Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions

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.