Non-syn hesizable a ie ies
G´abo Ho ´a h
Ins i u e o Ma hema ics,
Uni e si y o Deb ecen,
P . 12, Deb ecen, 4010, Hunga y
L´aszl´o Sz´ekelyhidi
Depa men o Ma hema ics
Uni e si y o Bo swana
P i a e Bag UB 0022
4775 Gabo one
Bo swana
Be ina Wilkens
Depa men o Ma hema ics
Uni e si y o Bo swana
P i a e Bag UB 0022
4775 Gabo one
Bo swana
Abs ac
In 2004 a coun e example was gi en o a 1965 esul o R. J. Ellio claiming
ha disc e e spec al syn hesis holds on e e y Abelian g oup. He e we p esen
a ing- heo e ical app oach o his p oblem, and show ha some a ie ies ail
o ha e spec al syn hesis. In pa icula , we gi e a new p oo o he esul o
he second au ho ha spec al syn hesis does no hold on Abelian g oups wi h
in ini e o sion ee ank.
Keywo ds: spec al syn hesis, A in ing, exponen ial monomial
2010 MSC: 43A45, 43A70, 16P20
1. In oduc ion
Spec al analysis and spec al syn hesis deal wi h he desc ip ion o ans-
la ion in a ian unc ion spaces o e locally compac Abelian g oups. One con-
side s he space C(G) o all complex alued con inuous unc ions on a locally
compac Abelian g oup G, which is a locally con ex opological linea space wi h
espec o poin -wise linea ope a ions (addi ion, mul iplica ion wi h scala s) and
Email add esses: [email p o ec ed] (G´abo Ho ´a h),
[email p o ec ed] (L´aszl´o Sz´ekelyhidi), [email p o ec ed] (Be ina Wilkens)
P ep in submi ed o Else ie Ma ch 5, 2014
o he opology o uni o m con e gence on compac se s. The ansla e by yin
Go an elemen in C(G) is de ined by τy (x) = (x+y) o each xin G. A
subse in C(G) is called ansla ion in a ian i i con ains e e y ansla es o all
o i s elemen s. A closed ansla ion in a ian linea subspace o he space C(G) is
called a a ie y on G. Con inuous homomo phisms o Gin o he addi i e [mul-
iplica i e] opological g oup o [nonze o] complex numbe s a e called addi i e
[exponen ial] unc ions. A unc ion is a polynomial i i belongs o he algeb a
gene a ed by he addi i e unc ions and cons an s. Usually, he p oduc o a poly-
nomial and an exponen ial is called an exponen ial monomial. This is equi alen
o he p ope y ha he unc ion gene a es a ini e dimensional indecomposable
a ie y (see e.g. [12]). We shall use his la e de ini ion he e.
I u ns ou ha exponen ial unc ions, o mo e gene ally, exponen ial mono-
mials can be conside ed as basic building blocks o a ie ies. A gi en a ie y may
o may no con ain any exponen ial unc ion o exponen ial monomial. I each
nonze o sub a ie y o a gi en a ie y con ains an exponen ial unc ion, hen we
say ha spec al analysis holds o he a ie y. Ano he p ope y is i he a i-
e y is syn hesizable, which means ha all exponen ial monomials in his a ie y
span a dense subspace in he a ie y. I each sub a ie y o a gi en a ie y is
syn hesizable, hen we say ha spec al syn hesis holds o he a ie y. I can
be shown ha spec al syn hesis o a a ie y implies spec al analysis, oo (see
[12], Theo em 1). I spec al analysis, espec i ely, spec al syn hesis holds o
e e y nonze o a ie y on an Abelian g oup, hen we say ha spec al analysis,
espec i ely, spec al syn hesis holds on he g oup. A amous and pionee esul
o L. Schwa z [8] exhibi s he si ua ion in a classical case by s a ing ha i he
unde lying g oup is he eals wi h he Euclidean opology, hen e e y nonze o
a ie y con ains an exponen ial unc ion, ha is, spec al analysis holds on he
eals. Mo eo e , spec al syn hesis also holds: he e a e su icien ly many expo-
nen ial monomials in each a ie y in he sense ha hei linea hull is dense in
he a ie y.
In his 1958 esul [7] M. Le anc p o ed ha spec al syn hesis holds on he
g oup Zn o each posi i e in ege n,Zbeing he in ege s. In his 1965 pape
[1] R. J. Ellio p esen ed a heo em on spec al syn hesis o a bi a y Abelian
g oups. Howe e , in his 1987 p i a e communica ion [2] Z. Gajda called he sec-
ond au ho ’s a en ion o he ac ha he p oo o Ellio ’s heo em had se e al
gaps. La e on se e al e o s ha e been made o sol e he p oblem o disc e e
spec al analysis and spec al syn hesis on a bi a y Abelian g oups. Finally, a
coun e example o Ellio ’s heo em was p esen ed a he 41s In e na ional Sym-
posium on Func ional Equa ions, Nosz aj, Hunga y, 2003 (see [9]). Fo basics,
u he de elopmen s and e e ences on spec al analysis and spec al syn hesis
he eade should e e o [3,11,12].
In his pape we gi e a new p oo o he ailu e o spec al syn hesis on
some ypes o disc e e Abelian g oups, which was shown by a coun e example
2
in [9]. Ou me hod is based on ing- heo e ical esul s and uses he annihila o
echnique. The basics o his me hod ha e been wo ked ou in [13].
2. Basic concep s
In his pape we conside disc e e commu a i e g oups only, and C(G) is he
se o all unc ions om G o C. Le Gbe an Abelian g oup and le CGdeno e
i s g oup algeb a, which is iden i ied wi h he se o all ini ely suppo ed complex
alued unc ions on G. Mo eo e , his se can be iden i ied wi h he se o all
ini ely suppo ed complex measu es Mc(G) on G, using he de ini ion
ZG
dµ =X
x∈G
(x)µ(x),
whene e µis in CGand is in C(G). This o mula exp esses he well-known
ac abou he dual C(G)∗o he opological ec o space C(G): i is iden i ied
wi h Mc(G), he pai ing gi en by he p e ious o mula.
Via hese iden i ica ions he mul iplica ion in he complex algeb a CGis gi en
by he con olu ion o measu es as
µ∗ν( ) = X
x,y∈G
(x+y)µ(x)ν(y)
o each µ, ν in CGand in C(G). Wi h his ope a ion CGis a commu a i e
uni al complex algeb a wi h iden i y δ0, which is he poin mass concen a ed
a 0, he ze o elemen o G. Mo e gene ally, we deno e by δx he cha ac e is ic
unc ion o he single on {x} o each xin G: i akes he alue 1 a he elemen
xand 0 o he wise.
Con olu ion is also de ined be ween elemen s o CGand C(G) in he ollowing
manne :
µ∗ (x) = X
y∈G
(x−y)µ(y),
whene e µis in CG, is in C(G) and xis in G. Wi h his ope a ion C(G)
u ns in o a CG-module, closed submodules being exac ly he a ie ies. The
in e sec ion o all a ie ies including a pa icula in C(G) is called he a ie y
o and is deno ed by τ( ).
Fo each subse Hin C(G) he annihila o H⊥o Hin CGis de ined by
H⊥={µ:µ∗ = 0 o each in H}.
I is easy o see ha H⊥is an ideal in CG. I H={ }is a single on, hen
H⊥=τ( )⊥and we call i he annihila o o .
3
Simila ly, he annihila o K⊥in C(G) o a subse Kin CGis de ined by
K⊥={ :µ∗ = 0 o each µin K}.
I is also easy o check ha K⊥is a a ie y in C(G).
The ollowing wo heo ems a e impo an echnical ools (see [6,13]).
Theo em 2.1. Le Gbe an Abelian g oup, Va a ie y on Gand Ian ideal in
CG. Then we ha e
V⊥⊥ =V, I⊥⊥ =I .
Theo em 2.2. Le Gbe an Abelian g oup, (Vγ)γ∈Γa amily o a ie ies on G
and (Iγ)γ∈Γa amily o ideals in CG. Then we ha e
(X
γ∈Γ
Vγ)⊥=
γ∈Γ
V⊥
γ,(
γ∈Γ
Iγ)⊥=X
γ∈Γ
I⊥
γ.
3. Exponen ials and maximal ideals
The basic building blocks o spec al analysis and spec al syn hesis a e ex-
ponen ial monomials. We call he eade ’s a en ion ha we shall use he wo d
”exponen ial” in se e al di e en meanings in he sequel. The gene alized cha -
ac e s o G, ha is, he homomo phisms o Gin o he mul iplica i e g oup o
nonze o complex numbe s will be called exponen ial unc ions, o simply expo-
nen ials. La e on we shall also use he e ms ”exponen ial maximal ideal”,
”exponen ial monomial”, and ”gene alized exponen ial monomial”, which e e
o di e en , howe e , ela ed concep s.
Exponen ials can be cha ac e ized by a numbe o p ope ies. We shall use
he ollowing esul (see [13, Theo ems 3 and 4], and [13, Co olla ies 1 and 2]).
Theo em 3.1. Le Gbe an Abelian g oup and m:G→Cbe an a bi a y unc-
ion. Then he ollowing condi ions a e equi alen :
1. mis an exponen ial.
2. The a ie y o mis one dimensional and m(0) = 1.
3. The annihila o τ(m)⊥is a maximal ideal in CG,CG/τ(m)⊥is isomo phic
o Cand m(0) = 1.
Maximal ideals Min CGwi h he p ope y ha CG/M ∼
=Cplay an impo -
an ole, and hey will be called exponen ial maximal ideals. They a e closely
ela ed o modi ied di e ences de ined as ollows. Fo each unc ion :G→C
and yin Gwe de ine
∆ ;y=δ−y− (y)δ0.
The measu e ∆ ;yis called modi ied di e ence. Fo p oduc s o modi ied di e -
ences we shall use he no a ion
∆ ;y1,y2,...,yn+1 = Πn+1
i=1 ∆ ;yi,
4
whene e y1, y2, . . . , yn+1 a e in G. The p oduc on he igh side is mean as a
con olu ion.
Gi en in C(G) he ideal in CGgene a ed by all modi ied di e ences o he
o m ∆ ;ywi h yin Gis deno ed by M . I is easonable o ask whe he M is
p ope . We ha e he ollowing esul .
Theo em 3.2. Le Gbe an Abelian g oup and :G→Cbe a unc ion. The
ideal M is p ope i and only i is an exponen ial. In his case M =τ( )⊥,
hence M is an exponen ial maximal ideal.
P oo . Suppose i s ha M is p ope . Then M⊥
is a nonze o a ie y, by Theo-
em 2.1, hence he e is a nonze o gannihila ed by all modi ied di e ences o he
o m ∆ ;y. Fo x, y in Gwe ha e
0=∆ ;y∗g(x) = g(x+y)− (y)g(x).(3.1)
Pu ing x= 0 we ha e g(y) = g(0) · (y). In pa icula , g(0) 6= 0, 6= 0 and we
ob ain (x+y) = (x) (y). As is nonze o, i ollows (0) = 1, hence is an
exponen ial.
Con e sely, suppose ha =mis an exponen ial. Then mis in M⊥
m, as
ob iously ∆m;y∗m(x) = m(x+y)−m(y)m(x) = 0 o each x, y in G. Hence Mm
is p ope . Mo eo e , τ(m)⊥is an exponen ial maximal ideal in CG, by Theo em
3.1. I gis in M⊥
, hen, by (3.1), i is a cons an mul iple o m, hence i belongs
o τ(m). I ollows M⊥
m⊆τ(m), hus τ(m)⊥⊆Mm. As τ(m)⊥is maximal and
Mmis p ope , we ha e τ(m)⊥=Mm, and he heo em is p o ed.
These la e wo esul s ha e been used in [13] o p o e he ollowing cha ac-
e iza ion esul s.
Theo em 3.3. Le Gbe an Abelian g oup and Vbe a a ie y on G. Then
spec al analysis holds o Vi and only i each maximal ideal con aining V⊥is
exponen ial.
Co olla y 3.4. Le Gbe an Abelian g oup. Then spec al analysis holds on Gi
and only i each maximal ideal in CGis exponen ial.
Co olla y 3.5. Le Gbe an Abelian g oup and Vbe a a ie y on G. Then spec al
analysis holds o Vi and only i each maximal ideal in CG/V ⊥is exponen ial.
4. Exponen ial monomials
Le Gbe an Abelian g oup. The a ie y Von Gis called decomposable, i
i is he sum o wo p ope sub a ie ies, which means ha he algeb aic sum
o wo p ope sub a ie ies is a dense submodule in i . O he wise i is called
indecomposable. The ollowing heo em is ob ious, by Theo em 2.2.
5

Theo em 4.1. Le Gbe an Abelian g oup. A a ie y on Gis decomposable i
and only i i s annihila o is he in e sec ion o wo ideals, which a e di e en
om i .
Le Gbe an Abelian g oup. The unc ion :G→Cis called a gene alized
exponen ial monomial, i he e exis s an exponen ial mand a na u al numbe n
such ha o each y1, y2, . . . , yn+1 we ha e
∆m;y1,y2,...,yn+1 ∗ = 0 (4.1)
holds. We e o mula e his de ini ion in e ms o he annihila o o .
Theo em 4.2. Le Gbe an Abelian g oup. The unc ion :G→Cis a gene -
alized exponen ial monomial i and only i i s annihila o con ains some posi i e
powe o an exponen ial maximal ideal.
P oo . The condi ion o he heo em is equi alen o he ollowing condi ion:
he e exis s an exponen ial mand a na u al numbe nsuch ha
Mn+1
m⊆τ( )⊥.(4.2)
As he modi ied di e ences ∆m;ywi h yin Ggene a e Mm, hence he modi ied
di e ences ∆m;y1,y2,...,yn+1 gene a e Mn+1
m, ha is, (4.1) and (4.2) a e equi alen
o .
I is easy o check (see [13, Theo em 7]) ha condi ion (4.2) can hold o a
mos one exponen ial m.
Theo em 4.3. Le Gbe an Abelian g oup and :G→Cbe a nonze o gene alized
exponen ial monomial. Then he e is a unique exponen ial msa is ying (4.1) o
some na u al numbe n. In o he wo ds, he e is a unique exponen ial maximal
ideal Msa is ying Mn+1 ⊆τ( )⊥ o some na u al numbe n.
Now we ha e he ollowing cha ac e iza ion esul s (see [13, Theo em 8]).
Theo em 4.4. Le Gbe an Abelian g oup. The unc ion :G→Cis a nonze o
gene alized exponen ial monomial i and only i CG/τ( )⊥is a local ing wi h
nilpo en exponen ial maximal ideal.
Theo em 4.5. Le Gbe an Abelian g oup. The unc ion :G→Cis an expo-
nen ial monomial i and only i CG/τ( )⊥is a local A in ing wi h exponen ial
maximal ideal.
P oo . By he p e ious heo em CG/τ( )⊥is a local ing wi h exponen ial max-
imal ideal. Any descending chain o ideals in CG/τ( )⊥induces a descending
chain o ideals con aining τ( )⊥in CG, which induces an ascending chain o
sub a ie ies in τ( ), hence, by ini e dimensionali y, i e mina es.
Fo he con e se see [13, Theo em 8].
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5. The ailu e o spec al syn hesis
Theo em 5.1. Le Gbe an Abelian g oup and Vbe a a ie y on G. I Vis
indecomposable, and spec al syn hesis holds o V, hen CG/V ⊥is a local A in
ing.
P oo . I Vis syn hesizable, hen i is he opological sum o all sub a ie ies
gene a ed by exponen ial monomials belonging o V, by de ini ion. This means
ha we ha e, by Theo em 2.2,
V⊥=
ϕ∈V
τ(ϕ)⊥,(5.1)
whe e he in e sec ion is ex ended o all exponen ial monomials ϕin V. As Vis
indecomposable, hence, by Theo em 4.1,V⊥is equal o one o he ac o s o he
in e sec ion on he igh side, ha is V⊥=τ(ϕ)⊥ o some exponen ial monomial
ϕin V. By Theo em 4.5,CG/V ⊥is a local A in ing.
Theo em 5.2. Le Gbe an Abelian g oup, and le :G→Cbe a gene alized ex-
ponen ial monomial. Then τ( )is syn hesizable i and only i is an exponen ial
monomial.
P oo . The s a emen is ob ious by he de ini ion o exponen ial monomials and
by he p e ious heo em.
The ollowing heo em ollows immedia ely.
Theo em 5.3. Le Gbe an Abelian g oup and Vbe a a ie y on G. I he e is
a gene alized exponen ial monomial in V, which is no an exponen ial monomial,
hen spec al syn hesis does no hold o V.
As a consequence we ob ain he ollowing esul (see [9]).
Theo em 5.4. Le Gbe an Abelian g oup wi h in ini e o sion ee ank. Then
spec al syn hesis ails o hold on G.
P oo . Indeed, by [10, Theo em 3], he o sion ee ank o an Abelian g oup is
in ini e i and only i he e is a gene alized exponen ial monomial on he g oup,
which is no an exponen ial monomial.
Acknowledgmen s
The i s au ho was pa ially suppo ed by he Hunga ian Na ional Foun-
da ion g an no. K109185, and by he J´anos Bolyai Resea ch Schola ship o he
Hunga ian Academy o Sciences. The second au ho ’s esea ch was suppo ed
by he Hunga ian Na ional Founda ion o Scien i ic Resea ch (OTKA), G an
No. NK-81402.
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7
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