Non-separability of the Gelfand space of measure algebras (Q517242)

Let \(G\) denote a locally compact abelian group with dual group \(\Gamma\). Let also \(M(G)\) be the algebra of all complex-valued Borel regular measures on \(G\) equipped with a total variation norm and convolution product. The Fourier-Stieljtes transform of \({\mu\in M(G)}\) is denoted by \(\hat{\mu}\) and is defined by \[ \hat{\mu}(\gamma)=\int_G\gamma(-x)d\mu(x) \] for all \(\mu\in\Gamma\). Let \(M_0(G)\) be the subspace of \(M(G)\) consisting of measures with Fourier-Stieltjes transform vanishing at infinity. In the following, let \(G\) be a compact abelian group or a non-discrete locally compact group. The authors prove that the Gelfand space of \(M_0(G)\) contains continuum pairwise disjoint open subsets and so the Gelfand space of \(M_0(G)\) is not separable. These results together with the fact that \(M_0(G)\) is a closed ideal in \(M(G)\) show that the Gelfand space of \(M(G)\) is not separable.