On the automorphisms of the spectral completion of the algebraic numbers field

Let G=Gal(Q¯/Q) be the absolute Galois group of QQ and let A=C(G,C)A=C(G,C) be the Banach algebra of all continuous functions defined on GG with values in CC. Let e¯ be the conjugation automorphism of CC and let BB be the RR-Banach subalgebra of AA consisting of continuous functions ff such that f(e¯σ)=e¯f(σ) for all σ∈Gσ∈G. Let ‖x‖=sup{|σ(x)|:σ∈G}‖x‖=sup{|σ(x)|:σ∈G} be the spectral norm on Q¯ and let Q˜ be the spectral completion of Q¯. Using a canonical isometry between Q˜ and BB we study the structure of the group of RR-algebras automorphisms of Q˜ and the structure of its subgroup Alg(Q˜) of all automorphisms of Q˜ which when restricted to Q¯ give rise to elements of GG. We introduce a topology on Alg(Q˜) and prove that this last one is homeomorphic and group isomorphic to GG.