A class of Banach algebras with a unique norm topology

Abstract

The purpose of this paper is to produce a class of nonsemisimple Banach algebras with a unique norm topology. That is, a class of Banach algebras such that each member B of the class has the property that any two Banach algebra norms on B are equivalent (cf. [9, Chapter II]). (Throughout this paper, an algebra will denote any commutative algebra over the complex field C which possesses an identity e.) To do this, we investigate an algebraic extension of a semisimple algebra A which is similar to the Arens-Hoffman extension of a normed algebra (cf. [1]). We let a(x) be a monic polynomial in A [x], the algebra of all polynomials in the indeterminate x with coefficients in A, and denote by (a(x)) the principal ideal in A [xI generated by a(x). If B =A [xI/ (a(x)) is a Banach algebra with respect to some norm, then A is a normed algebra with respect to the norm on B restricted to A and we ask whether or not A is a closed subalgebra of B. If for any Banach algebra norm on B, A is a closed subalgebra of B, then we show that B has a unique norm topology. The main result of this paper is that if A is a regular (in the language of [9], completely regular), semisimple Banach algebra and if a (x) is any monic polynomial in A [x], then B =A [x]/(a(x)) has a unique norm topology, where B is the Arens-Hoffman extension of A. (See below for a discussion of the Arens-Hoffman extension.) An example is given which shows that the condition of semisimplicity is essential. It is an open question whether or not the main result is true if A is not a regular Banach algebra. In the event that A is a semisimple Banach algebra, a(x) is a monic polynomial in A [x] and B =A [x]/(a(x)) is semisimple, then by [9, Corollary 2.5.18], B has a unique norm topology. Therefore our results have meaning only if B is not semisimple. R. Arens and K. Hoffman have shown that if the discriminant d of a(x) is not a zero divisor in A, then B is semisimple [1, Theorem 4.3]. (See [1, p. 207] for the definition of d.) The converse of this latter result is also valid