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

TL;DR

This paper characterizes the amenability and cohomological properties of vector-valued Lipschitz algebras and their second duals, linking these properties to the discreteness of the underlying space and amenability of the algebra.

Contribution

It provides new characterizations of amenability and cohomological properties for Lipschitz and related Banach algebras, including their second duals, with conditions based on space discreteness and algebra amenability.

Findings

$rak{F}(X, A)$ is amenable iff $X$ is uniformly discrete and $A$ is amenable

Amenability of $rak{F}(X, A)^{**}$ depends on $X$ being discrete and $A^{**}$ being amenable

Biprojectivity and cyclic weak amenability of $A^{**}$ imply the same for $A$

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.