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Description of closed maximal ideals in topological algebras of continuous vector-valued functions. (English) Zbl 1318.46031

Summary: 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.

MSC:

46H10 Ideals and subalgebras
46J10 Banach algebras of continuous functions, function algebras
46J20 Ideals, maximal ideals, boundaries
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