ABSTRACT
p-adic valued almost periodic functions were introduced by Rangan [1], [2] and [3] and their properties such as the existence of mean for such functions were also studied. Analogous to the classical case it was shown that the continuous almost periodic functions on a totally disconnected topological group G (not necessarily locally-compact and throughout this paper G stands for such a group) are the same as the continuous functions on the non-archimedean Bohr compactification β>(G) of G. A = Ac(G) of all continuous almost periodic functions on G form a Banach algebra with convolution (∗) as multiplication. In this paper we show that the characteristic functions χ H of closed normal subgroups H of finite index when suitably normalised give rise to the functions UH which are central idempotents in the algebra A. A ∗ UH is then a finite-dimensional two sided ideal of A. The direct sum ∑ U H ∗ A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq1805.tif"/> , as H varies over the closed normal subgroups of finite indices is shown to be dense in the algebra A. When the mean exists for continuous almost periodic functions on a group G then any continuous almost periodic functions on G can be uniformly approximated by linear combinations of matrix coefficients of the finite-dimensional representations on G. We have made at the end of the paper a conjecture regarding the structure of the algebra A which can possibly be proved affirmatively using the results on idempotents of Ain section 4 of this paper or using the structure theory of the group algebra of the compact group β>(G) of G.
