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Banach algebra of \(p\)-adic valued almost periodic functions. (English) Zbl 0776.46037
p-adic functional analysis, Proc. Int. Conf., Laredo/Spain 1990, Lect. Notes Pure Appl. Math. 137, 141-150 (1992).
[For the entire collection see Zbl 0746.00058.]
It is shown that the characteristic function \(\chi_ H\) of closed normal subgroups \(H\) of finite index when suitably normalised give rise to the functions \(U_ H\) which are central idempotents in the algebra \(A\). \(A*U_ H\) is then a finite dimensional two sided ideal of \(A\). The direct sum \(\sum U_ H*A\), 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 \(G\). A conjecture regarding the structure of the algebra \(A\) which can possibly be proved affirmatively using the results on idempotents of \(A\) or using the structure theory of the group algebra of the compact group \(\beta(G)\), the Bohr compactification of \(G\), is also made in the paper.

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46J05 General theory of commutative topological algebras
22C05 Compact groups