A functional calculus for Banach PI-algebras
Let A be a unital Banach algebra. Take \(a_ 1,...,a_ n\in A\) and let B be the closed subalgebra of A they generate. The algebras \({\mathcal D}(\Omega)\) of entire matrix-valued functions were introduced by J. L. Taylor, who asked if they led to a functional calculus, generalizing the Shilov-Waelbroeck-Arens-Calderon theorem. We show that a necessary condition for a functional calculus map \({\mathcal D}(\Omega)\to A\) to exist is that B satisfy a polynomial identity; sufficient conditions are that B be a topological subquotient of a Banach Azumaya algebra, or that \(n=2\) and B satisfy all identities of \(2\times 2\) matrices. For closed subalgebras of Banach Azumaya algebras, we obtain a functional calculus on polynomial polyhedra containing the joint spectrum. Various properties of algebras of matrix-valued functions are studied, including domains of holomorphy.
Shilov-Wealbroeck-Arens-Calderon theorem
scientific article; zbMATH DE number 3923447