Invariant means on almost periodic functions and equicontinuous actions
Anthony To Ming Lau · Proceedings of the American Mathematical Society · 1975
Let S S be a topological semigroup such that the almost periodic functions on S S have a left invariant mean (this is the case, for example, when S S has finite intersection property for closed right ideals). Then whenever S S acts equicontinuously on a compact Hausdorff space X X , there exists a compact group G G of homeomorphisms acting equicontinuously on a retract Y Y of X X such that S S has a common fixed point in X X if and only if G G has a common fixed point in Y Y . This result generalises some recent work of T. Mitchell. As an application, we show that whenever S S acts equicontinuously on the closed unit interval I I , then I I contains a common fixed point for S S .