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 $Li{p}_{\alpha }(X)$ and $\ell ip_{\alpha}(X)$areapproximatelybiflatifandonlyif$X$ 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.