Complexity of Weakly Almost Periodic Functions

Given a topological group G let C(G) denote the Banach space of bounded, continous real valued function on G. Eberlein [1] defined a function f ∈ C(G) to be weakly almost periodic if the weak closure of all of its translates is compact in the weak topology on C(G) — in other words, if fx(y) is defined to be f(yx−1) then the weak closure of {fx | x ∈ G} is weakly compact. The set of weakly almost periodic function on G will be denoted by WAP(G). In the case that G is a countable discrete group C(G) = l∞(G). If G is infinite then l∞(G) is not separable, but considered with the pointwise topology l∞(G) becomes a Polish space and it is natural to ask for the descriptive set theoretic complexity of WAP(G) as a subset of l∞(G). It has been shown [4, 5] that if G is the countable Boolean group then WAP(G) is a non-Borel Π1 set. However, the question of whether or not WAP(Z) is a non-Borel Π1 set was not solved by the methods of [4, 5]. The goal of the current work is to show that not only is WAP(Z) a non-Borel Π1, but it is also Π1-complete. Recall that a subset A of a zero-dimensional Polish space X is Π1-complete if it is Π 1 1 and if for every other zero-dimensional Polish space Y and B ⊆ Y that is Π1 there is a continuous function f : Y → X such that B = f−1A. In other words, no Π1 set is more complicated than A.