On the alpha-Amenability of Hypergroups

0.13 MB·English

by  Ahmadreza Azimifard

On the α-Amenability of Hypergroups
Ahmadreza Azimifard
Abstract
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0 Let UC(K) denote the Banach space of all bounded uniformly continuous functions on a
0
2 hypergroup K. The main results of this article concern on the α-amenability of UC(K) and

n quotients andproductsofhypergroups. ItisalsoshownthataSturm-Liouvillehypergroup witha
a
J positiveindexisα-amenable ifandonlyifα= 1.

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1
Keywords. Hypergroups:Sturm-Liouville,Che´bli-Trime`che,Bessel-Kingman.

] α-AmenableHypergroups.
A
F AMSSubjectClassification(2000):primary43A62,43A07,46H20,secondary33C10.
.
h
t
a
m
1 Introduction
[

3
In [20]M. Skantharajah systematicallystudiedtheamenabilityofhypergroups. Amongotherthings,
v
6
he obtained various equivalent statements on the amenability of hypergroups. Let K be a locally
7
9
compact hypergroup. Let L1(K) and UC(K) denote the hypergroup algebra and the Banach space
2
.
4 of all bounded uniformly continuous functions on K, respectively. He showed that K is amenable
0
7 if and only if there exists an invariant mean on UC(K). Observe that, contrary to the group case,
0
UC(K) may fail to be an algebra in general. Commutative or compact hypergroups are amenable,
:
v
i and the amenability of L1(K) implies the amenability of K; however, the converse is not valid any
X
r longerevenifK iscommutative(seealso [2, 7]).
a
Recently the notion of α-amenable hypergroups was introduced and studied in [7]. Suppose that
K is commutative, let α ∈ K, and denote by I(α) the maximal ideal in L1(K) generated by α.
As shown in [7], K is α-amenable if and only if either I(α) has a bounded approximate identity
b
or K satisfies the modified Reiter condition of P -type in α. The latter condition together with the
1
recursionformulasfororthogonalpolynomialsyieldsasufficientconditionfortheα-amenabilityofa
polynomial hypergroup. However, this condition is not available for well known hypergroups on the
non-negativereal axis.
The purpose of this article is to generalize the notion of α-amenability for K to the Banach space
UC(K). It then turns out that the α-amenability of K is equivalentto the α-amenability of UC(K),
1
and a α-mean on UC(K) is unique if and only if α belongs to L1(K)∩L2(K). Furthermore, some
results are obtained on the α-amenability of quotients and products of hypergroups. Given a Sturm-
Liouvillehypergroup K with a positiveindex, it is also shownthat there existnon-zero point deriva-
tionsonL1(K). Therefore,L1(K)isnotweaklyamenable,{α}(α 6= 1)isnotaspectralset,andK is
notα-amenableifα 6= 1. However,anexample(consistingofacertainBessel-Kingmanhypergroup)
showsthatin general K isnotnecessarily α-amenableif{α}is aspectral set.
Thisarticleisorganizedasfollows: Section2collectspertinentconceptsconcerningonhypergroups.
Section 3 considers the α-amenability of UC(K). Section 4 contains the α-amenability of quotients
and productsofhypergroups,andSection5 isconsideredonthequestionofα-amenabilityofSturm-
Liouvillehypergroups.
2 Preliminaries
Let (K,ω,∼) be a locally compact hypergroup, where ω : K × K → M1(K) defined by (x,y) 7→
ω(x,y), and ∼: K → K defined by x 7→ x˜, denote the convolution and involution on K, where
M1(K) stands for all probability measures on K. K is called commutativeif ω(x,y) = ω(y,x), for
everyx,y ∈ K.
ThroughoutthearticleK isacommutativehypergroup. LetC (K),C (K),andCb(K)bethespaces
c 0
ofall continuousfunctions,thosewhich havecompactsupport,vanishingat infinity,and boundedon
K respectively; both Cb(K) and C (K) will be topologized by the uniform norm k·k . The space
0 ∞
of complex regular Radon measures on K will be denoted by M(K), which coincides with the dual
space of C (K) [11, Riesz’s Theorem (20.45)]. The translation of f ∈ C (K) at the point x ∈ K,
0 c
T f, is defined by T f(y) = f(t)dω(x,y)(t), for every y ∈ K. Being K commutative ensures
x x K
the existence of a Haar measuRre on K which is unique up to a multiplicative constant [21]. Thus,
according to the translation T, let m be the Haar measure on K, and let (L1(K),k·k ) denote the
1
usual Banach space of all integrable functions on K [12, 6.2]. For f,g ∈ L1(K) the convolution
and involution are defined by f ∗ g(x) := f(y)T g(x)dm(y) (m-a.e. on K) and f∗(x) = f(x˜),
K y˜
respectively, that (L1(K),k·k ) becomes aRcommutative Banach ∗-algebra. If K is discrete, then
1
L1(K) has an identity;otherwiseL1(K) has a b.a.i. (boundedapproximateidentity),i.e. there exists
a net {e } of functionsin L1(K) with ke k ≤ M, for someM > 0, such that kf ∗e −fk → 0 as
i i i 1 i 1
i → ∞ [3]. The dual space L1(K)∗ can be identified with the space L∞(K) of essentially bounded
Borel measurable complex-valued functions on K. The bounded multiplicativelinear functionals on
L1(K) can beidentifiedwith
Xb(K) := α ∈ Cb(K) : α 6= 0, ω(x,y)(α) = α(x)α(y), ∀ x,y ∈ K ,
(cid:8) (cid:9)
where Xb(K) is a locally compact Hausdorffspace with the compact-open topology. Xb(K) with its
subset
K := {α ∈ Xb(K) : α(x˜) = α(x), ∀x ∈ K}
b
2
are consideredas thecharacter spaces ofK .
The Fourier-Stieltjes transform of µ ∈ M(K), µ ∈ Cb(K), is µ(α) := α(x)dµ(x), which by
K
restrictionon L1(K) it iscalled Fouriertransformand f ∈ C (K), foreveryRf ∈ L1(K).
b b 0 b
There existsauniqueregularpositiveBorel measureπ on K withthesupportS suchthat
b b
b 2
|f(x)|2dm(x) = f(α) dπ(α),
ZK ZS (cid:12) (cid:12)
(cid:12) (cid:12)
b
(cid:12) (cid:12)
for all f ∈ L1(K) ∩ L2(K). π is called Plancherel measure and its support, S, is a nonvoid closed
subsetofK. Observethattheconstantfunction1isingeneralnotcontainedinS. WehaveS ⊆ K ⊆
Xb(K), whereproperinclusionsare possible;see[12, 9.5].
b b
3 α-Amenability of UC(K)
Definition 3.1. Let K be a commutative hypergroup and α ∈ K. Let X be a subspace of L∞(K)
withα ∈ X whichisclosedundercomplexconjugationandistranslationinvariant. ThenX iscalled
b
α-amenableifthereexistsam ∈ X∗ withthefollowingproperties:
α
(i) m (α) = 1,
α
(ii) m (T f) = α(x)m (f), foreveryf ∈ X andx ∈ K.
α x α
The hypergroup K is called α-amenable if X = L∞(K) is α-amenable; in the case α = 1, K
respectivelyL∞(K) is called amenable. As shownin [7], K is α-amenableifand only ifeither I(α)
has a b.a.i. or K has the modified Reiter’s condition of P type in the character α. The latter is
1
also equivalentto theα-left amenabilityofL1(K), ifα is real-valued [2]. Forinstance, commutative
hypergroupsareamenable, and compacthypergroupsareα-amenableforeverycharacter α.
IfK isalocallycompactgroup,thentheamenabilityofK isequivalenttotheamenabilityofdiverse
subalgebras of L∞(K), e.g. UC(K) the algebra of bounded uniformly continuous functions on K
[17]. The same is true for hypergroups although UC(K) fails to be an algebra in general [20]. We
nowprovethisfact in termsofα-amenability.
Let UC(K) := {f ∈ Cb(K) : x 7→ T f iscontinuousfrom K to(Cb(K),k·k )}. The function
x ∞
spaceUC(K)isanormclosed,conjugateclosed,translationinvariantsubspaceofCb(K)containing
theconstantsandthecontinuousfunctionsvanishingatinfinity[20,Lemma2.2]. Moreover,Xb(K) ⊂
UC(K) andUC(K) = L1(K)∗L∞(K). LetB beasubspaceofL∞(K)suchthatUC(K) ⊆ B. K
is amenableifand onlyifB isamenable[20,Theorem 3.2].
Thefollowingtheoremprovidesafurtherequivalentstatementtotheα-amenabilityofK.
Theorem 3.2. Letα ∈ K. ThenUC(K) isα-amenableifandonlyifK is α-amenable.
b
Proof: Let UC(K) be α-amenable. There exists a muc ∈ UC(K)∗ such that muc(α) = 1 and
α α
muc(T f) = α(x)muc(f)forallf ∈ UC(K) andx ∈ K. Letg ∈ L1(K)suchthatg(α) = 1. Define
α x α
b
3
m : L∞(K) −→ C by
α
m (ϕ) = muc(ϕ∗g) (ϕ ∈ L∞(K))
α α
that m | = muc. Since ϕ ∗ g ∈ UC(K), m is a well-defined bounded linear functional on
α UC(K) α α
L∞(K), m (α) = 1,and forall x ∈ K wehave
α
m (T ϕ) = muc((T ϕ)∗g) (1)
α x α x
= muc((δ ∗ϕ)∗g))
α x˜
= muc(δ ∗(ϕ∗g))
α x˜
= muc(T (ϕ∗g))
α x
= α(x)muc(ϕ∗g)
α
= α(x)m (ϕ).
α
The latter shows that every α-mean on UC(K) extends on L∞(K). Plainly the restriction of any
α-mean ofK onUC(K) isaα-mean onUC(K). Therefore, thestatementisvalid.
Corollary 3.3. Let α ∈ K and UC(K) ⊆ B ⊆ L∞(K). Then K is α-amenable if and only if B is
α-amenable.
b
TheBanach spaceL∞(K)∗ withtheArens productdefined as followsisaBanach algebra:
hm·m′,fi = hm,m′ ·fi, in whichhm′ ·f,gi = hm′,f ·gi,
and hf ·g,hi = hf,g ∗hi for all m,m′ ∈ L∞(K)∗, f ∈ L∞(K) and g,h ∈ L1(K) where hf,gi :=
f(g).
The Banach space UC(K)∗ with the restriction of the Arens product is a Banach algebra, and it can
beidentified withaclosedrightideal oftheBanach algebraL1(K)∗∗ [14].
If m,m′ ∈ UC(K)∗, f ∈ UC(K), and x ∈ K, thenm′ ·f ∈ UC(K) andwe mayhave
hm·m′,fi = hm,m′ ·fi, hm′ ·f,xi = hm′,T fi.
x
If y ∈ K, then T fdω(x,y)(t) = T (T f) whichimpliesthat
K t y x
R
T (m·f) = m·T f,
x x
as
T (m·f)(y) = m·f(t)dω(x,y)(t)
x
Z
K
= hm,T fidω(x,y)(t)
t
Z
K
= hm, T fdω(x,y)(t)i.
t
Z
K
4
Let X = UC(K), f ∈ X and g ∈ C (K) (g ≥ 0) with kgk = 1. Since the mapping x → T f is
c 1 x
continuous from K to (Cb(K),k·k ) and the point evaluation functionals in X∗ separate points of
∞
X, wehave
g ∗f = g(x)T fdm(x).
x˜
Z
K
Therefore, for every 1-mean m on X we have m(f) = m(g ∗f). Hence, two 1-means m and m′ on
L∞(K) are equal iftheyare equal onUC(K), as
m(f) = m(g ∗f) = m′(g ∗f) = m′(f) (f ∈ L∞(K))
and g is assumed as above. The latter together with [20, Theorem 3.2] show a bijection between
means on UC(K) and L∞(K). So, if UC(K) is amenable with a unique mean, then its extension
on L∞(K) is also unique which implies that K is compact [13], therefore the identity character is
isolatedin S, thesupportofthePlancherel measure. Wehavethefollowingtheoremingeneral:
Theorem 3.4. If UC(K) is α-amenable with the unique α-mean muc, then muc ∈ L1(K)∩L2(K),
α α
{α} is isolated in S and muc = π(α), where π : L1(K) → L1(K)∗∗ is thecanonical embedding. If α
α kαk22
is positive,thenK is compact.
Proof: Let muc be the unique α-mean on UC(K), n ∈ UC(K)∗ and {n } be a net converging to n
α i i
in thew∗-topology. Let x ∈ K and f ∈ UC(K). Then
hmuc ·n ,T fi = hmuc,n ·(T f)i (2)
α i x α i x
= hmuc,T (n ·f)i
α x i
= α(x)hmuc,n ·fi
α i
= α(x)hmuc ·n ,fi.
α i
For λ = hn ,αi =6 0, sincethe associated functional to the character α on UC(K)∗ is multiplicative
i i
[27],muc·n /λ isaα-meanonUC(K)whichisequaltomuc. Thenthemappingn → muc·ndefined
α i i α α
on UC(K)∗ isw∗-w∗ continuous,hencemuc isin thetopologicalcentreofUC(K)∗, i.e. M(K); see
α
[13, Theorem 3.4.3]. Since muc(β) = δ (β) and muc ∈ Cb(K), {α} is an open-closed subset of K.
α α α
The algebra L1(K) is a two-sided closed ideal in M(K) and the Fourier transform is injective, so
d d b b
muc andαbelongtoL1(K)∩L2(K). Theinversiontheorem,[3,Theorem2.2.36],indicatesα = αˇ,
α
accordinglyα ∈ S. Letm := π(α)/kαk2. Obviouslym isaα-meanonL∞(K),andtherestriction
α 2 α b
ofm onUC(K) yieldsthedesired uniqueα-mean.
α
If αispositive,then
α(x) α(t)dm(t) = T α(t)dm(t) = α(t)dm(t)
x
Z Z Z
K K K
which impliesthatα = 1, henceK iscompact.
5
Observe that in contrast to the case of locally compact groups, there exist noncompact hypergroups
withuniqueα-means. Forexample,forlittleq-Legendrepolynomialhypergroups,wehaveK\{1} ⊂
L1(K)∩L2(K);see[8]. Therefore,byTheorem3.4,UC(K)andK areα-amenablewiththeunique
b
α-mean m ,whereas K hasinfinitelymany1-means[20].
α
Let Σ (X) be theset ofall α-meanson X = L∞(K) orUC(K). If α = 1, then Σ (X) is nonempty
α 1
(as K is commutative) weak∗-compact convex set in X∗ [20]. If α 6= 1 and X is α-amenable, then
thesameistrueforΣ (X).
α
Theorem 3.5. Let X be α-amenable (α 6= 1). Then Σ (X) is a nonempty weak∗-compact convex
α
subset of X∗. Moreover, Σ (X)·M/hM,αi ⊆ Σ (X), for all M ∈ X∗ with hM,αi =6 0. Further-
α α
more, ifm ∈ Σ (X), thenmn = m foralln ∈ N.
α α α α
Proof: Let 0 ≤ λ ≤ 1 and m , m′ ∈ Σ (X). If m′′ := λm + (1 − λ)m′ , then m′′(α) = 1 and
α α α α α α α
m′′(T f) = α(x)m′′(f),foreveryf ∈ X andx ∈ K;hence m′′ ∈ Σ (X).
α x α α α
w∗
If {m } ⊂ Σ (X) such thatm −→ m, then m ∈ Σ (X). We havem(α) = 1 and
i α i α
m(T f) = limm (T f) = α(x)limm (f) = α(x)m(f),
x i x i
i→∞ i→∞
for all f ∈ X and x ∈ K. Moreover, Σ (X) is w∗-compact by Alaoglu’s theorem [6, p.424]. Let
α
M ∈ X∗ withλ = hM,αi =6 0. Then M′ := m ·M/λ isaα-mean onX, as
α
hm ·M,T fi = hm ,M ·T fi = hm ,T (M ·f)i = α(x)hm ,M ·fi = α(x)hm ·M,fi.
α x α x α x α α
Since g ·m = m ·g = g∗(α)m for all g ∈ L1(K), thecontinuityof theArens product in thefirst
α α α
variableonX togetherwithGoldstein’stheoremyieldm2 = m ;hence,mn = m foralln ∈ N.
b α α α α
Remark 3.6. Observe that if K is α and β-amenable, then m · m = m · m if and only if
α β β α
m ·m = δ (β)m ,as
α β α α
α(x)hm ·m ,fi = α(x)hm ,m ·fi
α β α β
= hm ,T (m ·f)i
α x β
= hm ,m ·T fi
α β x
= hm ·m ,T fi
α β x
= hm ·m ,T fi
β α x
= hm ,m ·T fi
β α x
= hm ,T (m ·f)i
β x α
= β(x)hm ,m ·fi
β α
= β(x)hm ·m ,fi (f ∈ X,x ∈ K).
β α
6
4 α-Amenability of Quotient Hypergroups
A closed nonempty subset H of K is called a subhypergroup if H · H = H and H˜ = H, where
H˜ := {x˜ : x ∈ H}. Let H be a subhypergroup of K. Then K/H := {xH : x ∈ K} is a locally
compact spacewithrespectto thequotienttopology. IfH isasubgrouporacompact subhypergroup
ofK, then
ω(xH,yH) := δ dω(x,y)(z) (x,y ∈ K)
zH
Z
K
defines a hypergroup structure on K/H, which agrees with the double coset hypergroup K//H; see
[12]. The properties and duals of subhypergroups and qoutient of commutative hypergroups have
been intensivelystudiedby M.Voit[23, 24].
Theorem 4.1. Let H be a subgroup or a compact subhypergroup of K. Suppose p : K → K/H is
[
the canonical projection, and p : K/H −→ K is defined by γ 7−→ γop. Then K/H is γ-amenableif
and onlyifK isγop-amenable.
b b
Proof: Let K/H be γ-amenable. Then there exists a M : Cb(K/H) −→ C such that M (γ) = 1,
γ γ
and M (T f) = γ(xH)M (f).
γ xH γ
Since H isamenable[20], letm beamean onCb(H). Forf ∈ Cb(K), define
1
f1 : K −→ C by f1(x) := hm ,T f| i.
1 x H
Thefunctionf1 iscontinuous,bounded,and sincem isamean forH,wehave
1
T f1(x) = f1(t)dω(h,x)(t)
h
Z
K
= hm ,T f| idω(h,x)(t)
1 t H
Z
K
= hm , T f| dω(h,x)(t)i
1 t H
Z
K
= hm ,T [T f| ]i = hm ,T f| i = f1(x),
1 h x H 1 x H
for all h ∈ H. Then according to theassumptionson H, [24, Lemma1.5]impliesthat f1 is constant
on thecosets ofH in K. Wemaywritef1 = Fof,F ∈ Cb(K/H).Define
m : Cb(K) −→ C by m(f) = hM ,Fi.
γ
Wehave
F(yH) = T f1(y) = hm ,T f| idω(x,y)(u) = hm , T (T f)| i,
xH x 1 u H 1 y x H
Z
K
asu → T f| iscontinuousfromK to(Cb(H),k·k )andthepointevaluationfunctionalsinCb(H)∗
u H ∞
separates thepointsofCb(H);henceT Fop = (T f)1.Therefore,
xH x
m(T f) = hM ,T Fi = γ(xH)hM ,Fi = α(x)m(f).
x γ xH γ
7
Moreover, hm,αi = hM ,γi = 1, where γop = α. Then m(T f) = α(x)m(f) for all f ∈ Cb(K)
γ x
and x ∈ K.
To showtheconverse, letm beaα-mean onCb(K), and define
α
M : Cb(K/H) −→ C by hM,fi = hm ,fopi.
α
Since
T f(yH) = T fop(y) = fdω(xH,yH) = fopdω(x,y) = T fop(y),
xH xH x
Z Z
K/H K
so M(T f) = hm ,T fopi = α(x)hm ,fopi = α(x)hM,fi. Since p is an isomorphism, [23,
xH α x α
Theorem 2.5],and γop = α, wehave
b
hM,γi = hm ,γopi = hm ,αi = 1.
α α
Therefore, M is thedesired γ-mean on Cb(K/H).
4.2. Example: Let H be compact and H′ be discrete commutativehypergroups. Let K := H ∨H′
denotes thejointhypergroupthatH isasubhypergroupofK and K/H ∼= H′ [25]. Thehypergroups
K, H and H′ are amenable, and H is β-amenable for every β ∈ H [7]. By Theorem 4.1, K is
α-amenableifandonlyifH′ isγ-amenable(α = γop).
b
Remark4.3. LetGbea[FIA] -group[16]. ThenthespaceG ,B-orbitsinG,formsahypergroup
B B
[12, 8.3]. If G is an amenable [FIA] -group, then the hypergroup G is amenable [20, Corollary
B B
3.11]. However, G may not be α-amenable for α ∈ G \ {1}. For example, let G := Rn and B
B B
be thegroup of rotations which acts on G. Then the hypergroup K := G can be identified with the
c B
Bessel-Kingman hypergroup R := [0,∞) of order ν = n−2. Theorem 5.5 will show that if n ≥ 2
0 2
then R is α-amenableif and only if α = 1. Observe that L1(R ) is an amenable Banach algebra for
0 0
n = 1[26];henceeverymaximalidealofL1(R )hasab.a.i. [4],consequentlyG isα-amenablefor
0 B
everyα ∈ G [7].
B
LetK andHcbehypergroupswithleftHaarmeasures. ThenitisstraightforwardtoshowthatK×H
isahypergroupwithaleftHaarmeasure. IfK andH arecommutativehypergroups,thenK×H isa
commutativehypergroupwithaHaarmeasure. Asinthecaseoflocallycompactgroups[5],wehave
thefollowingisomorphism
φ : L∞(K)×L∞(H) −→ L∞(K ×H) by (f,g) −→ φ , (3)
(f,g)
where φ (x,y) = f(x)g(y)for all (x,y) ∈ K ×H. Let (x′,y′) ∈ K ×H and (f,g) ∈ L∞(K)×
(f,g)
L∞(H). Then
T(x′,y′)φ(f,g)(x,y) = φ(f,g)(t,t′)dω(x′,x)×ω(y′,y)(t,t′)
Z
K×H
= f(t)g(t′)dω(x′,x)(t)dω(y′,y)(t′)
Z Z
K H
= Tx′f(x)Ty′g(y)
= φ (x,y).
(T ′f,T ′g)
x y
8
Theorem 4.4. LetK and H becommutativehypergroups. Then
\
(i) the map φ defined in (3) is a homeomorphism between K × H and K ×H, where the dual
spaces bearthecompact-opentopologies.
b b
(ii) K ×H is φ -amenableifand onlyifK and H are αand β-amenablerespectively.
(α,β)
Proof: (i)It isthespecialcase of[4,Proposition19].
(ii) As in the case of locally compact groups [5], we have L1(K × H) ∼= L1(K) ⊗ L1(H), where
p
⊗ denotes the projection tensor product of two Banach algebras. If K is α-amenable and H is
p
β-amenable, then I(α) and I(β), the maximal ideals of L1(K) and L1(H) respectively, have b.a.i.
[7]. Since L1(K) and L1(H) have b.a.i., L1(K) ⊗ I(β) + I(α) ⊗ L1(H) , the maximal ideal in
p p
L1(K × H) associated to the character φ , has a b.a.i. [5, Proposition 2.9.21], that equivalently
(α,β)
K ×H isφ -amenable.
(α,β)
To provetheconverse, supposem isaφ -mean onL∞(K ×H), define
(α,β) (α,β)
m : L∞(K) −→ C byhm ,fi := hm ,φ i.
α α (α,β) (f,β)
Wehave
hm ,T fi = hm ,φ i
α x (α,β) (Txf,β)
= hm ,T φ i
(α,β) (x,e) (f,β)
= α(x)β(e)hm ,φ i
(α,β) (f,β)
= α(x)hm ,φ i
(α,β) (f,β)
= α(x)hm ,fi,
α
for all f ∈ L∞(K) and x ∈ K. Since hm ,αi = hm ,φ i = 1, K is α-amenable. Similarly
α (α,β) (α,β)
m : L∞(H) −→ C defined by m (g) := hm ,φ i is a β-mean on L∞(H), hence H is
β β (α,β) (α,g)
β-amenable.
Remark4.5. TheproofoftheprevioustheoremmayalsofollowfromTheorem3.2withamodifying
[20, Proposition3.8].
5 α-Amenability of Sturm-Liouville Hypergroups
SupposeA : R → Riscontinuous,positive,andcontinuouslydifferentiableonR \{0}. Moreover,
0 0
assumethat
A′(x) γ (x)
= 0 +γ (x), (4)
1
A(x) x
forall xin aneighbourhoodof0,withγ ≥ 0 such that
0
SL1 oneofthefollowingadditionalconditionsholds.
9
SL1.1 γ > 0 andγ ∈ C∞(R), γ beingan oddfunction,or
0 1 1
SL1.2 γ = 0 andγ ∈ C1(R ).
0 1 0
SL2 Thereexistsη ∈ C1(R )suchthatη(0) ≥ 0, A′ −β isnonnegativeanddecreasingonR \{0},
0 A 0
and q := 1η′ − 1η2 + A′η is decreasingon R \{0}.
2 4 2A 0
The function A is called Che´bli-Trime`che if A is a Sturm-Liouville function of type SL1.1 satisfy-
ing the additional assumptions that the quotient A′ ≥ 0 is decreasing and that A is increasing with
A
limA(x) = ∞.In thiscase SL2 isfulfilled withη := 0.
x→∞
Let A be a Sturm-Liouvillefunction satisfying (4) and SL2. Then there exists always a uniquecom-
mutativehypergroupstructureonR suchthatA(x)dxistheHaarmeasure. Ahypergroupestablished
0
by this way is called a Sturm-Liouvillehypergroup and it will be denoted by (R ,A(x)dx). If A is a
0
Che´bli-Trime`chefunction,then thehypergroup(R ,A(x)dx) iscalled Che´bli-Trime`chehypergroup.
0
Thecharacters of(R ,A(x)dx) can beconsidered as solutionϕ ofthedifferentialequation
0 λ
d2 A′(x) d
+ ϕ = −(λ2 +ρ2)ϕ , ϕ (0) = 1, ϕ′ (0) = 0,
(cid:18)dx2 A(x) dx(cid:19) λ λ λ λ
where ρ := lim A′(x), and λ ∈ R := R ∪ i[0,ρ]; see [3, Proposition 3.5.49]. As shown in [3,
2A(x) ρ 0
x→∞
Sec.3.5], ϕ isastrictlypositivecharacter, and ϕ has thefollowingintegralrepresentation
0 λ
x
ϕ (x) = ϕ (x) e−iλtdµ (t) (5)
λ 0 x
Z
−x
whereµ ∈ M1([−x,x]) foreveryx ∈ R andall λ ∈ C . Intheparticularcaseλ := iρ, theequality
x 0
(5)yields|ϕ (x)| ≤ e−ρx, as ϕ = 1.
0 iρ
Proposition5.1. Let ϕ beas above. Then
λ
dn
ϕ (x) ≤ xne(|Imλ|−ρ)x
(cid:12)dλn λ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
forall x ≥ 0, λ ∈ C and n ∈ N. (cid:12) (cid:12)
Proof: For all λ ∈ C\{0} and x > 0, applying the Lebesgue dominated convergence theorem [11]
yields
dn x
ϕ (x) = ϕ (x) (−it)ne−iλtdµ (t),
dλn λ 0 Z x
−x
hence
dn x
ϕ (x) ≤ ϕ (x)xn e−iλt dµ (t)
(cid:12)dλn λ (cid:12) 0 Z x
(cid:12) (cid:12) −x(cid:12) (cid:12)
which impliesthat ddλnnϕλ(x)(cid:12)(cid:12) ≤ xne(|Im(cid:12)(cid:12)λ|−ρ)x. (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
To study the α-amenability of Sturm-Liouville hypergroups, we may require the following fact in
general. ThefunctionalD ∈ L1(K)∗ iscalled aα-derivation(α ∈ K) onL1(K) if
D(f ∗g) = f(α)D (g)+g(α)D (f) (f,gb∈ L1(K)).
α α
b
Observethatifthemaximalideal I(α)has an apbproximateidentity,then D| = 0.
I(α)
10