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
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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.
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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).
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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)),
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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.
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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.
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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.
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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 − .
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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.
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