Completion of norms for $C(X, Q)$

Abstract:

Let $C(X,Q)$ denote the algebra of all continuous quaternion-valued functions vanishing at infinity on a locally compact Hausdorff space $X$. Under the natural norm (the sup norm) and under the spectral radius norm, $r(f)$, which is equivalent to the sup norm, $C(X,Q)$ is a Banach algebra. Let $\delta (f)$ be any multiplicative norm for $C(X,Q)$; i.e., one under which it is a normed algebra. It is shown that $\delta (f)$, whether or not it is complete, majorizes the natural norm and $r(f)$. Under certain conditions on the radical of the completion of $\delta (f),\delta (f)$ is equivalent to the natural norm and $r(f)$.