Weakly Almost Periodic Functions on N with a Negative Base

Abstract

Weakly almost periodic compactifications have been seriously studied for over 30 years. In the pioneering papers of de Leeuw and Glicksberg [ 4 ] and [ 5 ], the approach adopted was operator-theoretic. The current definition is more likely to be created from the perspective of universal algebra (see [ 1 , Chapter 3]). For a discrete group or semigroup S , the weakly almost periodic compactification wS is the largest compact semigroup which (i) contains S as a dense subsemigroup, and (ii) has multiplication continuous in each variable separately (where largest means that any other compact semigroup with the properties (i) and (ii) is a quotient of wS ). A third viewpoint is to envisage wS as the Gelfand space of the C*-algebra of bounded weakly almost periodic functions on S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup (ℕ, +). The three approaches described above give three methods of obtaining information about w ℕ. An early striking result about w ℕ, that it contains more than one idempotent, was obtained by T. T. West using operator theory [ 13 ]. He considered the weak operator closure of the semigroup { T , T 2 , T 3 , …} of iterates of a single operator T on the Hilbert space L 2 (μ) for a particular measure μ on [0, 1]. Brown and Moran, in a series of papers culminating in [ 2 ], used sophisticated techniques from harmonic analysis to produce measures μ that permitted the detection of further structure in w ℕ; in particular, they found 2 [cfr ] distinct idempotents. However, for many years, no other way of showing the existence of more than one idempotent in w ℕ was found. The breakthrough came in 1991, and it was made by Ruppert [ 11 ]. In his paper, he created a direct construction of a family of weakly almost periodic functions which could detect 2 [cfr ] different idempotents in w ℕ. His method was very ingenious (he used a unique variant of the p -adic expansion of integers) and rather complicated. Our main aim in this paper is to construct weakly almost periodic functions which are easy to describe and so appear more ‘natural’ than Ruppert's. We also show that there are enough functions of our type to distinguish 2 [cfr ] idempotents in w ℕ.