Uniqueness of the uniform norm and adjoining identity in Banach algebras

Abstract

Let A_e be the algebra obtained by adjoining identity to a non-unital Banach algebra (A,║ · ║). Unlike the case for a C^*-norm on a Banach *-algebra, A_e admits exactly one uniform norm (not necessarily complete) if so does A. This is used to show that the spectral extension property carries over from A to A_e. Norms on A_e that extend the given complete norm ║ · ║ on A are investigated. The operator seminorm ║ · ║_op on A_e defined by ║ · ║ is a norm (resp. a complete norm) iff A has trivial left annihilator (resp. ║ · ║_op restricted to A is equivalent to ║ · ║).