1 Introduction

In the sequel \({\mathbb {C}}\) denotes the set of complex numbers. We recall that the measure algebra of a locally compact Abelian group G is the set \({\mathcal {M}}_c (G)\) of all compactly supported complex Borel measures on G, which can be identified with the topological dual space of the topological vector space \({\mathcal {C}}(G)\) of all continuous complex valued functions on G, when the latter is equipped with the topology of uniform convergence on compact sets. The space \({\mathcal {M}}_c (G)\) turns into a commutative unital involutive complex algebra when equipped with the convolution defined by

$$\begin{aligned} \langle \mu *\nu ,f\rangle =\int _{G} f(x+y)d\mu (x)\,d\nu (y), \end{aligned}$$

and with the involution

$$\begin{aligned} \langle {\mu }^{*},f\rangle =\langle \mu ,{f}^{*}\rangle \end{aligned}$$

for each \(\mu ,\nu \) in \({\mathcal {M}}_c (G)\) and f in \({\mathcal {C}}(G)\). Here \({f}^{*}(x)=\overline{f(-x)}\) whenever x is in G. The unit element of \({\mathcal {M}}_c (G)\) is \(\delta _o\), where, in general, \(\delta _x\) denotes the point mass with support set \(\{x\}\). We call \({\mathcal {M}}_c (G)\) the measure algebra of the group G.

The locally convex topological vector space \({\mathcal {C}}(G)\) is a topological vector module over the measure algebra when we define

$$\begin{aligned} \mu *f(x)=\int _G f(x-y)\,d\mu (y) \end{aligned}$$

for f in \({\mathcal {C}}(G)\), \(\mu \) in \({\mathcal {M}}_c (G)\), and x in G.

In the special case, when G is a discrete group, the measure algebra is called group algebra and is denoted by \({\mathbb {C}}G\). The algebraic properties of the measure algebra, resp. the group algebra play a basic role in spectral analysis and synthesis on G. In particular, if G is a discrete group, then it is proved in [1] that \({\mathbb {C}}G\) is semisimple. The purpose of the present note is to show that this holds in the non-discrete case as well. Our approach here is completely different from that of [1].

2 Exponential maximal ideals

From now on we always denote by G a locally compact commutative topological group. An ideal in \({\mathcal {M}}_c(G)\) is called exponential if the residue algebra is topologically isomorphic to the complex field (see [2]). Clearly, in this case the ideal is weak*-closed and maximal. We will show that the intersection of all exponential ideals is zero. As a consequence we obtain that the Jacobson radical of \({\mathcal {M}}_c(G)\), i.e. the intersection of all maximal ideals, is zero.

Recall that the nonzero continuous function \(m:G\rightarrow {\mathbb {C}}\) is called an exponential, if

$$\begin{aligned} m(x+ y)=m(x)m(y) \end{aligned}$$

holds for each x, y in K. In this case \(m(0)=1\).

We shall use the following lemma.

Lemma 1

A necessary and sufficient condition for the ideal I is exponential is that there exists an exponential \(m:G\rightarrow {\mathbb {C}}\) such that \(\mu \) is in I if and only if \(\langle \mu ,{\check{m}}\rangle =0\).

In general, we use the notation \({\check{f}}(x)=f(-x)\) for each f in \({\mathcal {C}}(G)\) and x in G.

Proof

First we show the sufficiency. We define \(F:{\mathcal {M}}_c(G)\rightarrow {\mathbb {C}}\) by

$$\begin{aligned} F(\mu )=\langle \mu ,{\check{m}}\rangle \end{aligned}$$

for each \(\mu \) in \({\mathcal {M}}_c(G)\). We show that F is a multiplicative functional of the algebra \({\mathcal {M}}_c(G)\), i.e. F is a weak*-continuous linear functional satisfying

$$\begin{aligned} F(\mu *\nu )=F(\mu )F(\nu ) \end{aligned}$$
(1)

for each \(\mu ,\nu \) in \({\mathcal {M}}_c(G)\). The linearity and weak*-continuity is obvious, we need to show (1) only. We have

$$\begin{aligned} F(\mu *\nu )= & {} \langle \mu *\nu ,{\check{m}}\rangle =\int _G {\check{m}}(x*y)\,d\mu (x)\,d\nu (y) \\= & {} \int _G {\check{m}}(x)\,d\mu (x)\int _G {\check{m}}(y)\,d\nu (y)= F(\mu )F(\nu ), \end{aligned}$$

which proves our statement.

By assumption, the ideal I coincides with the kernel of F: \(I={\mathrm {Ker}}\,F\), and \({\mathcal {M}}_c(G)/{\mathrm {Ker}}\,F\cong {\mathbb {C}}\), hence I is an exponential ideal.

To prove the converse, let I be an exponential ideal, then I is maximal and \({\mathcal {M}}_c(G)/I\cong {\mathbb {C}}\). Let \(F: {\mathcal {M}}_c(G)\rightarrow {\mathbb {C}}\) be the natural homomorphism and we define

$$\begin{aligned} m(x)=F(\delta _{-x}) \end{aligned}$$

for x in G. Then we have

$$\begin{aligned} m(x+y)=F(\delta _{-x-y}) =F(\delta _{-x}*\delta _{-y})= F(\delta _{-x})F(\delta _{-y})=m(x)m(y). \end{aligned}$$

Using the fact that finitely supported measures in \({\mathcal {M}}_c(G)\) form a weak*-dense subspace, we have

$$\begin{aligned} \langle \mu ,{\check{m}}\rangle =F(\mu ) \end{aligned}$$

for each \(\mu \) in \({\mathcal {M}}_c(G)\). As \(m(0)=1\) and m is clearly continuous, we have that m is an exponential. If \(\mu \) is in I, then \(\mu \) is in \({\mathrm {Ker}}\,F\), hence

$$\begin{aligned} \langle \mu ,{\check{m}}\rangle =F(\mu )=0. \end{aligned}$$

Conversely, if \(\langle \mu ,{\check{m}}\rangle =0\), then \(F(\mu )=0\), hence \(\mu \) is in \({\mathrm {Ker}}\,F=I\). The theorem is proved. \(\square \)

Closed submodules of the module \({\mathcal {C}}(G)\) are called varieties.

The orthogonal complement\(X^{\perp }\) of a subset X in \({\mathcal {C}}(G)\) is defined as

$$\begin{aligned} X^{\perp }=\{\mu \in {\mathcal {M}}_c(G):\, \langle \mu ,f\rangle =0\text {for each}f\in X\}. \end{aligned}$$

Similarly, the orthogonal complement\(Y^{\perp }\) of a subset Y in \({\mathcal {M}}_c(G)\) is defined as

$$\begin{aligned} Y^{\perp }=\{f\in {\mathcal {C}}(G):\, \langle \mu ,f\rangle =0\text {for each}\mu \in Y\}. \end{aligned}$$

A standard application of the Hahn–Banach Theorem gives the relations

$$\begin{aligned} V^{\perp \perp }=V, \quad I^{\perp \perp }=I \end{aligned}$$

for each variety V in \({\mathcal {C}}(G)\) and weak*-closed ideal I in \({\mathcal {M}}_c(G)\).

Also, the following relations are important, and can be proved easily (see [2]):

Theorem 1

For each family \((V_i)\) of varieties and \((I_i)\) of weak*-closed ideals we have

$$\begin{aligned} \Bigl (\sum V_i\Bigr )^{\perp }= & {} \bigcap V_i^{\perp },\quad \Bigl (\sum I_i\Bigr )^{\perp }=\bigcap I_i^{\perp }, \\ \Bigl (\bigcap V_i\Bigr )^{\perp }= & {} \sum V_i^{\perp },\quad \Bigl (\bigcap I_i\Bigr )^{\perp }=\sum I_i^{\perp }. \end{aligned}$$

3 The main result

Theorem 2

Let G be a commutative locally compact topological group. Then the measure algebra \({\mathcal {M}}_c(G)\) is semi-simple.

Proof

We show that the intersection of all exponential maximal ideals in \({\mathcal {M}}_c(G)\) is zero. By Theorem 1, this is equivalent to the relation

$$\begin{aligned} \sum I_i^{\perp }={\mathcal {C}}(G), \end{aligned}$$
(2)

where \(I_i\) runs through all exponential ideals in \({\mathcal {M}}_c(G)\). By Lemma 1, \(I_i^{\perp }\) is the one dimensional space in \({\mathcal {C}}(G)\) generated by an exponential \(m_i\), hence Eq. (2) states that the finite linear combinations of all exponentials on G form a dense subspace in \({\mathcal {C}}(G)\). To prove this we use the Stone–Weierstrass Theorem. Indeed, for a given compact set C in G let \({\mathcal {A}}_C\) denote the set of the restrictions of all finite linear combinations of exponentials on G to C. Clearly, \({\mathcal {A}}_C\) is a complex linear space in \({\mathcal {C}}(C)\). Moreover, \({\mathcal {A}}_C\) is a unital algebra: indeed, the product of two exponentials is an exponential, and 1 is an exponential. Also, \({\mathcal {A}}_C\) is closed under complex conjugation as the complex conjugate of an exponential is an exponential again. Finally, \({\mathcal {A}}_C\) is a separating family: indeed, for any two elements \(x\ne y\) in C there exists an exponential m with \(m(x)\ne m(y)\). It follows that \({\mathcal {A}}_C\) is uniformly dense in \({\mathcal {C}}(C)\), which implies that the finite linear combinations of all exponentials on G form a dense subspace in \({\mathcal {C}}(G)\). The theorem is proved. \(\square \)