Amenability properties of Banach algebra valued continuous functions
Let \(X\) be a compact Hausdorff space and \(A\) a Banach algebra. It is well known that \(C(X)=C(X, \mathbb{C}) \) is amenable for any compact Hausdorff space \(X\). The authors investigate amenability properties of the Banach algebra \(C(X,A)\) of all \(A\)-valued continuous functions on \(X\) endowed with the uniform norm \(\|f\|_{\infty} = \sup \left\{ \|f(x) \|_A : x \in X \right\},\) \(f \in C(X,A)\). As results, they prove that \(C(X,A)\) is amenable if and only if \(A\) is amenable; \(C(X,A)\) has a compactly approximate diagonal if \(A\) has a central compactly approximate diagonal; and \(C(X,A)\) is weakly amenable if \(A\) is a commutative weakly amenable Banach algebra with a bounded approximate identity.
scientific article; zbMATH DE number 6362878
scientific article