Almost periodic functions on groupoids

Abstract

In this paper we generalize the notion of almost periodic functions on groups to the corresponding notion for groupoids. We prove a number of theorems about almost periodic functions in this general setting. We show that the set of almost periodic functions on a groupoid $G,AP(G),$ is a C*-subalgebra of ${\ell }^{\infty }(G)$ . We investigate some topological properties of the maximal ideal space of $AP(G),\mathfrak{b}(G),$ and we obtain a continuous partial operation on $\mathfrak{b}G$ . Also, we study almost periodic functions on groupoids defined by an equivalence relation on a set $X$ and obtain a compactification of $X.$