Note on the semi-simplicity of measure algebras

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 C denotes the set of complex numbers. We recall that the measure algebra of a locally compact Abelian group G is the set 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 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 M c (G) turns into a commutative unital involutive complex algebra when equipped with the convolution defined by and with the involution where, in general, δ x denotes the point mass with support set {x}. We call M c (G) the measure algebra of the group G.
The locally convex topological vector space C(G) is a topological vector module over the measure algebra when we define In the special case, when G is a discrete group, the measure algebra is called group algebra and is denoted by CG. 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 CG 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].

Exponential maximal ideals
From now on we always denote by G a locally compact commutative topological group. An ideal in 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 M c (G), i.e. the intersection of all maximal ideals, is zero.
Recall that the nonzero continuous function m : G → 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
for each μ, ν in 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 = Ker F, and M c (G)/Ker F ∼ = C, hence I is an exponential ideal.
To prove the converse, let I be an exponential ideal, then I is maximal and M c (G)/I ∼ = C. Let F : M c (G) → C be the natural homomorphism and we define Using the fact that finitely supported measures in M c (G) form a weak*-dense subspace, we have for each μ in M c (G). As m(0) = 1 and m is clearly continuous, we have that m is an exponential. If μ is in I , then μ is in Ker F, hence Conversely, if μ,m = 0, then F(μ) = 0, hence μ is in Ker F = I . The theorem is proved.
Closed submodules of the module C(G) are called varieties.
The orthogonal complement X ⊥ of a subset X in C(G) is defined as Similarly, the orthogonal complement Y ⊥ of a subset Y in M c (G) is defined as Also, the following relations are important, and can be proved easily (see [2]):

The main result
Theorem 2 Let G be a commutative locally compact topological group. Then the measure algebra M c (G) is semi-simple.