton6. First we need to recall some definitions.

Definition 4.1. A Banach spaceXhas the WORTH property if for every weakly null sequence xn⊂Xand everyx∈X, the following equality holds:

lim

n xn−x − xnx 0. 4.1

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Definition 4.2. A Banach spaceX∗has the WORTH∗property if for every weak∗null sequence x∗n⊂X∗and everyx∗∈X∗, the following equality holds:

lim

n x ∗

n−x∗ − x∗nx∗ 0. 4.2

IfXis separable andX∗has WORTH∗, this coincides with the propertyau∗defined in 6.

Definition 4.3. A Banach spaceXhas the Weak Alternating Banach-SaksWABSproperty if every bounded sequencexninXhas a convex block sequenceynsuch that

lim

n _r₁_<rsup₂_<···<r_n

n1

n

j1

−1j_y rj

0. 4.3

Cowell and Kalton in6proved the following three results.

Theorem 4.4. IfXis a separable real Banach space, thenX∗has the property WORTH∗if and only if for anyδ >0there is a Banach spaceY with a shrinking1-unconditional basis and a subspaceXδof

Y such thatdX, Xδ<1δ.

Dalby5observed that property WORTH∗in a spaceX∗ implies property WORTH inXand it follows that ifX 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. IfXis a separable real reflexive space, thenXhas property WORTH if and only if for anyδ > 0there is a reflexive Banach spaceY with a1-unconditional basis and a subspaceXδofY such thatdX, Xδ<1δ.

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

Theorem 4.6. IfX is a separable real Banach space, thenX has both the properties WORTH and WABS if and only if for anyδ >0there is a Banach spaceY with a shrinking1-unconditional basis and a subspaceXδofYsuch thatdX, Xδ<1δ.

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

Theorem 4.7. IfXis a real separable space such that either

IX∗has property WORTH∗,

IIXis reflexive and has property WORTH, or

IIIXhas both the properties WORTH and WABS, thenXhas the property AMC and thus the wFPP.

It is known that reflexivity implies WABS, and thusIIimpliesIII, but we want to includeIIin order to deduce the next corollary. Properties WORTH and WABS are inherited by subspaces, and ifX∗has property WORTH∗andBX∗isw∗sequentially compact, then the

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Corollary 4.8. LetXbe a real Banach space.

1IfXis reflexive and has property WORTH, thenXandX∗both have the FPP.

2IfXhas properties WORTH and WABS, thenXhas the wFPP.

3IfX∗has WORTH∗andBX∗isw∗sequentially compact, thenXhas the wFPP.

Proof. IfX is a Banach space that satisfies1,2, or3, every separable subspace has the wFPP and hence, since the wFPP is separably determined,Xhas the wFPP. IfXis separable and reflexive and has property WORTH, then it has property WORTH∗ as well and this implies by definition thatX∗also has property WORTH. Therefore both have the FPP and hence the result follows for nonseparable reflexive spaces.

Acknowledgment

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

References

1 B. Sims, “Orthogonality and fixed points of nonexpansive maps,” in Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), vol. 20 ofProceedings of the Centre for Mathematical Analysis, Australian National University, pp. 178–186, The Australian National University, Canberra, Australia, 1988.

2 J. Garc´ıa-Falset, “The fixed point property in Banach spaces whose characteristic of uniform convexity is less than 2,”Journal of the Australian Mathematical Society. Series A, vol. 54, no. 2, pp. 169–173, 1993. 3 E. M. Mazcu ñán Navarro,Geometr´ıa de los espacios de Banach en teor´ıa métrica del punto fijo, Tesis doctoral,

Universitat de Valencia, Valencia, Spain, 2003.

4 B. Sims, “A class of spaces with weak normal structure,”Bulletin of the Australian Mathematical Society, vol. 49, no. 3, pp. 523–528, 1994.

5 T. Dalby, “The eﬀect of the dual on a Banach space and the weak fixed point property,”Bulletin of the Australian Mathematical Society, vol. 67, no. 2, pp. 177–185, 2003.

6 S. R. Cowell and N. J. Kalton, “Asymptotic unconditionality,”http://arxiv.org/abs/0809.2294. 7 J. Garc´ıa-Falset and E. Llor´ens Fuster, “A geometric property of Banach spaces related to the fixed point

property,”Journal of Mathematical Analysis and Applications, vol. 172, no. 1, pp. 39–52, 1993.