Normed algebras with involution

We show that most of the theory of Hermitian Banach algebras can be proved for normed *-algebras without the assumption of completeness. The conditionr(x)≤p(x) for allx (wherep(x)=r(x ^* x) ^1/2 is the Pták function), which is essential in the theory of Hermitian Banach algebras, is replaced for normed *-algebras by the conditionr(x+y)≤p(x)+p(y) for allx, y. In case of Banach *-algebras these conditions are equivalent. The research has been supported by a grant from La Junta de Andalucía and by the Department of Applied Mathematics, University of Seville This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990