Abstract
Let K be a subfield of (Q) over bar, a fixed algebraic closure of Q, the field of rational numbers. Let G(K) = Gal((Q) over bar /K) be the absolute Galois group of K. For any x is an element of (Q) over bar, we consider the K-spectral norm: parallel to x parallel to(K) = max{|sigma(x)| : sigma is an element of G(K)}. Let (e) over bar be the conjugation automorphism of (Q) over bar and let C(G(K)) be the Banach algebra of all continuous mappings defined on the compact group G(K) with values in C. Let C(sym)(G(K)) be the Banach algebra over (K) over tilde of all symmetric functions of C(G(K)). Here K = R or C is the completion of K relative to the usual absolute value |.|. A function f is said to be symmetric if f((e) over bar sigma) = (e) over bar (f(sigma)) for any sigma is an element of G(K) (when (e) over bar is an element of G(K)). Let (Q) over tilde (K) be the completion of (Q) over bar with respect to parallel to.parallel to(K). In this paper we prove that (Q) over tilde (K) congruent to C(sym) (G(K)) if (e) over bar is an element of G(K) and (Q) over tilde (K) congruent to C(G(K)) if (e) over bar is not an element of G(K). These last isomorphisms are (K) over tilde -isomorphisms of Banach algebras. We give some other properties of the closed subalgebras of C(G(K)) in connection with some subfields of algebraic numbers.