On left \phi-Connes biprojectivity of dual Banach algebras

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^۱(S) is left \phi_{S}-Connes biprojective if and only if S is a finite group, where  \phi_{S}\in\Delta_{w^*}(\ell^۱(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

Keywords:

Semigroup algebras , Matrix algebras , Connes amenability , Left \phi-Connes biprojectivity