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On algebras of bounded continuous functions valued in a topological algebra. (English) Zbl 1416.46052

Summary: Let \(X\) be a completely regular space. We denote by \(C\left(X,A\right) \) the locally convex algebra of all continuous functions on \(X\) valued in a locally convex algebra \(A\) with a unit \(e.\) Let \(C_{b}\left(X,A\right) \) be its subalgebra consisting of all bounded continuous functions and endowed with the topology given by the uniform seminorms of \(A\) on \(X.\) It is clear that \(A\) can be seen as the subalgebra of the constant functions of \(C_{b}\left(X,A\right)\). We prove that if \(A\) is a Q-algebra, that is, if the set \(G\left( A\right) \) of the invertible elements of \(A\) is open, or a Q-álgebra with a stronger topology, then the same is true for \(C_{b}\left( X,A\right) \).

MSC:

46J10 Banach algebras of continuous functions, function algebras
46E40 Spaces of vector- and operator-valued functions
46H05 General theory of topological algebras
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References:

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