Prüfer’s ideal numbers as Gelfand’s maximal ideals

Abstract

Abstract. Polyadic arithmetics is a branch of mathematics related to p-adic theory. The aim of the present paper is to show that there are very close relations between polyadic arithmetics and the classic theory of commutative Banach algebras. Namely, let A be the algebra consisting of all complex periodic functions on Z with the uniform norm. Then the polyadic topological ring can be defined as the ring of all characters A → C with convolution operations and the Gelfand topology. ProlegomenaLet us start from an elementary polynomial identity. Let {p j } be an arbitrary sequence of natural numbers and z be a complex variable. It is easily seen that for each k ∈ N (1+z+· · ·+zIn particular, if the sequence p j is constant, i.e. p j = p for all j, the identity will be transformed to the formMathematics Subject Classification. Primary: 43A60 Almost periodic functions on groups, Secondary: 43A75 Analysis on specific compact groups.