$C^{\ast }$-algebras generated by Fourier-Stieltjes transforms

Abstract:

For G a locally compact group and $\hat G$ its dual, let ${\mathcal {M}_d}(\hat G)$ be the ${C^ \ast }$-algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on ${\mathcal {M}_d}(\hat G)$ is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in ${\mathcal {M}_d}(\hat G)$ are the discrete measures, and also the sup and lim sup norms are identical on ${\mathcal {M}_d}(\hat G)$. These results are extensions of classical theorems on almost periodic functions on locally compact abelian groups.