In thispaper,wecontinue thestudyof ∗∗biprojectivityofBanachalgebras,was introducedin[19],andtherelationbetweenthisnew notionandφ-amenabilityofBanachalgebras isinvestigated. ∗∗-biprojectivityofSegalalgebrasandlower triangularmatrixalgebrasis studied. Also,we introduce thenotionofφ∗∗-biprojectivityofBanachalgebras. Some examples indicate that this notion isweaker than ∗∗-biprojectivity.Weobtaintherelation betweenthisnotionandφ-amenabilityandφinneramenability. Finally,weinvestigatethis newnotiononcertainBanachalgebrassuchas groupalgebras,measurealgebras, and lower triangularBanachalgebras.
