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

Let image be the absolute Galois group of image and let image be the Banach algebra of all continuous functions defined on G with values in image. Let image be the conjugation automorphism of image and let image be the image-Banach subalgebra of image consisting of continuous functions f such that image for all σset membership, variantG. Let double vertical barxdouble vertical bar=sup{σ(x):σset membership, variantG} be the spectral norm on image and let image be the spectral completion of image. Using a canonical isometry between image and image we study the structure of the group of image-algebras automorphisms of image and the structure of its subgroup image of all automorphisms of image which when restricted to image give rise to elements of G. We introduce a topology on image and prove that this last one is homeomorphic and group isomorphic to G.