Generalizations of a theorem of Mutylin

Abstract

We generalize Mutylin’s theorem that the only complete, locally bounded, additively generated topological fields are R and C by showing: (1) the only complete, locally bounded, additively generated topological division rings with left bounded commutator subgroup are R, C, and H; (2) a commutative, Hausdorff topological ring A with identity is a Banach algebra over R, equipped with the absolute value | . . | p |..{|^p} for some p ∈ ( 0 , 1 ] p \in (0,1] , if (and only if) A is complete, locally bounded, additively generated, and possesses an invertible topological nilpotent.