Abstract
LetA e be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, ∥ · ∥). Unlike the case for aC*-norm on a Banach *-algebra,A e admits exactly one uniform norm (not necessarily complete) if so doesA. This is used to show that the spectral extension property carries over fromA to A e . Norms onA e that extend the given complete norm ∥ · ∥ onA are investigated. The operator seminorm ∥ · ∥op onA e defined by ∥ · ∥ is a norm (resp. a complete norm) iffA has trivial left annihilator (resp. ∥ · ∥op restricted toA is equivalent to ∥ · ∥).
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