Approximate biflatness and Johnson pseudo-contractibility of some Banach algebras

In this paper, 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 $Lip_{\alpha}(X)$ and $\ell ip_{\alpha}(X)$ are approximately biflat if and only if $X$ is finite, provided that $0<\alpha<1$. We give an enough and sufficient condition that a vector-valued Lipschitz algebras is Johnson pseudo-contractible for each $\alpha>0.$ We also show that some triangular Banach algebras are not approximately biflat.