Cohomological properties of vector-valued Lipschitz algebras and their second duals

Abstract

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.