Rank in Banach Algebras: A Generalized Cayley-Hamilton Theorem

TL;DR

This paper extends the Cayley-Hamilton theorem to elements in the socle of semisimple Banach algebras, introducing a characteristic polynomial and demonstrating its validity in infinite-dimensional contexts.

Contribution

It defines a characteristic polynomial for socle elements and proves a generalized Cayley-Hamilton theorem in the setting of semisimple Banach algebras.

Findings

Established a characteristic polynomial for socle elements.

Proved a generalized Cayley-Hamilton theorem in infinite-dimensional Banach algebras.

Provided a spectral approach to handle infinite-dimensional socles.

Abstract

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.