Generalized Numerical Index of Function Algebras

Abstract

Let X be a complex Banach space and C-b(Omega : X) be the Banach space of all bounded continuous functions from a Hausdorff space to X, equipped with sup norm. A closed subspace A of C-b(Omega : X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A:{x*circle f : f is an element of A,x* is an element of X*} is a closed subalgebra of Cb(Omega), the Banach space of all bounded complex-valued continuous functions; (ii) phi circle times x is an element of A for all phi is an element of A and x is an element of X; and (iii) phi f is an element of A for every phi is an element of A and for every f is an element of A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.