Rank in Banach Algebras: A Generalized Cayley-Hamilton Theorem

Let $A$ be a semisimple Banach algebra with non-trivial, and possibly infinite-dimensional socle. Addressing a problem raised by Harte and Hernandez, we first define a characteristic polynomial for elements belonging to the socle, and we then show that a Generalized Cayley-Hamilton Theorem holds for the associated polynomial. The key arguments leading to the main result follow from the observation that a purely spectral approach to the theory of the socle carries alongside it an efficient method of dealing with relativistic problems associated with infinite-dimensional socles.