Let A be a unital Banach
algebra. Take a1,…,an∈ A and let B be the closed subalgebra of A they generate.
The algebras 𝒟(Ω) 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 𝒟(Ω) → 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 × 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.
