On algebras of bounded continuous functions valued in a topological algebra

Let $X$ be a completely regular space. We denote by $C(X,A)$ the locally convex algebra of all continuous functions on $X$ valued in a locally convex algebra $A$ with a unit $e.$ Let $C_b(X,A)$ 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 $C_b(X,A)$. 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 $C_b(X,A)$.