The Szlenk index and the fixed point property under renorming

Hindawi Publishing Co po a ion Fixed Poin Theo y and Applica ions Volume 2010, A icle ID 268270, 9pages doi:10.1155/2010/268270 Resea ch A icle The Szlenk Index and he Fixed Poin P ope y unde Reno ming T. Dom´ ınguez Bena ides Facul ad de Ma em´ a icas, Uni e sidad de Se illa, P.O. Box 1160, Se illa 41080, Spain Co espondence should be add essed o T. Dom´ ınguez Bena ides, [email p o ec ed] Recei ed 25 No embe 2009; Accep ed 19 Janua y 2010 Academic Edi o : Tomona i Suzuki Copy igh q2010 T. Dom´ ınguez Bena ides. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. Assume ha Xis a Banach space such ha i s Szlenk index SzXis less han o equal o he i s in ini e o dinal ω. We p o e ha Xcan be eno med in such a way ha Xwi h he esul an no m sa is ies RX<2, whe e R·is he Ga c´ ıa-False coefficien . This leads us o p o e ha i Xis a Banach space which can be con inuously embedded in a Banach space Ywi h SzY≤ω, hen, Xcan be eno med o sa is y he w-FPP. This esul can be applied o Banach spaces which can be embedded in CK,whe eKis a sca e ed compac opological space such ha Kω∅. Fu he mo e, o a Banach space X, ·, we conside a dis ance in he space Po all no ms in X which a e equi alen o · o which Pbecomes a Bai e space.I SzX≤ω, we show ha o almos all no ms in he sense o po osi yin P,Xsa is ies he w-FPP. Fo gene al e lexi e spaces independen ly o he Szlenk index, we p o e ano he s ong gene ic esul in he sense o Bai e ca ego y. 1. In oduc ion Assume ha X, ·is a Banach space. The mos common aim o he Reno ming Theo y is o ind an equi alen no m which sa is ies o which does no sa is yce ain speci ic p ope ies. A de ailed accoun o his opic can be ound in he monog aphs 1–3. This pape ocuses on he Reno ming Theo y in connec ion wi h he Fixed Poin Theo y. I is usually said ha a Banach space Xsa is ies he weak Fixed Poin P ope y w-FPPi o e e y con ex weakly compac subse Co X, each nonexpansi e mapping T:C→Chas a ixed poin . Many geome ical p ope ies o Xuni o m con exi y, uni o m smoo hness, uni o m con exi y in e e y di ec ion, uni o m non-squa eness, no mal s uc u e, e c.a e known o imply he w-FPP see, e.g., 4–6and e e ences he ein. Howe e , no cha ac e iza ion o he w-FPP in e ms o hese p ope ies is known. The e o e, we can ega d he w-FPP as an in insic p ope y o a Banach space. Since he w-FPP is no p ese ed unde isomo phisms, a e y 2 Fixed Poin Theo y and Applica ions na u al ques ion in Reno ming Theo y and Fixed Poin Theo y would be he ollowing: le Xbe a Banach space. Is i possible o eno m Xso ha he esul an space has he w-FPP? This is no gene ally he case. Indeed, Pa ing on 7,8has p o ed ha e e y eno ming o ∞Γ o an uncoun able se Γand any eno ming o ∞/c0con ains an isome ic copy o ∞and, consequen ly, i ails he w-FPP due o Alspach example 9. Thus, i would be in e es ing o iden i y some classes o Banach spaces which can be eno med o sa is y he w-FPP. Fo ins ance, Day e al. 10ha e p o ed ha e e y sepa able Banach space has a UCED eno ming. Since uni o m con exi y in e e y di ec ion implies no mal s uc u e and his p ope y implies he w-FPP see, e.g., 4, we ob ain ha any sepa able Banach space can be eno med o sa is y he w-FPP. These a gumen s do no wo k o nonsepa able spaces because, as men ioned abo e, he e a e some Banach spaces which canno be eno med o sa is y he w-FPP. In ac , in 10, i is shown ha c0Γ has no UCED eno ming i Γis uncoun able. Since in 11an example is gi en o a e lexi e Banach spaces which does no admi any UCED eno ming, he ollowing ques ion, which appea s in 12, Open Ques ion VIand 1, P oblem VII.3and which emained unanswe ed o a long ime, seems o be e y na u al: can any e lexi e Banach space be eno med o sa is y he w-FPP? In 13i is shown ha his is indeed he case. Ac ually, he ollowing esul is p o ed in 13: assume ha Xis a Banach space such ha he e exis s a bounded one-one linea ope a o om Xin o c0Γ. Then, Xhas an equi alen no m which sa is ies he w-FPP. This embedding p ope y is sa is ied by a e y gene al class o Banach spaces, o ins ance subspaces o a space wi h Ma kushe ich basis, as WCG spaces and so sepa able and e lexi e spaces, dual o sepa able spaces as ∞, and so o h. The p oo o he esul in 13is s ongly based upon some speci ic p ope ies o he space c0Γ, specially he equali y Rc0Γ  1, whe e R·is Ga c´ ıa-False ’s coefficien 14. I mus be no ed ha any Banach space Ysuch ha RY<2 sa is ies he w-FPP see 15. Thus, i would be na u al o ex end he abo e esul o any Banach space which can be embedded in mo e gene al Banach spaces han c0Γ, bu s ill sa is ying RY<2. In 16 we p o e his ex ension in he ollowing sense: assume ha Yis a Banach space such ha RY<2, whe e R·is Ga c´ ıa-False ’s coefficien , and Xis a Banach space which can be con inuously embedded in Y. Then, Xcan be eno med o sa is y he w-FPP. In his pape we will use he Szlenk index o show a wide class o Banach spaces X which can be eno med o sa is y RX<2. The Szlenk index SzX17is an o dinal numbe which was in oduced o p o e ha he e is no sepa able e lexi e Banach space uni e sal o he class o all sepa able e lexi e Banach spaces. La e , his index has been used in a ious a eas o he geome y o Banach spaces see 18 o a su ey abou i . Recen ly, Raja 19 has p o ed ha i Xis an Asplund space and SzX≤ω, hen he e is an equi alen no m on Xsuch ha he dual no m on X∗is UKK∗. We will show in his pape ha his ac leads us o p o e RX<2 when Xis endowed wi h his no m. On he o he hand, i we endow Γwi h he disc e e opology and deno e by K he one-poin compac i ica ion o Γ, hen c0Γ is isome ically con ained in CK, whe e K is a opological compac space which sa is ies K2∅. Thus, i a Banach space can be con inuously embedded in c0Γ hen, i can also be embedded in CK, whe e Kis a sca e ed compac opological space such ha Kω∅. Since CKsa is ies he w-FPP 20when Kis a sca e ed compac opological space Ksuch ha Kω∅, ano he na u al ques ion would be he ollowing: assume ha Xis a Banach space which can be con inuously embedded in CK o some Kas abo e. Can Xbe eno med o sa is y he w-FPP? Using he esul s abou he Szlenk index and he main esul in 16, we can p o e ha his is indeed he case. Nominally, since SzCK ≤ωi and only i Kis as abo e, we ob ain he ollowing: le CKbe he Fixed Poin Theo y and Applica ions 3 space o eal con inuous unc ions de ined on a sca e ed compac opological space Ksuch ha Kω∅. Then, i can be eno med in such a way ha RCK,·<2whe e ·is he new no mand he dual no m is UKK∗. In o de o be e unde s and he ele ance o his esul , no e ha in he me izable case, i Kω∅, hen CKis isomo phic o c0and, consequen ly, he e exis s an equi alen no m ·such ha RCK,·1. F om his esul and he main esul in 16, we can easily deduce ha i a Banach space can be con inuously embedded in CK,Kas abo e, hen i can be eno med o sa is y he w-FPP. In 16 he same esul o CKwas ob ained by a di ec and e y echnical me hod. This is a s ic imp o emen o he esul in 13, because, as p o ed in 21, when Kis a Ciesielski-Pol’s compac , hen K3∅,bu CKcanno be con inuously embedded in c0Γ o any se Γ. In he las sec ion, o a Banach space X, ·, we conside a me ic in he space Po all no ms in Xwhich a e equi alen o ·, and no e ha Pbecomes a Bai e space o he co esponding me ic opology. I SzX≤ω, we show ha o almos all no ms in he sense o po osi yin P,Xsa is ies he w-FPP. We inish wi h ano he s ong gene ic esul in he sense o Bai e ca ego y o gene al e lexi e spaces wi hou any assump ion on he Szlenk index. 2. Szlenk Index and Fixed Poin s We s a eminding some de ini ion and s a ing he p e ious esul s which we will use. De ini ion 2.1. Le Mbe a opological space and Aasubse o M.These Ais said o be pe ec i i is closed and has no isola ed poin , ha is, Ais equal o he se o i s own accumula ion poin s. The space Mis said o be sca e ed i i con ains no pe ec non oid subse . I Ais a subse o a opological space M, he de i ed se o Ais he se A1o all accumula ion poin s o A.I αis an o dinal numbe , we de ine he α h-de i ed se by ans ini e induc ion: A0A, Aα1Aα1,A λ α<λ Aα,2.1 whe e λis a limi o dinal. Le us ecall he de ini ion o Ga c´ ıa-False ’s coefficien . De ini ion 2.2 see 14.Le Xbe a Banach space. The coefficien RXis de ined by RXsuplim in xnx:xnis weakly null wi h xn≤1,x1.2.2 Theo em 2.3 see 15.Le Xbe a Banach space such ha RX<2. Then, Xsa is ies he w-FPP. Theo em 2.4 see 16.Le Ybe a Banach space such ha RY<2. Assume ha Xis ano he Banach space, such ha he e exis s a con inuous one- o-one mapping J:X→Y. Then, Xcan be eno med o sa is y he w-FPP. De ini ion 2.5. Le Xbe a Banach space wi h dual X∗. We say ha he dual no m is UKK∗i o e e y ε>0 he e is θε>0 such ha e e y u∈BX∗wi h u>1−θεhas a weak∗open neighbo hood Uwi h diam BX∗∩U<ε. 4 Fixed Poin Theo y and Applica ions We emind he de ini ion o he Szlenk index. Following he su ey 18, we conside a mo e gene al de ini ion han ha in 17. Howe e , bo h de ini ions a e iden ical o sepa able spaces which do no con ain 1. De ini ion 2.6. Le Xbe a Banach space and X∗i s dual. Fo any bounded subse A⊂X∗,we de ine a Szlenk de i a ion by A ε{u∈A: o e e y w∗-neighbo hood Uo u,diamA∩ U≥ε}. By i e a ion, he se s Aγ εa e de ined o any o dinal numbe γ, aking in e sec ion in he case o limi o dinals. The indices SzXεa e o dinal numbe s de ined as SzXεin γ:BX∗γ ε∅2.3 i such an o dinal exis s. O he wise, we w i e SzXε∞. Finally he Szlenk index is de ined by SzXsupε>0SzXε. Rema k 2.7. I is known see 18, Theo em 2o 1, Theo em 5.2 ha SzX/ ∞i and only i Xis an Asplund space. Since ou esul s apply o Banach spaces sa is ying SzX≤ω, om now on, we will only conside Asplund spaces. Theo em 2.8 see 19.Le Xbe an Asplund space wi h SzX≤ω. Then, he e is an equi alen no m on Xsuch ha he dual no m on X∗is UKK∗. Le Kbe a compac opological space. I is known see, e.g., 1, Lemma 8.3 ha CK is an Asplund space i and only i Kis sca e ed. Fo special sca e ed se s, we ha e a mo e p ecise esul . Theo em 2.9 see 18, Theo em 24.Le Kbe a sca e ed compac space. The ollowing asse ions a e equi alen : iSzCK ≤ω, iiKω∅. We will use he equi alen de ini ion o he UKK∗p ope y gi en by he ollowing lemma. Lemma 2.10. Assume ha Xis a Banach space. Then he dual no m is UKK∗i and only i o e e y ε>0, he eexis sδ>0such ha i {uα}is a ne in he uni ball o X∗con e gen o uin he weak∗ opology such ha limαuα−u>ε, henu<1−δ. P oo . Assume ha he abo e condi ion is sa is ied and le ε>0. Suppose ha diam U∩B∗ X>ε o e e y open neighbo hood o uin he weak∗- opology. We can choose uU∈BX∗∩Usuch ha uU−u>ε/3. Then, {uU}is a ne in BX∗con e gen o uin he weak∗- opology. Taking a subne {uα}o {uU}such ha limαuα−uexis s, we ob ain u≤1−δε/3. Con e sely, assume ha he dual no m is UKK∗.Le {uα}be a ne in BX∗con e gen o uin he weak∗- opology such ha limαuα−u>ε.Le Ube an open neighbo hood o uin he weak∗- opology. The e exis α0such ha o e e y α≥α0we ha e uα−u>εand uα∈U.Thus diam U∩BX∗>ε, which implies u≤1−θε. Rema k 2.11. No e ha he abo e no ion implies he sequen ial-UKK∗condi ion, ha is, he dual no m is sequen ially-UKK∗i o e e y ε>0, he e exis s δ>0 such ha i {un}is a Fixed Poin Theo y and Applica ions 5 sequence in he uni ball o X∗con e gen o uin he weak∗ opology such ha un−u>ε, hen u<1−δ. Bo h condi ions a e equi alen i ei he Xis sepa able and, consequen ly, he weak∗- opology es ic ed o bounded subse s o X∗is me izableo Xis e lexi e due o he angelici y o weak compac se s. Theo em 2.12. Le Xbe an Asplund space wi h SzX≤ω. Then, he e is an equi alen no m |·| on Xsuch ha RX, |·| <2and, hence, X, |·|sa is ies he w-FPP. P oo . By Theo em 2.8, he e exis s an equi alen no m on X, such ha he dual no m sa is ies he UKK∗p ope y. We ollow an a gumen inspi ed on ha in he p oo o P oposi ion III.11 in 15. Assume ha {xn}is a weakly null sequence in BXand x∈BX. Fo e e y n∈N, choose un∈SX∗such ha unxxnxxn. Taking a subsequence, i necessa y, we can assume ha limnxxndoes exis . Le {unα}be a subne o {un}which is weak∗-con e gen o uand such ha limαunα−uexis s. Assume ≤1/2 and choose an a bi a y η>0. Since {xnα}is a weakly null ne , he e exis s α0such ha |uxnα|<η/2, unα−u<1/2η and |unαx−ux|<η/2 o e e y α≥α0. Thus, we ha e xnαxunαxnαunαx uxunα−uxnαunα−uxuxnα ≤uunα−u2η ≤11 22η, 2.4 which implies ha limnxnx≤3/2. I >1/2, om Lemma 2.10 we ha e ha u< 1−δ1/2. Since xnαxunαxnαx≤|unαx||unαxnα|≤1|unαx|,2.5 we ha e lim in nxnxlim αxnαx≤1|ux|≤1u≤11−δ1 2<2−δ1 2.2.6 Thus, RX<max{3/2,2−δ1/2}. Rema ks 2.13. 1Following an a gumen as in he p oo o P oposi ion III.11 in 15, we can also ob ain he condi ion RX<2 unde he ollowing mo e gene al assump ion which is usually deno ed as w-UKK∗p ope y: he e exis ∈0,1and δ>0 such ha i {uα}is a ne in he uni ball o X∗con e gen o uin he weak∗- opology and such ha limαuα−u>, hen u<1−δ. Howe e , his condi ion does no yield o an imp o emen o he abo e heo em, because i X∗sa is ies he w-UKK∗p ope y, he e is a eno ming o Xsuch ha he dual no m sa is ies he UKK∗p ope y. Indeed, i is easy o check ha he w-UKK∗p ope y implies ha he Szlenk index SzXis ini e o some ∈0,1. Since he unc ion SzXis submul iplica i e 18,P oposi ion4, we ha e ha SzXn≤SzXnand hus SzXis ini e o e e y posi i e . Thus, he exis ence o an equi alen no m in Xsuch ha he dual no m sa is ies he UKK∗p ope y is a consequence o Theo em 2.8. 6 Fixed Poin Theo y and Applica ions 2We can also deduce some ixed poin p ope ies o he dual no m. Fi s o all, we should men ion ha i Xis an Asplund space, hen X∗can be con inuously embedded in c0Γ o some se Γ22. Thus, by he main esul in 13,X∗has an equi alen in gene al non- dualno m which sa is ies he w-FPP. On he o he hand, we know see 23, Co olla y 5.10 ha p ope y UKK∗implies ha he coefficien w∗CSX∗is g ea e han 1, whe e w∗CSX∗in limn/ mun−um limnun,2.7 and he in imum is aken o e all weak∗-null sequences {un}in X∗such ha bo h limi s exis and limnun/ 0. This condi ion implies ha e e y sepa able weak∗-compac subse o X∗ has no mal s uc u e see 24, Theo em 2o 23, P oposi ion 5.3.Thus,X∗admi s a dual equi alen no m such ha i Tis a nonexpansi e mapping de ined om a sepa able weak∗- compac con ex subse Co X∗in o C, hen Thas a ixed poin see 24, Theo em 1.I Xis e lexi e, he sepa abili y assump ion can be emo ed, because he condi ion WCSX∗> 1 implies no mal s uc u e o weakly compac subse s o X∗and we eco e he i s men ioned eno ming esul now, o a dual no m because any equi alen no m is a dual no m in a e lexi e space 25. Howe e , in his case we ob ain a s onge esul because we ha e an equi alen no m in Xsuch ha Xendowed wi h he new no m sa is ies he w- FPP and X∗endowed wi h he dual no m sa is ies he w-FPP ei he Theo em 3.4 in he las sec ion will show a diffe en way o p o e a s onge esul . Also in he e lexi e case, since X∗is nea ly uni o m con ex, we can also assu e ha X∗sa is ies he w-FPP o nonexpansi e mul i alued mappings wi h compac con ex aluessee, e.g., 26. Theo em 2.12 join ly wi h 16, Theo em 2.5yields o he main esul in his pape . Theo em 2.14. Le Ybe a Banach space wi h SzY≤ω. Assume ha Xis ano he Banach space, such ha he e exis s a con inuous one- o-one mapping J:X→Y. Then, Xcan be eno med o sa is y he w-FPP. Assume ha Γis an uncoun able se . We can conside ha Γis endowed wi h he disc e e opology. Le Kbe he one-poin compac i ica ion o Γ. Then, c0Γ,· ∞is isomo phic o CK,· ∞by de ining S:CK→c0Γ by Sxγ  xγ−x∞. Thus any space which can be con inuously embedded in c0Γ,·∞, can be also embedded in CK,· ∞, whe e K2∅. F om Theo ems 2.9 and 2.14, we ob ain he ollowing esul which s ic ly imp o es he main esul in 13, because as men ioned in he in oduc ion and p o ed in 21, he e exis s a compac se Ciesielski-Pol’s compac , such ha K3∅,bu CKcanno be con inuously embedded in c0Γ o any se Γ. The same esul is p o ed in 16using a di ec bu e y echnical a gumen . Co olla y 2.15. Le Xbe a Banach space which can be con inuously embedded in CK,· ∞ o some compac se Ksuch ha Kω∅. Then, Xcan be eno med o sa is y he w-FPP. 3. Gene ici y o he w-FPP and Szlenk Index Following he app oach in 27, o a Banach space X, ·, wi h closed uni ball B, we deno e by P he Bai e space o all equi alen no ms wi h he me ic ρp, qsup{|px−qx|:x∈ B}. Fixed Poin Theo y and Applica ions 7 In a Bai e space, we can ega d i s ca ego y se s as negligible se s. Howe e , we can also conside a deepe no ion o negligible se . We should emembe ha a se Ain a opological space Xis nowhe e dense i i s closu e has emp y in e io . I Xis a me ic space, his ac means ha o e e y x∈Aand >0, he e exis s y∈Xand >0 such ha By, ⊂Bx,  A. A mo e s ic condi ion is he ollowing. De ini ion 3.1. Le Mbe a me ic space. A subse Ao Mis said o be po ous i he e exis 0<β≤1and 0>0 such ha o e e y x∈Aand 0 < ≤ 0, he e exis s y∈Xsuch ha By,β ⊂Bx, ∩M A.Asubse Ao Mis called σ-po ous i Ais he union o a coun able amily o po ous se s. Po ous and σ-po ous se can be conside ed “small” in M. In pa icula a σ-po ous se is ob iously o Bai e i s ca ego y and, o MRn,aσ-po ous se is a null se wi h espec o he Lebesgue measu e. In 28, Theo em 14,i isp o ed ha i Xis a Banach space such ha RX<2, hen he e exis s a σ-po ous se A⊂Psuch ha i q∈P A he space X, qsa is ies he w-FPP. F om his and Theo em 2.12, we easily ob ain he ollowing gene ic esul . Co olla y 3.2. Assume ha Xis a Banach space wi h SzX≤ωand Pis he se o all no ms in Xwhich a e equi alen o he o iginal no m wi h he me ic ρp, qsup{|px−qx|:x∈B}. Then, he e exis s a σ-po ous se A⊂Psuch ha i q∈P A he space X, qsa is ies he w-FPP. In pa icula , we ob ain he ollowing gene ic esul , which can be ega ded as an imp o emen o he esul in 20abou he w-FPP in CK. Co olla y 3.3. Assume ha Kω∅and Pis he se o all no ms in CKwhich a e equi alen o he sup emum no m wi h he me ic ρp,qsup{|px−qx|:x∈B}. Then, he e exis s a σ-po ous se A⊂Psuch ha i q∈P A, he space CK,qsa is ies he w-FPP. Fo gene al e lexi e spaces independen ly o he Szlenk index, we can use he main esul in 29 o p o e a s ong gene ic esul in he sense o he Rema ks 2.13.I pis a no m in a Banach space X, we will deno e by p∗ he dual no m on he dual space X∗and by Q he Bai e space o all equi alen no ms o · ∗wi h he me ic ρ , ssup{| u−su|:u∗≤ 1}. Theo em 3.4. Le X, ·be a e lexi e space. The e exis s a esidual subse R0o P(i.e., P R0is o Bai e is ca ego y) such ha o e e y p∈R 0, he spaces X, pand (X∗,p∗sa is y bo h he w-FPP. P oo . By 29, Co olla y 2.5, he e exis a esidual subse Ro Pand ano he esidual subse Sin Qsuch ha i p∈Rand s∈S, he spaces X, pand X∗,ssa is y he w-FPP. We claim ha he mapping h:P→Qde ined by hpp∗is an homeomo phism om Pon o Q. Indeed, his mapping is clea ly one-one. Mo eo e , his on o because any equi alen no m in a e lexi e space is a dual no m 25. I is enough o p o e ha his con inuous because h−1 is simila o h. Fixed p∈Pand >0. Deno e by a he posi i e numbe in {px:x1}. Assume ha ρp, q<δ:min{a2/4,a/2}.No e ha qx≥px−δx≥ax/2 o e e y x∈X. Fu he mo e, px≤1 implies qx 1δx≤1,3.1 8 Fixed Poin Theo y and Applica ions and, analogously, qx≤1 implies px 1δx≤1.3.2 Assume ha u∗≤1andqx≤1. We ha e |ux|≤   ux 1δx       ux−x 1δx    ≤sup uy :py≤1   x−x 1δx    ≤p∗uδx2≤p∗uε. 3.3 Thus q∗u<p ∗uε. Analogously, p∗u<q ∗uεwhich implies |p∗u−q∗u|<ε o e e y uin he uni ball o X∗,·∗, ha is,ρp∗,q∗<ε. Finally, de ining R0R∩h−1S, we conclude he p oo . Rema k 3.5. We do no know i a po ous e sion o he abo e heo em does hold. In ac , we do no know ei he i Co olla y 2.5 in 29holds in he sense o po osi y. Fu he mo e, he mapping h:P→Qde ined in he p oo o Theo em 3.4 is a homeomo phism, bu i is no uni o mly con inuous. Indeed, he sequence o no ms in R, de ined by pn | |/n, is a Cauchy sequence, bu he dual sequence pn∗un|u| is no . Thus, he σ-po osi y o Q Sdoes no , in gene al, imply he σ-po osi y o P h−1S. Acknowledgmen s The au ho is e y g a e ul o M. Fabian o some aluable commen s. The au ho is pa ially suppo ed by DGES, G an BFM2006-13997-C02-01 and Jun a de Andaluc´ ıa, G an FQM-127. 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