New Results on Spectral Synthesis

In our former paper we introduced the concept of localization of ideals in the Fourier algebra of a locally compact Abelian group. It turns out that localizability of a closed ideal in the Fourier algebra is equivalent to the synthesizability of the annihilator of that closed ideal which corresponds to this ideal in the measure algebra. This equivalence provides an eﬀective tool to prove synthesizability of varieties on locally compact Abelian groups. In this paper we utilize this tool to show that when investigating synthesizability of a variety, roughly speaking compact elements of the group can be neglected. Our main result is that spectral synthesis holds on a locally compact Abelian group G if and only if it holds on G { B , where G is the closed subgroup of all compact elements. In particular, spectral synthesis holds on compact Abelian groups. Also we obtain a simple proof for the characterization theorem of spectral synthesis on discrete Abelian groups.


Introduction
Let G be a locally compact Abelian group.Spectral synthesis deals with uniformly closed translation invariant linear spaces of continuous complex valued functions on G.Such a space is called a variety.We say that spectral analysis holds for a variety, if every nonzero subvariety contains a one dimensional subvariety.We say that a variety is synthesizable, if its finite dimensional subvarieties span a dense subspace in the variety.Finally, we say that spectral synthesis holds for a variety, if every subvariety of it is synthesizable.On commutative topological groups finite dimensional varieties of continuous functions are completely characterized: they are spanned by exponential monomials.Exponential polynomials on a topological Abelien group are defined as the elements of the complex algebra of continuous complex valued functions generated by all continuous homomorphisms into the multiplicative group of nonzero complex numbers (exponentials), and all continuous homomorphisms into the additive group of all complex numbers (additive functions).An exponential monomial is a function of the form x Þ Ñ P `a1 pxq, a 2 pxq, . . ., a n pxq ˘mpxq, where P is a complex polynomial in n variables, the a i 's are additive functions, and m is an exponential.Every exponential polynomial is a linear combination of exponential monomials.For more about spectral analysis and synthesis on groups see [5,6].
In [4], the authors characterized those discrete Abelian groups having spectral synthesis: spectral synthesis holds for every variety on the discrete Abelian group G, if and only if G has finite torsion free rank.In particular, from this result it follows, that if spectral synthesis holds on G and H, then it holds on G ' H. Unfortunately, such a result does not hold in the non-discrete case.Namely, by the fundamental result of L. Schwartz [1], spectral synthesis holds on R, but D. I. Gurevich showed in [3] that spectral synthesis fails to hold on R ˆR.In this paper we prove that spectral synthesis holds on the locally compact Abelian group G assuming that it holds on the factor group G{B, where B is the closed subgroup of G formed by all compact elements.We recall, that in a topological group an element is called compact element, if the closure of the cyclic subgroup generated by the element is compact.In a locally compact Abelian group all compact elements form a closed subgroup (see [2, (9.10) Theorem]).

Localization
In our former paper [7] we introduced the concept of localization of ideals in the Fourier algebra of a locally compact Abelian group.We recall this concept here.
Let G be a locally compact Abelian group and let ApGq denote its Fourier algebra, that is, the algebra of all Fourier transforms of compactly supported complex Borel measures on G.This algebra is topologically isomorphic to the measure algebra M c pGq.For the sake of simplicity, if the annihilator Ann I of the closed ideal I in M c pGq is synthesizable, then we say that the corresponding closed ideal p I in ApGq is synthesizable.For each derivation D on ApGq, we introduced (see [ We note that if m is not a root of p I, then D p I,m " P p I,m " t0u, consequently p I D p I,m ,m " p I P p I,m ,m " ApGq, hence those terms have no effect on the intersection.The ideal p I is called localizable, if we have equalities in (1).Roughly speaking, localizability of an ideal means, that the ideal is completely determined by the values of "derivatives" of the functions belonging to this ideal.The main result in [7] is that p I is synthesizable if and only if it is localizable.We shall use this result in the subsequent paragraphs.

The main result
Our main result in this paper is the following.

Theorem 1. Let G be a locally compact Abelian group and let B denote the closed subgroup of G consisting of all compact elements. Then spectral synthesis holds on G if and only if it holds on G{B.
Proof.If spectral synthesis holds on G, then it obviously holds on every continuous homomorphic image of G, in particular, it holds on G{B.
Conversely, we assume that spectral synthesis holds on G{B.This means, that every closed ideal in the Fourier algebra of G{B is localizable, and we need to show the same for all closed ideals of the Fourier algebra of G.
First we remark that the polynomial rings over G and over G{B can be identified.Indeed, polynomials on G are built up from real characters on G, which clearly vanish on compact elements, as the topological group of the reals has no nontrivial compact subgroups.Consequently, if a is a real character, and x, y are in the same coset of B, then x ´y is in B, and apx ´yq " 0, which means apxq " apyq.So, the real characters on G arise from the real characters of G{B, hence the two polynomial rings can be identified.xµ, q m ˝Φy " xµ B , q my " 0, hence m ˝Φ, which is clearly an exponential on G, is a root of p I. Suppose that D is a derivation in P p I,m˝Φ , then it has the form Dp µpm ˝Φq " ż p ¨p q m ˝Φq dµ with some polynomial p on G (see [7]).According to our remark above, the polynomial p can uniquely be written as p B ˝Φ, where p B is a polynomial on G{B.In other words,

Now we define a projection of the
which defines a derivation D B on ApG{Bq with generating function f DB ,m " p B (see [7]).[4,7]).
Proof.If the torsion free rank of G is infinite, then there is a non-polynomial derivation on its Fourier algebra, hence we have the chain of inclusions Conversely, let G have finite torsion free rank.The subgroup B of compact elements coincides with the set T of all elements of finite order, and G{T is a (continuous) homomorphic image of Z n with some nonnegative integer n.As spectral synthesis holds on Z n , it holds on its homomorphic images.

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7]) the set p I D,m as the set of all functions p µ in ApGq for which Dp µpmq " ż ∆ x,y1,y2,...,y k ˚fD,m p0q q mpxq dµpxq " p µpmq " 0 holds for each k " 1, 2, . . .and y 1 , y 2 , . . ., y k in G. Then p I D,m is a closed ideal in ApGq.For a family D of derivations we write p I D,m " č DPD p I D,m .Clearly, p I D,m is a closed ideal as well.The dual concept is the following: given an ideal p I in ApGq and an exponential m, the set of all derivations on ApGq which annihilate p I at m is denoted by D p I,m .The subset of D p I,m consisting of all polynomial derivations is denoted by P p I,m .We have the basic inclusion p Fourier algebra of G into the Fourier algebra of G{B as follows.Let Φ : G Ñ G{B denote the natural mapping.For each measure µ in M c pGq we define µ B as the linear functional xµ B , ϕy " xµ, ϕ ˝Φy whenever ϕ : G{B Ñ C is a continuous function.It is straightforward that the mapping p µ Þ Ñ p µ B is a continuous algebra homomorphism of the Fourier algebra of G into the Fourier algebra of G{B.As Φ is an open mapping, closed ideals are mapped onto closed ideals.For a given closed ideal p I in ApGq, we denote by p I B the closed ideal in ApG{Bq which corresponds to p I under the above homomorphism.If m is a root of the ideal p I B , then p µ B pmq " 0 for each p µ in p I. In other words,

p
I Ď p I D p I,m ,m Ĺ p I P p I,m ,m , which implies that p I ‰ p I P p I,m ,m , hence p I is not synthesizable.
I which is annihilated at M by all polynomial derivations in P p I,M , for each exponential M on G.In particular, p ν is annihilated at m ˝Φ by all polynomial derivations in P p I,m˝Φ , for each exponential m on G{B.We have seen above that this implies that p ν B is annihilated at m by all polynomial derivations in P p