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´en-Lindel¨of 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.

Key words:

• Asymptotically intertwined
• commutator
• spectral distance
• quasinilpotent equivalent

Mathematics Subject Classification (2010):

• 46H05
• 46L05
• 47B47