Abstract:

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X,A) be the algebra of all continuous A-valued functions on X equipped with the topology of ${\mathcal{K}(X)}$ -convergence. Moreover, let ${\mathfrak{M}_{\ell}(A)}$ and ${\mathfrak{M}(A)}$ denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X,A) and show that there exist bijections from ${\mathfrak{M}_{\ell}(C(X, A))}$ onto ${X \times \mathfrak{M}_{\ell}(A)}$ and ${\mathfrak{M}(C(X, A))}$ onto ${X \times \mathfrak{M}(A)}$ . We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand–Mazur algebra with jointly continuous multiplication.