Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions |
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Authors: | Mati Abel Mortaza Abtahi |
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Affiliation: | 1. Institute of Pure Mathematics, University of Tartu, 2 J. Liivi Str., room 614, 50409, Tartu, Estonia 2. School of Mathematics and Computer Sciences, Damghan University, P.O. Box 36715-364, Damghan, Iran
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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. |
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