## Abstract

In this paper we prove that the measure algebra of a locally compact abelian group is semi-simple. This result extends the corresponding result of S. A. Amitsur in the discrete group case using a completely different approach.

## 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

and with the involution

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

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

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

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

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

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

for *x* in *G*. Then we have

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

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

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

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

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

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

## 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

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. \(\square \)

## References

Amitsur, S.A.: On the semi-simplicity of group algebras. Mich. Math. J.

**6**, 251–253 (1959)Székelyhidi, L.: Harmonic and Spectral Analysis. World Scientific Publishing Co. Pte. Ltd., Hackensack (2014)

## Acknowledgements

Open access funding provided by University of Debrecen (DE). The author expresses his special thanks to Prof. Bettina Wilkens for calling attention to S. A. Amitsur’s paper.

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Communicated by John S. Wilson.

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Székelyhidi, L. Note on the semi-simplicity of measure algebras.
*Monatsh Math* **192**, 935–938 (2020). https://doi.org/10.1007/s00605-019-01356-9

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DOI: https://doi.org/10.1007/s00605-019-01356-9