Non-synthesizable varieties

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]. 6 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. [1] R. J. Ellio , Two no es on spec al syn hesis o disc e e Abelian g oups, Ma h. P oc. Camb idge Phil. Soc. 61(1965), 617–620. 7 [2] Z. Gajda, P i a e communica ion, Hambu g–Rissen, (1987) [3] D. I. Gu e iˇc, Coun e examples o a p oblem o L. Schwa z, Funkcional. Anal. i P iloˇzen 9(2)(1975), 29–35. (English ansla ion: Funkcional Anal. Appl. 9(2)(1975), 116–120.) [4] E. Hewi and K. Ross, Abs ac Ha monic Analysis I, II. Die G undleh- en de Ma hema ischen Wissenscha en. ol. 115. Sp inge Ve lag, Be lin, G¨o ingen, Heidelbe g, 1963. [5] N. Jacobson. The adical and semi-simplici y o a bi a y ings, Ame . J. Ma h. 67:300–320, 1945. [6] M. Laczko ich and L. Sz´ekelyhidi, Spec al syn hesis on disc e e Abelian g oups, Ma h. P oc. Camb. Phil. Soc. 143(01)(2007), 103–120. [7] M. Le anc, Analyse spec ale su Zn, C. R. Acad. Sci. Pa is. 246(1958), 1951–1953. [8] L. Schwa z, Th´eo ie g´ene ale des onc ions moyenne-p´e iodiques, Annals o Ma h. 48(4)(1947), 857–929. [9] L. Sz´ekelyhidi, The ailu e o spec al syn hesis on some ypes o disc e e Abelian g oups, Jou . Ma h. Anal. Appl. 291(2004), 757–763. [10] L. Sz´ekelyhidi, Polynomial unc ions and spec al syn hesis, Aequa iones Ma h. 70(1–2)(2005), 122–130. [11] L. Sz´ekelyhidi, Disc e e Spec al Syn hesis and I s Applica ions, Sp inge Monog aphs in Ma hema ics, Sp inge , Do d ech , The Ne he lands, 2006. [12] L. Sz´ekelyhidi, Spec al syn hesis p oblems on locally compac g oups, Mona s. Ma h. 161(2)(2010), 223–232. [13] L. Sz´ekelyhidi, Annihila o me hods in disc e e spec al syn hesis, o appea in Ac a Ma h. Hung. 8