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 in [5, p. 1399], 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.