Approximate biflatness and Johnson pseudo-contractibility of some Banach algebras

Abstract

We study the structure of Lipschitz algebras under the notions of approximate biflatness and Johnson pseudo-contractibility. We show that for a compact metric space $X$, the Lipschitz algebras ${\rm Lip}_{\alpha}(X)$ and ${\rm lip}_{\alpha}(X)$ are approximately biflat if and only if $X$ is finite, provided that $0<\alpha<1$. We give a necessary and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible. We also show that some triangular Banach algebras are not approximately biflat.