Algebraic Characterization of Rings of Continuous p-Adic Valued Functions

Abstract

The aim of this article is to characterize among the class of all commutative rings containing ℚ the rings C(X, ℚ_p) of all continuous ℚ_p-valued functions on a compact space X. The characterization is similar to that of M. Stone from 1940 (see [9]) for the case of ℝ-valued functions. The Characterization Theorem 4.6 is a consequence of our main result, the p-adic Representation Theorem 4.5.

Key Words:

• Banach algebras
• Normed rings
• p-adic representations
• p-adic spectrum
• p-valuations

2010 Mathematics Subject Classification:

• 13A18
• 46H15
• 12J12

Notes

Although pre-orderings on commutative rings have already been used by M. Stone, the notation “pre-ordering” was introduced much later by Krivine in a systematic study [4].

Compared with the real situation, one could as well call them “pre-p-valuations.”

Note that every element of F has a canonical expansion as a power series in the uniformizer p with coefficients from {0, 1,…, p − 1} (cf. [2], Proposition 1.3.5).