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
wℕ. An early striking result about wℕ,
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, T2, T3, …}
of iterates
of a single operator T on the Hilbert space L2(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
wℕ; 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 wℕ
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 wℕ.
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 wℕ.
