Hermitian operators on Banach algebras of Lipschitz functions

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Numbe 10, Oc obe 2014, Pages 3469–3481 S 0002-9939(2014)12048-X A icle elec onically published on May 30, 2014 HERMITIAN OPERATORS ON BANACH ALGEBRAS OF LIPSCHITZ FUNCTIONS FERNANDA BOTELHO, JAMES JAMISON, A. JIM´ ENEZ-VARGAS, AND MOIS´ ES VILLEGAS-VALLECILLOS (Communica ed by Thomas Schlump ech ) Abs ac . Fo compac me ic spaces (X, d), we show ha he Lipschi z spaces Lip(X, d) and he li le Lipschi z spaces lip(X, dα)wi h0<α<1, equipped wi h he sum no m, suppo only i ial he mi ian ope a o s, ha is, eal mul iples o he iden i y ope a o . 1. In oduc ion Le Abe a complex Banach algeb a wi h uni y Iand le A∗be i s dual space. Gi en a∈A, ecall ha he algeb aic nume ical ange V(a)isgi enby V(a)={F(a): F∈A∗,F=F(I)=1}. An elemen a∈Ais said o be he mi ian i V(a)⊂R.I isknown ha a∈Ais he mi ian i and only i exp(i a)=1 o all ∈R;see[3]. Le Ebe a complex Banach space and B(E) he Banach algeb a o all bounded linea ope a o s on Eequipped wi h he ope a o no m. I is well-known ha an ope a o T∈B(E) is he mi ian i and only i exp(i T)isanisome y o each ∈R; see [6, Theo em 5.2.6]. The se o he mi ian ope a o s on Eis a eal subspace o B(E) which con ains all ope a o s o he o m λI,whe eλis a eal numbe . A he mi ian ope a o is said o be i ial i i is a eal mul iple o he iden i y ope a o . Some impo an Banach spaces only suppo i ial he mi ian ope a o s, as o example, he Be gman spaces Lp a(Δ) (1 ≤p<∞,p= 2) [9, Co olla y 5.4] and he Ha dy spaces Hp(Δ) (1 ≤p<∞,p= 2) [1]. Also, he he mi ian ope a o s on se e al spaces o scala - alued con inuous unc ions defined on he in e al [0,1] a e known o be jus eal scala mul iples o he iden i y. Such spaces include he space o con inuously diffe en iable unc ions C1[0,1]; he space o absolu ely con inuous unc ions AC[0,1]; and he spaces o Lipschi z unc ions: Lip[0,1] and lip α,0<α<1. We ecall ha lip αconsis s o all pe iod 1 unc ions on R sa is ying | (x)− (y)|=o(|x−y|α) uni o mly as |x−y|→0; c . [2, Theo em 3.1]. Recei ed by he edi o s Feb ua y 6, 2012 and, in e ised o m, Sep embe 5, 2012; Sep em- be 13, 2012; and Oc obe 9, 2012. 2010 Ma hema ics Subjec Classifica ion. P ima y 46E15, 47B15, 47B38. Key wo ds and ph ases. Spaces o Lipschi z unc ions, he mi ian ope a o , de i a ion, bi- ci cula p ojec ion. The hi d and ou h au ho s we e pa ially suppo ed by Jun a de Andaluc´ıa g an s FQM-3737 and FQM-194 and by MICINN p ojec MTM 2010-17687. c 2014 Ame ican Ma hema ical Socie y Re e s o public domain 28 yea s om publica ion 3469 Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use 3470 FERNANDA BOTELHO ET AL. In his pape we in es iga e he he mi ian ope a o s on spaces o Lipschi z unc- ions defined on a compac me ic space. Mo e p ecisely, o a compac me ic space (X,d) and a posi i e eal pa ame e α∈(0,1], we conside he space o all α-Lipschi z unc ions :X→Csuch ha pα( ):=sup| (x)− (y)| d(x, y)α:x, y ∈X, x =y<∞, and also he subspace o all α-Lipschi z unc ions :X→Csa is ying he addi- ional local fla ness condi ion: lim d(x,y)→0 | (x)− (y)| d(x, y)α=0. These wo spaces wi h he s anda d ope a ions o addi ion, mul iplica ion and scala mul iplica ion a e complex algeb as, and when equipped wi h he no m  α=pα( )+ ∞ become Banach algeb as. These wo algeb as a e deno ed by Lip(X,dα)and lip(X,dα), espec i ely. I is impo an o obse e ha Lip(X,dα) and lip(X, dα) a e uni al semi-simple commu a i e complex Banach algeb as, and lip(X,dα) is a closed subalgeb a o Lip(X,dα). No ice ha lip(X,d) may con ain only cons an unc ions, o example lip[0,1] wi h he usual me ic. When X=[0,1] o X=Twi h he usual me ics, Lip(X,dα) and lip(X, dα) a e among he classical algeb as conside ed by de Leeuw in [4]. These algeb as we e fi s s udied by She be in [14,15]. In [2], i was shown ha he he mi ian ope a o s on he Lipschi z spaces Lip[0,1] and lip α,0<α<1, a e eal mul iples o he iden i y ope a o . In his pape we p o e ha he same p ope y holds o he spaces Lip(X,d) and he spaces lip(X,dα)wi h0<α<1, o (X,d) a compac me ic space. This gene alizes he a o emen ioned esul . We also men ion he na u al connec ion be ween he mi ian ope a o s and he class o bi-ci cula p ojec ions. A p ojec ion Pon a complex Banach space is bi- ci cula i eisP+ei (I−P) is an isome y o all s, ∈R. This ype o p ojec ion was in oduced by S ach´o and Zala in [17]. Jamison [10] showed ha hese p ojec ions a e exac ly he he mi ian p ojec ions. Ou esul implies ha he only bi-ci cula p ojec ions on Lip(X, d) and lip(X,dα)wi h0<α<1 a e he i ial p ojec ions, 0andI. 2. P elimina ies In his sec ion we gi e a ep esen a ion o all su jec i e linea isome ies on Lip(X,d) o lip(X, dα)(0<α<1) ha fix he cons an unc ion e e ywhe e equal o 1. Then we cha ac e ize he he mi ian elemen s o Lip(X,d) and lip(X,dα) (0 <α<1). The las esul p o ides a use ul desc ip ion o he con inuous linea unc ionals on bo h spaces. Th oughou his pape (X,d) is a compac me ic space, 1Xdeno es he cons an unc ion equal o 1 on X,IX ep esen s he iden i y unc ion on Xand Iis he iden i y ope a o on Lip(X, d) o lip(X,dα), 0 <α<1. Fo each x∈X,δxs ands o he e alua ion unc ional a he poin xdefined on Lip(X, d) o lip(X,dα), 0<α<1. Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use HERMITIAN OPERATORS 3471 Ou app oach equi es ha he su jec i e linea isome ies on he spaces Lip(X, d) and lip(X,dα)(0<α<1) ha e a sui able ep esen a ion. Rao and Roy [13] p o ed ha any su jec i e linea isome y o Lip[0,1] can be exp essed as a weigh ed com- posi ion ope a o → τ ◦ϕ( ∈Lip[0,1]) whe e τis a scala o modulus 1 and ϕ is a su jec i e isome y o [0,1]. They asked whe he e e y isome y on he Banach spaces Lip(X,d) and lip(X,dα)(0<α<1) a e induced by he isome ies o he me ic space X. Nex we de i e a cha ac e iza ion o su jec i e linea isome ies on hese spaces ha fix 1X. This cha ac e iza ion ollows om a heo em due o Ja osz in [11], a heo em in [8] (page 144) and a esul by She be in [14]. Theo em 2.1. Le Xbe a compac me ic space. Then T: Lip(X,d)→Lip(X,d) is a su jec i e linea isome y such ha T(1X)=1 Xi and only i he e exis s a su jec i e isome y ϕ:X→Xsuch ha Tis o he o m T( )= ◦ϕ o all ∈Lip(X,d). The same cha ac e iza ion holds o a su jec i e linea isome y T on lip(X, dα)(0<α<1) such ha T(1X)=1 X. P oo . I is s aigh o wa d o check ha an ope a o To he o m desc ibed in he heo em is a su jec i e isome y. Then we jus p o e he e e sed implica ion. Le A(X) ep esen ei he Lip(X,d) o lip(X, dα)wi h0<α<1 and le C(X) be he algeb a o con inuous complex- alued unc ions on X.Wefi s obse e ha A(X) is a egula subspace o C(X) and he sum no m is a p-no m o he no m on R2gi en by p(s, )=|s|+| |. Le us ecall (see [11]) ha gi en a compac Hausdo ff space X, a complex linea subspace Ao C(X) ha con ains he unc ion 1X,is said o be egula i o any ε>0, any x0∈ChAand any open neighbo hood Uo x0, he eisan ∈Awi h  ∞≤1+ε, (x0) = 1, and | (x)|<ε o x∈X U.ChAdeno es he se o ex eme poin s Fo he uni ball o (A, ·∞)∗ such ha F(1X) = 1, and we iden i y ChAwi h a subse o X. Suppose ha Tis a su jec i e linea isome y on A(X) such ha T(1X)=1 X. An applica ion o he main heo em in [11] o A(X) yields ha Tis a su jec i e isome y on (A(X),·∞). Nex we quo e a heo em om Hoffman’s book [8, p. 44]: Le Xbe a compac Hausdo ff space and le Bbe a complex linea subalgeb a o C(X) ha con ains he unc ion 1X. Suppose ha Sis a linea map o Bon o Bsuch ha S( )∞=  ∞ o all ∈B.I S(1X)=1 X, henSis mul iplica i e. The e o e Tis an au omo phism o A(X). By She be ’s heo em [14, Co olla y 5.2], e e y au omo phism To Lip(X,d) ha ca ies1 Xin o 1Xis o he o m T( )= ◦ϕ,whe eϕ:X→Xis a homeomo phism. Simila ly, we can p o e ha his is also ue o hose au omo phisms o lip(X, dα)(0<α<1) ha fix 1X. We now show ha ϕis an isome y o X. Obse e ha gi en any α∈(0,1], we ha e pα(T( )) = pα( ) o all ∈A(X)sinceTis an isome y o bo h no ms ·αand ·∞. Fo he case A(X)=Lip(X,d), fix y∈Xand define y:X→Rby y(z)= d(z,ϕ(y)) o all z∈X. Clea ly, y∈Lip(X,d)andp1( y)≤1. Fo any x, y ∈X, we ha e d(ϕ(x),ϕ(y)) = | y(ϕ(x)) − y(ϕ(y))| =|T( y)(x)−T( y)(y)| ≤p1(T( y))d(x, y) ≤d(x, y). Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use 3472 FERNANDA BOTELHO ET AL. Fo he case A(X) = lip(X,dα)(0<α<1), fix x, y ∈X,x=y,choose β∈(α, 1) and define xy(z)=d(z,ϕ(y))β−d(z,ϕ(x))β 2d(ϕ(x),ϕ(y))β−α,∀z∈X. I is no ha d o check ha xy ∈lip(X,dα)andpα( xy) = 1 (see, o example, [12, p. 62]). An easy calcula ion gi es d(ϕ(x),ϕ(y))α=| xy(ϕ(x)) − xy(ϕ(y))| =|T( xy)(x)−T( xy)(y)| ≤pα(T( xy))d(x, y)α =d(x, y)α. In ei he case we ha e d(ϕ(x),ϕ(y)) ≤d(x, y) o all x, y ∈X. Since T−1is also a su jec i e linea isome y on A(X) such ha T−1(1X)=1 X, he same a gumen used abo e implies he exis ence o a homeomo phism φ:X→ Xsuch ha T−1( )= ◦φ o all ∈A(X). The e o e d(φ(x),φ(y)) ≤d(x, y) o all x, y ∈X.Gi enx∈X,weha e (ϕ−1(x)) = T(T−1( ))(ϕ−1(x)) = T−1( )(x)= (φ(x)) o all ∈A(X). Since A(X) sepa a es he poin s o X, hisimplies ha ϕ−1=φ. Consequen ly, ϕis a su jec i e isome y. This comple es he p oo o he heo em.  We will nex cha ac e ize he he mi ian elemen s o he spaces Lip(X,d)and lip(X,dα), 0 <α<1. Lemma 2.2. Le (X, d)be a compac me ic space and h∈Lip(X,d)(o lip(X,dα), 0<α<1). Then his a he mi ian elemen in Lip(X,d)(o lip(X,dα))i andonly i his a eal cons an unc ion. P oo . Assume ha his he mi ian in Lip(X, d). Then F(h)∈V(h)⊂R o all F∈Lip(X,d)∗such ha F=F(1X) = 1. In pa icula , h(x)=δx(h)∈R o all x∈X,andsohis eal- alued. Using ha ea−eb≤|a−b|exp (max {|a|,|b|}) o all a, b ∈C, we deduce ha exp(ih) is a unc ion in Lip(X,d). We also ha e ha , o each ∈R,exp(i h)1=1. Sinceexp(i h)∞= 1, i ollows ha p1(exp(i h)) = 0. Hence exp(i h) is a cons an unc ion on X o all ∈Rwhich implies ha his cons an . Con e sely, i his a eal cons an unc ion, hen his a eal mul iple o 1X. The e o e his he mi ian in Lip(X,d). The same p oo wo ks o lip(X,dα), 0 < α<1.  Following an idea o de Leeuw [4], we embed he Banach spaces Lip(X,d)and lip(X,dα)(0<α<1) isome ically in o some sui able spaces o complex- alued con inuous unc ions. Le Xbe a compac me ic space and le  Xbe he se (x, y)∈X2:x=y. I is easy o check ha  Xis comple ely egula ; we deno e by β X he S one-ˇ Cech compac ifica ion o  X.Le C(X∪β X) deno e he Banach space o all complex- alued con inuous unc ions on X∪β X, unde he no m  = |X∞+   |β  X  ∞ ( ∈C(X∪β X)), Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use HERMITIAN OPERATORS 3473 and le C0(X∪ X) deno e he Banach space o all complex- alued con inuous unc- ions on X∪ X anishing a infini y, endowed wi h he no m  = |X∞+  | X ∞( ∈C0(X∪ X)). We now ecall ha he Riesz ep esen a ion heo em s a es ha he map μ→ Fμ, gi en by Fμ( )=X∪β  X dμ ( ∈C(X∪β X)), defines an isome ic isomo phism om he Banach space M(X∪β X) o all complex- alued egula Bo el measu es on X∪β Xequipped wi h he no m o o al a ia ion: μ=|μ|(X∪β X)(μ∈M(X∪β X)) on o he dual space o (C(X∪β X),·∞). Simila ly, he map ν→ Gνdefined by Gν( )=X∪  X dν ( ∈C0(X∪ X)) is an isome ic isomo phism om he Banach space M(X∪ X) wi h he no m ν=|ν|(X∪ X)(ν∈M(X∪ X)) on o he dual space o (C0(X∪ X),·∞). Fo each ∈Lip(X, d)o ∈lip(X, dα), 0 <α<1, we se  : X→C o be he map gi en by  (x, y)= (x)− (y) d(x, y)α,∀(x, y)∈ X, whe e α=1when ∈Lip(X,d). I is easy o show ha  is con inuous on  Xand      ∞ =pα( )(0<α≤1). Hence he e exis s a unique con inuous unc ion β on β Xsuch ha (β ) X= and   β   ∞ =     ∞ .Fu he mo e,i ∈lip(X,dα), hen  anishes a infini y on  X. The maps Φ: Lip(X,d)→C(X∪β X)and Ψ: lip(X,dα)→C0(X∪ X), defined by (1) Φ( )(w)=⎧ ⎨ ⎩ (w)i w∈X, β (w)i w∈β X, and (2) Ψ( )(w)=⎧ ⎨ ⎩ (w)i w∈X,  (w)i w∈ X, a e isome ic linea embeddings om Lip(X,d) wi h he no m ·1in o C(X∪β X), and om lip(X, dα) wi h he no m ·αin o C0(X∪ X), espec i ely. The Hahn–Banach heo em and he Riesz ep esen a ion heo em yield he ol- lowing lemma. Lemma 2.3. Le (X,d)be a compac me ic space. Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use 3474 FERNANDA BOTELHO ET AL. (1) Fo each F∈Lip(X, d)∗, he eexis sμ∈M(X∪β X)wi h F≤μ sa is ying F( )=X∪β  X Φ( )(w)dμ(w),∀ ∈Lip(X, d). (2) Le α∈(0,1).Fo eachG∈lip(X, dα)∗, he eexis sν∈M(X∪ X)wi h G≤νsuch ha G( )=X∪  X Ψ( )(w)dν(w),∀ ∈lip(X,dα). P oo . Le F∈Lip(X,d)∗. The unc ional T: Φ(Lip(X, d)) →C, defined by T(Φ( )) = F( ) o all ∈Lip(X, d), is linea , con inuous and T=F.By he Hahn–Banach heo em, he e exis s a linea con inuous unc ional  T:C(X∪β X)→ Csuch ha  T(Φ( )) = T(Φ( )) o all ∈Lip(X,d)and   T  =T. Since g≤2g∞ o all g∈C(X∪β X), i ollows ha he linea unc ional  Tis con inuous on he space C(X∪β X) equipped wi h he no m ·∞.Wedeno e by ·∗ ∞ he no m on he dual Banach space o C(X∪β X),·∞.By heRiesz ep esen a ion heo em, he e exis s μ∈M(X∪β X) wi h    T   ∗ ∞ =μsa is ying  T(g)=X∪β  X g(w)dμ(w),∀g∈C(X∪β X). Since g∞≤g o all g∈C(X∪β X), we ha e    T  ≤   T   ∗ ∞ ,andsoF≤μ. Mo eo e , F( )=T(Φ( )) =  T(Φ( )) = X∪β  X Φ( )(w)dμ(w) o all ∈Lip(X,d), as we wan ed. Simila ly, we p o e s a emen (2).  Such a μis called a ep esen ing measu e o F(analogously, ν o G). We should no e ha a ep esen ing measu e o Fo Gis no always de e mined uniquely. 3. The main esul In his sec ion we desc ibe all he he mi ian ope a o s on Lip(X, d) o lip(X,dα) wi h 0 <α<1. We p oceed wi h he s a emen and p oo o ou main esul . Theo em 3.1. Le (X,d)be a compac me ic space. A bounded linea ope a o T: Lip(X,d)→Lip(X,d)is he mi ian i and only i Tis a eal mul iple o he iden i y ope a o on Lip(X, d). An analogous asse ion holds o T: lip(X,dα)→ lip(X,dα)wi h 0<α<1. Be o e p o ing his heo em we se no a ion and p o e some p elimina y lemmas. Le A(X) deno e ei he Lip(X, d) o lip(X,dα), 0 <α<1. Recall ha α=1in he case A(X)=Lip(X,d). Lemma 3.2. I T:A(X)→A(X)is a he mi ian bounded linea ope a o , hen he ollowing s a emen s hold: (i) The e exis s λ∈Rsuch ha T(1X)=λ1X. (ii) Fo each ∈R,exp(i (T−λI)) is a su jec i e linea isome y on A(X) fixing 1X. Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use HERMITIAN OPERATORS 3475 (iii) Fo each ∈R, he e exis s a su jec i e isome y ϕ on Xsuch ha exp(i (T−λI))( )(x)= (ϕ (x)),∀ ∈A(X),∀x∈X. (i ) {ϕ } ∈Ris a one-pa ame e g oup o su jec i e isome ies on Xsuch ha , o each x∈X, hemap → ϕ (x) om R o Xis con inuous. ( ) Fo e e y ∈A(X), (3) lim →0( ◦ϕ − )(x)=0,∀x∈X, and (4) lim →0 ( ◦ϕ − )(x)−( ◦ϕ − )(y) d(x, y)α=0,∀(x, y)∈ X. P oo . (i) Fo each F∈A(X)∗wi h F=F(1X) = 1, define ΦF:B(A(X)) →C by ΦF(S)=F(S(1X)),∀S∈B(A(X)). I is easy o check ha ΦFis a linea unc ional on B(A(X)), and since |ΦF(S)|=|F(S(1X))|≤FS(1X)α≤S1Xα=S, o all S∈B(A(X)), hen ΦFis con inuous and ΦF≤1. Mo eo e , ΦF(I)= F(1X) = 1; hence ΦF≥|ΦF(I)|=1and husΦF=Φ F(I)=1. Since T∈B(A(X)) is he mi ian, i ollows ha F(T(1X)) = ΦF(T)∈V(T)⊂R o all F∈A(X)∗such ha F=F(1X) = 1. This means ha T(1X)isa he mi ian elemen in A(X). Then, acco ding o Lemma 2.2, he e exis s λ∈R such ha T(1X)=λ1X. (ii) By (i), we ha e (T−λI)(1X) = 0 and so exp(i (T−λI))(1X)=1 X o all ∈R. Indeed, since exp(i (T−λI)) = I+ ∞  n=1 in n(T−λI)n, i ollows ha exp(i (T−λI))(1X)=1 X+ ∞  n=1 in n(T−λI)n(1X)=1 X. Since Tand λI a e he mi ian ope a o s in B(A(X)), i is easily seen ha so is T−λI. Indeed, using he ac ha exp(i (T−λI)) = exp(i T)exp(−i λI) o all ∈R,weha e 1=1Xα=exp(i (T−λI))(1X)α≤exp(i (T−λI)) ≤exp(i T)exp(−i λI)=1. The e o e, o each ∈R,exp(i (T−λI)) is a linea isome y om A(X)on o i sel , fixing 1X. (iii) In iew o (ii), by applying Theo em 2.1, o each ∈R he e exis s a su jec i e isome y ϕ on Xsuch ha (5) exp(i (T−λI))( )(x)= (ϕ (x)),∀ ∈A(X),∀x∈X. Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use 3476 FERNANDA BOTELHO ET AL. (i ) Using he ac ha A(X) sepa a es he poin s o X, i is easily de i ed ha ϕ(s+ )=ϕs◦ϕ o all s, ∈Rand ϕ0=IX. Mo e p ecisely, gi en ∈A(X)and x∈X,weha e (ϕs+ (x)) = exp(i(s+ )(T−λI))( )(x) =exp(i( +s)(T−λI))( )(x) =exp(i (T−λI))exp(is(T−λI))( )(x) =exp(i (T−λI))(exp(is(T−λI))( ))(x) =exp(is(T−λI))( )(ϕ (x)) = (ϕs(ϕ (x))) = (ϕs◦ϕ (x)) and (ϕ0(x)) = exp(i0(T−λI))( )(x)=exp(0)( )(x)=I( )(x)= (x). We nex p o e ha o each x∈X, hemap → ϕ (x) omR o Xis con inuous. No e fi s ha δ:X→A(X)∗defined by δ(x)=δxis a Lipschi z bijec ion om (X,dα)on oδ(X). Indeed, δis injec i e since A(X) sepa a es poin s; and gi en x, y ∈X,weha e |(δ(x)−δ(y)) ( )|=| (x)− (y)|≤ αd(x, y)α o all ∈A(X). Hence δ(x)−δ(y)≤d(x, y)α.SinceXis compac , we deduce ha δ−1:δ(X)→Xis con inuous. No ice ha δ−1(δx)=x o all x∈X. Fix x∈X. The maps → exp(i (T−λI)) om R o B(A(X)), U→ U∗ om B(A(X)) o B(A(X)∗)andS→ S(δ(x)) om B(A(X)∗) oA(X)∗a e clea ly con inuous. F om (5), we deduce ha (6) (exp(i (T−λI)))∗(δ(x)) = δ(ϕ (x)) ( ∈R,x∈X). Since ϕ (x)=δ−1(exp(i (T−λI)))∗(δ(x)) ( ∈R,x∈X), we conclude ha → ϕ (x) omR o Xis con inuous. ( ) Le ∈A(X). Gi en x∈X, we ha e lim →0( ◦ϕ )(x)=( ◦ϕ0)(x)= (x) by (i ), and hus lim →0( ◦ϕ − )(x)=0. Using his, o (x, y)∈ X, we deduce ha lim →0 ( ◦ϕ − )(x)−( ◦ϕ − )(y) d(x, y)α=0.  We ecall ha o ∈A(X) hemap  : X→Cis defined o be  (x, y)=( (x)− (y))/d(x, y)α. We ecall ha β X ep esen s he S one-ˇ Cech compac ifica ion o  X. This en ails ha e e y bounded, con inuous and scala - alued map defined on  Xhas a unique con inuous ex ension o β X. Lemma 3.3. I ∈A(X), hen (7) lim →0β (w)=0,∀w∈β X, whe e, o each ∈R, deno es he unc ion ◦ϕ − . Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use HERMITIAN OPERATORS 3477 P oo . We define g:[−1,1] × X→Cby g( , (x, y)) = ( ◦ϕ − )(x)−( ◦ϕ − )(y) d(x, y)α. The unc ion gis con inuous and bounded. In ac , we ha e |g( , (x, y))|≤pα( )+pα( )=2pα( ) o all ∈[−1,1] and (x, y)∈ X. Fo he con inui y o g, define σ:[−1,1] × X→ A(X)∗by σ( , (x, y)) = δ(ϕ (x)) −δ(x)−(δ(ϕ (y)) −δ(y)) d(x, y)α and no ice ha g( , (x, y)) = σ( , (x, y))( )( ∈[−1,1],(x, y)∈ X). Taking in o accoun he equali y (6), o any , s ∈[−1,1] and x, y ∈X,weha e δ(ϕ (x)) −δ(ϕs(y))=(exp(i (T−λI)))∗(δ(x)) −(exp(is(T−λI)))∗(δ(y)) ≤(exp(i (T−λI)))∗δ(x)−δ(y) +(exp(i (T−λI)))∗−(exp(is(T−λI)))∗δ(y) ≤d(x, y)+exp(i (T−λI)) −exp(is(T−λI))δ(y). Le us ecall now ha i Aand Ba e bounded commu ing ope a o s on a Banach algeb a, hen exp(iA)−exp(iB)≤A−Bexp (max {A,B}). Applying his o mula o A= (T−λI)andB=s(T−λI), we ob ain (8) exp(i (T−λI)) −exp(is(T−λI))≤| −s|k, whe e k=T−λIexp(T−λI)isacons an ,andso δ(ϕ (x)) −δ(ϕs(y))≤d(x, y)α+k| −s|. The e o e, o e e y , s ∈[−1,1] and x, y ∈X,weha e δ(ϕ (x)) −δ(x)−(δ(ϕs(y)) −δ(y))≤2d(x, y)α+k| −s|. Hence he mapping ( , (x, y)) → δ(ϕ (x)) −δ(x), defined on [−1,1] × Xand wi h alues in A(X)∗, is con inuous. Since (x, y)→ d(x, y)α om  X o Ris con inuous, i ollows ha σis con inuous. Hence, gi en ε>0and( 0,(x0,y 0)) ∈[−1,1] ×  X, he e is a neighbo hood Vo ( 0,(x0,y 0)) such ha i ( , (x, y)) ∈V, hen σ( , (x, y)) −σ( 0,(x0,y 0))<ε/(1 +  α). The e o e, o e e y ( , (x, y)) ∈V, we ha e |g( , (x, y)) −g( 0,(x0,y 0))|<ε 1+ α  α<ε, and his p o es ha gis con inuous. Licensed o Uni e si y de Alme ia. P epa ed on Thu Jul 31 21:03:52 EDT 2014 o download om IP 150.214.156.17. License o copy igh es ic ions may apply o edis ibu ion; see h p://www.ams.o g/jou nal- e ms-o -use