Fourier–Stieltjes algebras and measure algebras on compact right topological groups

Abstract

Let G be an admissible compact Hausdorff right topological group, that is, a group with a Hausdorff topology such that for each a∈G, the map g↦ga is continuous, and the set of a∈G such that the map g↦ag is continuous is dense in G. Such groups arise in the study of distal flows. In this paper we study the Fourier–Stieltjes algebra B(G), the linear span of the continuous positive definite functions on G. We show that B(G) is isomorphic with the Fourier–Stieltjes algebra of an associated compact topological group. This result is then applied to obtain some geometric properties including the weak and weak⁎-fixed point properties on B(G). We also study some related properties on the measure algebra M(G).