Double orbits of weakly almost periodic functions

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 WS(G) and UCS(G). Then WS(G)⊂UCS(G) and AP(G)⊂WS(G)⊂WAP(G). G is called a WS-group if 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)⊕C0(G). If G is minimally w.a.p., then 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≥2, the motion group M(n), and the special linear group SL(n,R) are minimally w.a.p. On the other hand, there exist locally compact groups G such that WS(G)=AP(G) but G is not minimally w.a.p. We will show that if G is an IN-group and K=KG is the intersection of all closed invariant neighborhoods of the identity of G, then UCS(G)=UCS(G/K) and WS(G)=WS(G/K). We will identify the strictly w.a.p. functions on the ax+b group. We will also show that UCS(SL(2,R)) only contains the constant functions.