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

Let G   be an admissible compact Hausdorff right topological group, that is, a group with a Hausdorff topology such that for each a∈Ga∈G, the map g↦gag↦ga is continuous, and the set of a∈Ga∈G such that the map g↦agg↦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)B(G), the linear span of the continuous positive definite functions on G  . We show that B(G)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⁎weak⁎-fixed point properties on B(G)B(G). We also study some related properties on the measure algebra M(G)M(G).