Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 6 1 2021 08 01 On Left φ-biflat and Left φ-biprojectivity of θ-lau Product Algebras 13 23 192 10.22128/gadm.2020.380.1033 EN Amir Sahami Department of Mathematics, Faculty of Basic Sciences Ilam University P.O. Box 69315-516 Ilam, Iran. 0000-0003-0041-509X Sayed Mehdi Kazemi Torbaghan Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran. Journal Article 2020 10 09 \textit{Monfared} defined $\theta$-Lau product structure $A\times_{\theta} B$ for two Banach algebras $A$ and $B$, where $\theta:B\rightarrow \mathbb{C} $ is a multiplicative linear functional. In this paper, we study the notion of left $\phi$-biflatness and left $\phi$-biprojectivity for the $\theta$ Lau product structure $A\times_{\theta} B$. For a locally compact group $G$, we show that $M(G)\times_{\theta}M(G)$ is left character biflat (left character biprojective) if and only if $G$ is discrete and amenable ($G$ is finite), respectively.<br />Also we prove that $\ell^{1}(\Bbb{N}_{\vee})\times_{\theta}\ell^{1}(\Bbb{N}_{\vee})$ is neither $(\phi_{\Bbb{N}_{\vee}}, \theta)$-biprojective nor $ (0, \phi_{\Bbb{N}_{\vee}})$-biprojective, where $\phi_{\Bbb{N}_{\vee}}$ is the augmentation character on $\ell^{1}(\Bbb{N}_{\vee}).$<br />Finally, we give an example among the Lau product structure of matrix algebras which is not left $\phi$-biflat. https://ansne.du.ac.ir/article_192_e0b08a20a6576b83cc53393bdae68fa9.pdf