Applications of the Growth Characteristics Induced by the Spectral Distance

Abstract

Let $A$ be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into $A$, we derive a generalization of Gelfand's famous Power Boundedness Theorem. Elaborating on these ideas, with the help of a Phragm\'{e}n-Lindel\"{o}f device for subharmonic functions, it is then shown, as the main result, that two normal elements of a $C^*$-algebra are equal if and only if they are quasinilpotent equivalent.