Generalized Numerical Index of Function Algebras

Let X be a complex Banach space andC b ( 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 ofC b ( 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 ofC b ( Ω ) , the Banach space of all bounded complex-valued continuous functions; (ii) ϕ ⊗ x ∈ A for all ϕ ∈ A and x ∈ X ; and (iii) ϕ f ∈ 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 .