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Weakly almost period functions on N with a negative base

Bordbar, Behzad (1998) Weakly almost period functions on N with a negative base. Journal of the London Mathematical Society, (57). pp. 706-720. ISSN 0024-6107. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:21648)

The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided.


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 ([open face N], +). The three approaches described above give three methods of obtaining information about w[open face N]. An early striking result about w[open face 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, T2, T3, …} of iterates of a single operator T on the Hilbert space L2(?) for a particular measure ? on [0, 1]. Brown and Moran, in a series of papers culminating in [2], used sophisticated techniques from harmonic analysis to produce measures ? that permitted the detection of further structure in w[open face N]; in particular, they found 2[fraktur c] distinct idempotents. However, for many years, no other way of showing the existence of more than one idempotent in w[open face 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[fraktur c] different idempotents in w[open face 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[fraktur c] idempotents in w[open face N].

Item Type: Article
Subjects: Q Science > QA Mathematics (inc Computing science) > QA 76 Software, computer programming,
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Computing
Depositing User: Mark Wheadon
Date Deposited: 25 Aug 2009 17:38 UTC
Last Modified: 16 Nov 2021 10:00 UTC
Resource URI: (The current URI for this page, for reference purposes)

University of Kent Author Information

Bordbar, Behzad.

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