On biprojectivity and Connes biprojectivity of a dual Banach algebra with respect to a $w^{*}$-closed ideal

Document Type : Original Article


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

2 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran

3 Department of Mathematics, Farhangian University of Kermanshah, Kermanshah, Iran


In 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 matrix algebras and Banach sequence algebras are studied under this new notion. Also with some mild assumptions, the relation between $I$-Connes biprojectivity and left $\varphi$-contractibility is given, where $\varphi$ is a $w^*$-continuous multiplicative linear functional on $A$. As an application, we characterize Connes biprojectivity of some Lipschitz algebras.


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