$$\varphi $$ φ -Biprojectivity of Banach Algebras with Applications to Hypergroup Algebras

Abstract

At the present paper, we study the notions of $$\varphi $$ -biprojectivity, $$\varphi $$ -Johnson contractibility, and $$\varphi $$ -contractibility of Banach algebras, where $$\varphi $$ is a nonzero character. We introduce the condition (Q) which is weaker than $$\varphi $$ -biprojectivity. For classes of Banach algebras with a left and right approximate identity, we obtain some relations between these notions. Moreover, we apply these results for the hypergroup algebra $$L^{1}(K)$$ and some Segal algebras with respect to the $$L^{1}(K)$$ . As a main result, for a hypergroup K, we prove that the hypergroup algebra $$L^{1}(K)$$ is $$\varphi $$ -biprojective (left $$\varphi $$ -contractible) if and only if K is compact.