On Lau–Loy’s decomposition of a measure algebra on CHART groups

Abstract

By using the Furstenberg–Ellis–Namioka structure theorem, we give a decomposition theorem for the Banach algebra ${\mathcal {M}}(G)$, i.e. the Banach algebra of those complex regular Borel measures on a compact Hausdorff admissible right topological (or simply CHART) group G for which the natural convolution product makes sense, generalizing an existing result due to Lau and Loy. Next, we characterize the Furstenberg–Ellis–Namioka structure theorem on a family of CHART groups, namely the groups $E({\mathbb {T}})^{k}$ where $E({\mathbb {T}})$ is the family of all endomorphisms of the unit circle ${\mathbb {T}}$, and then we apply the generalized decomposition theorem to these groups.