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In this short note, we give some counter examples which show that [11, Proposition 3.5] is not true. As a consequence, the arguments in [11, Proposition 4.10] is not valid.

We investigate the amenability and its related homological notions for a class of $I\times I$-upper triangular matrix algebra, say $\mathrm{UP}(I,A)$, where $A$ is a Banach algebra equipped with a nonzero character. We show that $\mathrm{UP}(I,A)$ is pseudo-contractible (amenable) if and only if $I$ is singleton and $A$ is pseudo-contractible (amenable), respectively. We also study pseudo-amenability and approximate biprojectivity of $\mathrm{UP}(I,A)$.

We study the notion of left $\varphi$-biflatness for Segal algebras and semigroup algebras. We show that the Segal algebra $S\left(G\right)$ is left $\varphi$-biflat if and only if $G$ is amenable. Also we characterize left $\varphi$-biflatness of semigroup algebra $l^1\left(S\right)$ in terms of biflatness, when $S$ is a Clifford semigroup.

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 ${\mathrm{Lip}}_{\alpha }\left(X\right)$ and ${\mathrm{lip}}_{\alpha }\left(X\right)$ 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.