Non-synthesizable varieties

Non-synthesizable varieties G´abor Horv´ath Institute of Mathematics, University of Debrecen, Pf. 12, Debrecen, 4010, Hungary L´aszl´o Sz´ekelyhidi Department of Mathematics University of Botswana Private Bag UB 0022 4775 Gaborone Botswana Bettina Wilkens Department of Mathematics University of Botswana Private Bag UB 0022 4775 Gaborone Botswana Abstract In 2004 a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Here we present a ring-theoretical approach to this problem, and show that some varieties fail to have spectral synthesis. In particular, we give a new proof for the result of the second author that spectral synthesis does not hold on Abelian groups with infinite torsion free rank. Keywords: spectral synthesis, Artin ring, exponential monomial 2010 MSC: 43A45, 43A70, 16P20 1. Introduction Spectral analysis and spectral synthesis deal with the description of translation invariant function spaces over locally compact Abelian groups. One considers the space C(G) of all complex valued continuous functions on a locally compact Abelian group G, which is a locally convex topological linear space with respect to point-wise linear operations (addition, multiplication with scalars) and Email addresses: [email protected] (G´abor Horv´ath), [email protected] (L´aszl´o Sz´ekelyhidi), [email protected] (Bettina Wilkens) Preprint submitted to Elsevier March 5, 2014 to the topology of uniform convergence on compact sets. The translate by yin Gof an element fin C(G) is defined by τyf(x) = f(x+y) for each xin G. A subset in C(G) is called translation invariant if it contains every translates of all of its elements. A closed translation invariant linear subspace of the space C(G) is called a variety on G. Continuous homomorphisms of Ginto the additive [multiplicative] topological group of [nonzero] complex numbers are called additive [exponential] functions. A function is a polynomial if it belongs to the algebra generated by the additive functions and constants. Usually, the product of a polynomial and an exponential is called an exponential monomial. This is equivalent to the property that the function generates a finite dimensional indecomposable variety (see e.g. [12]). We shall use this latter definition here. It turns out that exponential functions, or more generally, exponential monomials can be considered as basic building blocks of varieties. A given variety may or may not contain any exponential function or exponential monomial. If each nonzero subvariety of a given variety contains an exponential function, then we say that spectral analysis holds for the variety. Another property is if the variety is synthesizable, which means that all exponential monomials in this variety span a dense subspace in the variety. If each subvariety of a given variety is synthesizable, then we say that spectral synthesis holds for the variety. It can be shown that spectral synthesis for a variety implies spectral analysis, too (see [12], Theorem 1). If spectral analysis, respectively, spectral synthesis holds for every nonzero variety on an Abelian group, then we say that spectral analysis, respectively, spectral synthesis holds on the group. A famous and pioneer result of L. Schwartz [8] exhibits the situation in a classical case by stating that if the underlying group is the reals with the Euclidean topology, then every nonzero variety contains an exponential function, that is, spectral analysis holds on the reals. Moreover, spectral synthesis also holds: there are sufficiently many exponential monomials in each variety in the sense that their linear hull is dense in the variety. In his 1958 result [7] M. Lefranc proved that spectral synthesis holds on the group Znfor each positive integer n,Zbeing the integers. In his 1965 paper [1] R. J. Elliot presented a theorem on spectral synthesis for arbitrary Abelian groups. However, in his 1987 private communication [2] Z. Gajda called the second author’s attention to the fact that the proof of Elliot’s theorem had several gaps. Later on several efforts have been made to solve the problem of discrete spectral analysis and spectral synthesis on arbitrary Abelian groups. Finally, a counterexample for Elliot’s theorem was presented at the 41st International Symposium on Functional Equations, Noszvaj, Hungary, 2003 (see [9]). For basics, further developments and references on spectral analysis and spectral synthesis the reader should refer to [3,11,12]. In this paper we give a new proof for the failure of spectral synthesis on some types of discrete Abelian groups, which was shown by a counterexample 2 in [9]. Our method is based on ring-theoretical results and uses the annihilator technique. The basics of this method have been worked out in [13]. 2. Basic concepts In this paper we consider discrete commutative groups only, and C(G) is the set of all functions from Gto C. Let Gbe an Abelian group and let CGdenote its group algebra, which is identified with the set of all finitely supported complex valued functions on G. Moreover, this set can be identified with the set of all finitely supported complex measures Mc(G) on G, using the definition ZG fdµ =X x∈G f(x)µ(x), whenever µis in CGand fis in C(G). This formula expresses the well-known fact about the dual C(G)∗of the topological vector space C(G): it is identified with Mc(G), the pairing given by the previous formula. Via these identifications the multiplication in the complex algebra CGis given by the convolution of measures as µ∗ν(f) = X x,y∈G f(x+y)µ(x)ν(y) for each µ, ν in CGand fin C(G). With this operation CGis a commutative unital complex algebra with identity δ0, which is the point mass concentrated at 0, the zero element of G. More generally, we denote by δxthe characteristic function of the singleton {x}for each xin G: it takes the value 1 at the element xand 0 otherwise. Convolution is also defined between elements of CGand C(G) in the following manner: µ∗f(x) = X y∈G f(x−y)µ(y), whenever µis in CG,fis in C(G) and xis in G. With this operation C(G) turns into a CG-module, closed submodules being exactly the varieties. The intersection of all varieties including a particular fin C(G) is called the variety of fand is denoted by τ(f). For each subset Hin C(G) the annihilator H⊥of Hin CGis defined by H⊥={µ:µ∗f= 0 for each fin H}. It is easy to see that H⊥is an ideal in CG. If H={f}is a singleton, then H⊥=τ(f)⊥and we call it the annihilator of f. 3 Similarly, the annihilator K⊥in C(G) of a subset Kin CGis defined by K⊥={f:µ∗f= 0 for each µin K}. It is also easy to check that K⊥is a variety in C(G). The following two theorems are important technical tools (see [6,13]). Theorem 2.1. Let Gbe an Abelian group, Va variety on Gand Ian ideal in CG. Then we have V⊥⊥ =V, I⊥⊥ =I . Theorem 2.2. Let Gbe an Abelian group, (Vγ)γ∈Γa family of varieties on G and (Iγ)γ∈Γa family of ideals in CG. Then we have (X γ∈Γ Vγ)⊥=\ γ∈Γ V⊥ γ,(\ γ∈Γ Iγ)⊥=X γ∈Γ I⊥ γ. 3. Exponentials and maximal ideals The basic building blocks of spectral analysis and spectral synthesis are exponential monomials. We call the reader’s attention that we shall use the word ”exponential” in several different meanings in the sequel. The generalized characters of G, that is, the homomorphisms of Ginto the multiplicative group of nonzero complex numbers will be called exponential functions, or simply exponentials. Later on we shall also use the terms ”exponential maximal ideal”, ”exponential monomial”, and ”generalized exponential monomial”, which refer to different, however, related concepts. Exponentials can be characterized by a number of properties. We shall use the following result (see [13, Theorems 3 and 4], and [13, Corollaries 1 and 2]). Theorem 3.1. Let Gbe an Abelian group and m:G→Cbe an arbitrary function. Then the following conditions are equivalent: 1. mis an exponential. 2. The variety of mis one dimensional and m(0) = 1. 3. The annihilator τ(m)⊥is a maximal ideal in CG,CG/τ(m)⊥is isomorphic to Cand m(0) = 1. Maximal ideals Min CGwith the property that CG/M ∼ =Cplay an important role, and they will be called exponential maximal ideals. They are closely related to modified differences defined as follows. For each function f:G→C and yin Gwe define ∆f;y=δ−y−f(y)δ0. The measure ∆f;yis called modified difference. For products of modified differences we shall use the notation ∆f;y1,y2,...,yn+1 = Πn+1 i=1 ∆f;yi, 4 whenever y1, y2, . . . , yn+1 are in G. The product on the right side is meant as a convolution. Given fin C(G) the ideal in CGgenerated by all modified differences of the form ∆f;ywith yin Gis denoted by Mf. It is reasonable to ask whether Mfis proper. We have the following result. Theorem 3.2. Let Gbe an Abelian group and f:G→Cbe a function. The ideal Mfis proper if and only if fis an exponential. In this case Mf=τ(f)⊥, hence Mfis an exponential maximal ideal. Proof. Suppose first that Mfis proper. Then M⊥ fis a nonzero variety, by Theorem 2.1, hence there is a nonzero gannihilated by all modified differences of the form ∆f;y. For x, y in Gwe have 0=∆f;y∗g(x) = g(x+y)−f(y)g(x).(3.1) Putting x= 0 we have g(y) = g(0) ·f(y). In particular, g(0) 6= 0, f6= 0 and we obtain f(x+y) = f(x)f(y). As fis nonzero, it follows f(0) = 1, hence fis an exponential. Conversely, suppose that f=mis an exponential. Then mis in M⊥ m, as obviously ∆m;y∗m(x) = m(x+y)−m(y)m(x) = 0 for each x, y in G. Hence Mm is proper. Moreover, τ(m)⊥is an exponential maximal ideal in CG, by Theorem 3.1. If gis in M⊥ f, then, by (3.1), it is a constant multiple of m, hence it belongs to τ(m). It follows M⊥ m⊆τ(m), thus τ(m)⊥⊆Mm. As τ(m)⊥is maximal and Mmis proper, we have τ(m)⊥=Mm, and the theorem is proved. These latter two results have been used in [13] to prove the following characterization results. Theorem 3.3. Let Gbe an Abelian group and Vbe a variety on G. Then spectral analysis holds for Vif and only if each maximal ideal containing V⊥is exponential. Corollary 3.4. Let Gbe an Abelian group. Then spectral analysis holds on Gif and only if each maximal ideal in CGis exponential. Corollary 3.5. Let Gbe an Abelian group and Vbe a variety on G. Then spectral analysis holds for Vif and only if each maximal ideal in CG/V ⊥is exponential. 4. Exponential monomials Let Gbe an Abelian group. The variety Von Gis called decomposable, if it is the sum of two proper subvarieties, which means that the algebraic sum of two proper subvarieties is a dense submodule in it. Otherwise it is called indecomposable. The following theorem is obvious, by Theorem 2.2. 5 Theorem 4.1. Let Gbe an Abelian group. A variety on Gis decomposable if and only if its annihilator is the intersection of two ideals, which are different from it. Let Gbe an Abelian group. The function f:G→Cis called a generalized exponential monomial, if there exists an exponential mand a natural number n such that for each y1, y2, . . . , yn+1 we have ∆m;y1,y2,...,yn+1 ∗f= 0 (4.1) holds. We reformulate this definition in terms of the annihilator of f. Theorem 4.2. Let Gbe an Abelian group. The function f:G→Cis a generalized exponential monomial if and only if its annihilator contains some positive power of an exponential maximal ideal. Proof. The condition of the theorem is equivalent to the following condition: there exists an exponential mand a natural number nsuch that Mn+1 m⊆τ(f)⊥.(4.2) As the modified differences ∆m;ywith yin Ggenerate Mm, hence the modified differences ∆m;y1,y2,...,yn+1 generate Mn+1 m, that is, (4.1) and (4.2) are equivalent for f. It is easy to check (see [13, Theorem 7]) that condition (4.2) can hold for at most one exponential m. Theorem 4.3. Let Gbe an Abelian group and f:G→Cbe a nonzero generalized exponential monomial. Then there is a unique exponential msatisfying (4.1)for some natural number n. In other words, there is a unique exponential maximal ideal Msatisfying Mn+1 ⊆τ(f)⊥for some natural number n. Now we have the following characterization results (see [13, Theorem 8]). Theorem 4.4. Let Gbe an Abelian group. The function f:G→Cis a nonzero generalized exponential monomial if and only if CG/τ(f)⊥is a local ring with nilpotent exponential maximal ideal. Theorem 4.5. Let Gbe an Abelian group. The functionf:G→Cis an exponential monomial if and only if CG/τ(f)⊥is a local Artin ring with exponential maximal ideal. Proof. By the previous theorem CG/τ(f)⊥is a local ring with exponential maximal ideal. Any descending chain of ideals in CG/τ(f)⊥induces a descending chain of ideals containing τ(f)⊥in CG, which induces an ascending chain of subvarieties in τ(f), hence, by finite dimensionality, it terminates. For the converse see [13, Theorem 8]. 6 5. The failure of spectral synthesis Theorem 5.1. Let Gbe an Abelian group and Vbe a variety on G. If Vis indecomposable, and spectral synthesis holds for V, then CG/V ⊥is a local Artin ring. Proof. If Vis synthesizable, then it is the topological sum of all subvarieties generated by exponential monomials belonging to V, by definition. This means that we have, by Theorem 2.2, V⊥=\ ϕ∈V τ(ϕ)⊥,(5.1) where the intersection is extended to all exponential monomials ϕin V. As Vis indecomposable, hence, by Theorem 4.1,V⊥is equal to one of the factors of the intersection on the right side, that is V⊥=τ(ϕ)⊥for some exponential monomial ϕin V. By Theorem 4.5,CG/V ⊥is a local Artin ring. Theorem 5.2. Let Gbe an Abelian group, and let f:G→Cbe a generalized exponential monomial. Then τ(f)is synthesizable if and only if fis an exponential monomial. Proof. The statement is obvious by the definition of exponential monomials and by the previous theorem. The following theorem follows immediately. Theorem 5.3. Let Gbe an Abelian group and Vbe a variety on G. If there is a generalized exponential monomial in V, which is not an exponential monomial, then spectral synthesis does not hold for V. As a consequence we obtain the following result (see [9]). Theorem 5.4. Let Gbe an Abelian group with infinite torsion free rank. Then spectral synthesis fails to hold on G. Proof. Indeed, by [10, Theorem 3], the torsion free rank of an Abelian group is infinite if and only if there is a generalized exponential monomial on the group, which is not an exponential monomial. Acknowledgments The first author was partially supported by the Hungarian National Foundation grant no. K109185, and by the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences. The second author’s research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. NK-81402. [1] R. J. Elliott, Two notes on spectral synthesis for discrete Abelian groups, Math. Proc. Cambridge Phil. Soc. 61(1965), 617–620. 7 [2] Z. Gajda, Private communication, Hamburg–Rissen, (1987) [3] D. I. Gureviˇc, Counterexamples to a problem of L. Schwartz, Funkcional. 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