In this paper we continue to study the spectral norms and their completions in the case of the algebraic closure ℚ̄ of ℚ in ℂ. Let ℚ̄̃ be the completion of ℚ̄ relative to the spectral norm. We prove that ℚ̄̃ can be identified with the ℝ-subalgebra of all symmetric functions of C(G), where C(G) denotes the ℂ-Banach algebra of all continuous functions defined on the absolute Galois group G = Gal (ℚ̄/ℚ). We prove that any compact, closed to conjugation subset of ℂ is the pseudo-orbit of a suitable element of ℚ̄̃. We also prove that the topological closure of any algebraic number field in ℚ̄̃ is of the form ℚ[x]̃ with x in ℚ̄̃.
- Banach algebras of continuous functions
- Galois groups
- Number fields
- Spectral norms
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