On algebras of bounded continuous functions valued in a topological algebra

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-\'{a}lgebra with a stronger topology, then the same is true for \(C_{b}\left( X,A\right) \).