## 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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### References:

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