Approximate biflatness and Johnson pseudo-contractibility of some Banach algebras

TL;DR

This paper investigates the properties of Lipschitz algebras related to approximate biflatness and Johnson pseudo-contractibility, establishing conditions under which these properties hold for certain Banach algebras.

Contribution

It provides new characterizations of approximate biflatness and Johnson pseudo-contractibility in Lipschitz and vector-valued Lipschitz algebras, including necessary and sufficient conditions.

Findings

Lipschitz algebras are approximately biflat iff the underlying space is finite for 0<α<1.

A condition for vector-valued Lipschitz algebras to be Johnson pseudo-contractible.

Some triangular Banach algebras are not approximately biflat.

Abstract

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.