Ideals in a certain Banach algebra

Abstract

1. Introduction.The purpose of this paper is to improve a result of Bertram Yood in his paper [2].We are concerned with a complex commutative Banach algebra X and a compact Hausdorff space fi.Let C(fi, X) denote the set of all continuous functions defined over fi with values in X. C(fi, X) is a complex commutative 5-algebra if we define addition, multiplication, and scalar multiplication in the natural "pointwise" manner and if |/| c<a,X) = supoefi \f(a) \ x for/£C(fi,.X").Here, | | x denotes the norm in X. Yood proves, in Lemma 5.1 of his paper, that if X has a unit and if every maximal ideal in C(fi, X) is