Applications of the growth characteristics induced by the spectral distance
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 Phragmén–Lindelöf 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.
Keywords: Asymptotically intertwined, commutator, spectral distance, quasinilpotent equivalent