Completion of Normed Algebras of Polynomials

Let *3* be the algebra of polynomials in one indeterminate x over the complex field C. Suppose || • || is a norm on 9 such that the coefficient functionals c,: X a.oc' —* a, (j = 0,1,2, • • •) are all continuous with respect to || • ||, and let K CC be the set of characters on 9> which are || • ||-continuous. Then K is compact, C\K is connected, and 0 G K. Let A be the completion of 9 with respect to || • ||. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K. The functionals c, have unique extensions to bounded linear functionals on A, and the map a —»£Cj(a)jt' (a G .4) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a G A and a^O imply c,(a)^0 for some j . We are interested in the relationship between the propositions (S): A is semi-simple, and (P): A is an algebra of power series. Loy (1974; Theorem 5) has proved that if 0 G K ° (the interior of K), then (P) implies (S). With the further conditions that K° is connected and dense in K, it is easy to see that (S) and (P) are equivalent (Theorem 2). Examples show that without the given restrictions on K, (S) does not imply (P), and without the condition 0G K", (P) does not imply (S). The equivalence between (5) and (P) has a generalization to the case of a projective tensor product B ®&, where B is a commutative Banach algebra with identity and 9 is suitably normed (Theorem 5). For a discussion of tensor products of Banach algebras, and in particular of the question of semi-simplicity of B(g)A when B and A are semi-simple, see Gelbaum's paper (1962).