For a locally compact group \(G\) , let \( AP (G)\) and \( WAP (G)\) be respectively the \(C^{*}\) -algebras of almost periodic and weakly almost periodic functions on \(G\) . For a bounded continuous function \(f\) on \(G\) , \(f\) is said to be strictly w.a.p. if its double orbit \(O(f)\) is relatively weakly compact and \(f\) is said to be strictly uniformly continuous if its double orbit is uniformly equicontinuous on \(G\) . The \(C^{*}\) -algebras of such functions are denoted, respectively, by \(\textit{WS}(G)\) and \( UCS (G)\) . Then \(\textit{WS}(G) \subset UCS (G)\) and \( AP (G) \subset \textit{WS}(G) \subset WAP (G)\) . \(G\) is called a \( WS \) -group if \(\textit{WS}(G) = WAP (G)\) . We will show that if a discrete FC-group \(G\) is a \( WS \) -group, then its center is of finite index in \(G\) . A noncompact locally compact group \(G\) is minimally w.a.p., if \( WAP (G) = AP (G) \oplus C_{0}(G)\) . If \(G\) is minimally w.a.p., then \(\textit{WS}(G) = AP (G)\) , i.e., if the double orbit of a bounded continuous function \(f\) is relatively weakly compact then it is relatively norm compact. It is known that for \(n \ge 2\) , the motion group \(M(n)\) , and the special linear group \(\textrm{SL}(n,\,\mathbb {R})\) are minimally w.a.p. On the other hand, there exist locally compact groups \(G\) such that \(\textit{WS}(G) = AP (G)\) but \(G\) is not minimally w.a.p. We will show that if \(G\) is an IN-group and \(K = K_{G}\) is the intersection of all closed invariant neighborhoods of the identity of \(G\) , then \( UCS (G) = UCS (G/K)\) and \(\textit{WS}(G) = \textit{WS}(G/K)\) . We will identify the strictly w.a.p. functions on the \(ax + b\) group. We will also show that \( UCS (\textrm{SL}(2,\,\mathbb {R}))\) only contains the constant functions.
