On the Structure of Rank One Elements in Banach Algebras

Throughout this paper A denotes a complex, unital Banach algebra with the minimum requirement that A be semiprime, that is xAx = {0} implies x = 0 holds for all x ∈ A. It is obvious that any semisimple Banach algebra is also semiprime. We call an element a ∈ A spatially rank one if a is rank on