Generalized Numerical Index of Function Algebras

Abstract

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A, x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈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.

On the Lipschitz numerical index of Banach spaces

“…With the aid of Lemma 3.10, we can prove the following density result for the space A(K, X), which is inspired by the corresponding result for the space A(Ω, X) in [29]. Proposition 3.11.…”

“…By [29,Theorem 4], there exists h ∈ A(Ω, X) with h = 1 which strongly peaks at t 0 and satisfies that h…”

“…For the following result analogous to Theorem 3.6, we omit the detail of its proof since it is identical to the proof of Theorem 3.6 if we use Proposition 3.11 instead of [29,Theorem 4]. Theorem 3.13.…”