Vector-valued spectra of Banach algebra valued continuous functions

Abstract

Given a compact space X, a commutative Banach algebra A, and an A-valued function algebra A on X, the notions of vector-valued spectrum of functions f ∈ A are discussed. The A-valued spectrum sp A

About the joint spectrum of vector-valued functions

“…Notice that our definition of the general element of the set A λ,f is similar to the element used in the definition of the joint spectrum in [2]. As we did not assume the commutativity of our algebra, we had to add here also the elements b 1 , .…”

“…The joint spectrum or vector-valued spectrum of f ∈ A X is defined in [2]. There it is denoted by E-SP (f ).…”

A Characterization of Polynomially Convex Sets in Banach Spaces

“…Since f : X → A is continuous, the joint spectrum sp(f ) is a compact set in C (X); see e.g. [2,Theorem 3.3]. We just need to prove that Φ is a continuous bijection.…”

Vector-valued characters on vector-valued function algebras

“…An application of vector-valued characters are presented in the forthcoming paper [2] to identify the vector-valued spectrum of functions f ∈ A . It is known that the spectrum of an element a ∈ A is equal to sp(a) = {φ(a) : φ ∈ M(A)}.…”

“…It is known that the spectrum of an element a ∈ A is equal to sp(a) = {φ(a) : φ ∈ M(A)}. In [2] the A-valued spectrum sp A (f ) of functions f ∈ A are studied and it is proved that, under certain conditions,…”