### On approximately biprojective and approximately biflat Banach algebras

#### Abstract

In this paper, we study approximate biprojectivity and approximate biflatness of a Banach algebra $\mathcal A$ and find some relations between theses concepts with $\phi$-amenability and $\phi$-contractibility, where $\phi$ is a character on ${\mathcal A}$. Among many other things, we show that $\theta$-Lau product algebra $L^{1}(G)\times_{\theta}A(G)$ is approximately biprojective if and only if $G$ is finite, where $L^{1}(G)$ and $A(G)$ are the group algebra and the Fourier algebra of the locally compact group $G$, respectively. We also characterize approximately biprojective

and approximately biflat semigroup algebras associated with the inverse semigroups.

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