Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24495320240701On biprojectivity and Connes biprojectivity of a dual Banach algebra with respect to a $w^*$ -closed ideal225231516410.22060/ajmc.2023.22285.1149ENAmirSahamiDepartment of Mathematics, Faculty of Basic Sciences, Ilam University, P.O. Box 69315-516, Ilam, Iran0000-0003-0041-509XS. FatemehShariatiFaculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), IranMehdiRostamiFaculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran0000-0002-8989-0286MonaAjDepartment of Mathematics, Farhangian University of Kermanshah, Kermanshah, IranJournal Article20230322In this paper, we introduce a notion of Connes biprojectivity for a dual Banach algebra $A$ with respect to its $w^{*}$-closed ideal $I$, say $I$-Connes biprojectivity. Some Lipschitz algebras $Lip_{\alpha}(X)$ and some matrix algebras are studied under this new notion. Also, with some mild assumptions, the relation between $I$-Connes biprojectivity and left $\phi$-contractibility is given, where $\phi$ is a $w^{*}$-continuous multiplicative linear functional on $A$. As an application, we characterize Connes biprojectivity of some Lipschitz algebras.https://ajmc.aut.ac.ir/article_5164_8045577d0a3d1f2213c1003fc44fdfc6.pdf