# Note on the semi-simplicity of measure algebras

## 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.

## 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  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 .

## 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 ). 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 xy 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 ):

### 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}

## 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. $$\square$$

## References

1. Amitsur, S.A.: On the semi-simplicity of group algebras. Mich. Math. J. 6, 251–253 (1959)

2. 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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Correspondence to László Székelyhidi.