Let $\frak{ F}(X, A)$ be one of the Banach algebras $\hbox{ Lip}(X, A)$ or $\hbox{ lip}(X, A)$. In this paper, we show that $\frak{ F}(X, A)$ is amenable if and only if $X$ is uniformly discrete and $A$ is amenable. We also prove that the result holds for $\hbox{ lip}^\circ(X, A)$ instead of $\frak{ F}(X, A)$. In the case where $A^*$ is separable, we establish that $\frak{ F}(X, A)^{ **}$ is amenable if and only if $X$ is uniformly discrete and $A^{ **}$ is amenable, however, amenability of $\hbox{ lip}^\circ(X, A)^{ **}$ is equivalent to amenability of $A^{ **}$ and finiteness of $X$. We prove that if $\hbox{ Lip}(X, A)$ is point (respectively, weakly) amenable, then $X$ is uniformly discrete and $A$ is point (respectively, weakly) amenable. In particular, $\hbox{ Lip}X$ is weakly amenable if and only if $X$ is discrete. We then investigate cohomological properties for vector-valued Banach algebras $C_0(X, A)$ and $L^1(G, A)$. Finally, we prove that biprojectivity (respectively, cyclically weak amenability) of $A^{ **}$ implies biprojectivity (respectively, cyclically weak amenability) of $A$. This result holds for weak amenability and cyclic amenability when $A$ is commutative.
Categories: math.FA