# On Left φ-biflat and Left φ-biprojectivity of θ-lau Product Algebras

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Basic Sciences Ilam University P.O. Box 69315-516 Ilam, Iran.

2 Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran.

Abstract

\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.
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 $\ell^{1}(\Bbb{N}_{\vee}).$
Finally, we give an example among the Lau product structure of matrix algebras which is not left $\phi$-biflat.

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