Amenability properties of vector-valued function algebras

Let X be a compact Hausdorff space, A be a (commutative) Banach algebra and \mathcal{A} be a Banach A-valued function algebra on X. Let \mathfrak{A} be the function algebra on X, consisting of scalar-valued functions in \mathcal{A}. We study and characterize various amenability properties of the algebra \mathcal{A} in terms of cohomological properties of \mathfrak{A} and A. Containing some well-known examples, such as C(X,A) and Lip(X,A), the class of vector-valued function algebras also includes, in some sense, the tensor products \mathfrak{A} \hat \otimes_\gamma A. As consequences, some known results in this area are covered.

Keywords:

Vector-valued function algebras , tensor products , character amenability , character contractibility