On left $\phi$-Connes biprojectivity of dual Banach algebras

Document Type : Research Paper


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

2 Department of Mathematics, University of Kurdistan, Pasdaran Boulevard, Sanandaj 66177--15175, P. O. Box 416, Iran

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

4 Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, P.O.Box 94531, Iran


We introduce the notion of left (right) $\phi$-Connes biprojective for a dual Banach algebra $\mathcal{A}$, where $\phi$ is a non-zero $wk^*$-continuous multiplicative linear functional on $\mathcal{A}$. We discuss the relationship of left $\phi$-Connes biprojectivity with $\phi$-Connes amenability and Connes biprojectivity. For a unital weakly cancellative semigroup $S$, we show that $\ell^1(S)$ is left $\phi_{S}$-Connes biprojective if and only if $S$ is a finite group, where  $\phi_{S}\in\Delta_{w^*}(\ell^1(S))$. We prove that for a non-empty totally ordered set $I$ with the smallest element, the upper triangular $I\times I$-matrix algebra $UP(I,\mathcal{A})$ is right $\psi_\phi$-Connes biprojective if and only if $\mathcal{A}$ is right $\phi$-Connes biprojective and $I$ is a singleton, provided that $\mathcal{A}$ has a right identity and $\phi\in\Delta_{w^*}(\mathcal{A})$. Also for a finite set $I$,  if $Z({\mathcal A})\cap ({\mathcal A}-\ker\phi)\neq \emptyset$, then the dual Banach algebra $UP(I, {\mathcal A})$ under this new notion forced to have a singleton index


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Volume 14, Issue 6
June 2023
Pages 257-264
  • Receive Date: 04 June 2021
  • Revise Date: 05 September 2021
  • Accept Date: 08 October 2021