Topological algebras with subadditive boundedness radius

Let $A$ be a topological algebra and $\beta$ a subadditive boundedness radius on $A$. In this paper we show that $\beta$ is, under certain conditions, automatically submultiplicative. Then we apply this fact to prove that the spectrum of any element of $A$ is non-empty. Finally, in the case when $A$ is a normed algebra, we compare the initial normed topology with the normed topology $\tau_{\beta}$, induced by $\beta$ on $A$, where $\beta^{-1} (0)=0$.