Uniformly Continuous and Weakly Almost Periodic Functions on Some Topological Semigroups

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

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 .

The known theory for a discrete oid \(T\) shows that how to find a subset \(T^{\infty}\) of \(\beta T\) which is a compact right topological semigroup (see section 2 for details).In this paper we try to find an analogue of almost periodic functions for oids. We discover, new compact semigroups by using a certain subspace of functions \(u^{\infty}(T)\) of \(C(T)\) for an oid \(T\) for which \(f\beta\) is continuous on \(T^{\infty}\times(T\cup T^{\infty}\cup TT^{\infty})\),where \((T\cup T^{\infty}\cup TT^{\infty})\) is a suitable subspace of \(\beta T\) for a wide range.