Principal ideals in $F$-algebras

Abstract

This paper is concerned with generalizations to F-algebras of theorems which Gleason has proved for finitely generated maximal ideals in Banach algebras. Let A be a uniform commutative F-algebra with identity such that Spec (A) is locally compact; let x be a nonisolated point of Spec (A), and let ker(x) denote the maximal ideal of all elements of A which vanish at x. In this paper it is shown that: If f is an element of A vanishing only at x, then the principal ideal Af generated by f is closed in A. If the polynomials in the elementf are dense in A and if ker (x) is finitely generated, then there exists an open set U containing x such that ker(y) is generated by f f(y) for all y in U. An example is given which shows that if A is not uniform, the conclusion of the last result may not be true. In fact, the example shows that it is possible to have a nonisolated finitely generated maximal ideal in the algebra. A second example shows that in a uniform Falgebra with locally compact spectrum, ker(x) can be generated by an elementf such thatf f(y) generates no other ker(y) even when the ker(y) are principal. Introduction. The results in this paper generalize to F-algebras results which are known for Banach algebras (see Theorem 2.1(ii) and Theorem 2.2 of [4]). Although the results stated are only for principal maximal ideals, they should point out some of the difficulties in general for finitely generated maximal ideals. Suppose that B is a commutative Banach algebra with identity. Gleason [4] proved the following theorems dealing with the generators of an algebraically finitely generated maximal ideal. (G 1) If I is a maximal ideal of B which is generated by gl, ..., gn,X then there exists a neighborhood U of I in the maximal ideal space of B such that each maximal ideal M in U is generated by g1 ^ (M), ..., gn g (M). (G2) If the subalgebra generated by 1, Z1, . . , Zk is dense in B and if I is a finitely generated maximal ideal in B, then I is generated by z1 Zj (I), Zk -Zk (I ). One immediate consequence of these theorems is that in a commutative Banach algebra with identity, the set of finitely generated maximal ideals is an open set. We will see with an example that this need not be true for F-algebras. Furthermore, we will show that when one considers F-algebras the conclusion Presented to the Society, January 25, 1973; received by the editors February 18, 1976. AMS (MOS) subject classifications (1970). Primary 46H99; Secondary 46E25.