On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces

In this paper, we study the notion of approximate biprojectivity and left phi-biprojectivity of some Banach algebras, where phi is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra L-1(K) implies that K is compact. Moreover, we investigate left phi-biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra B is left phi-biprojective if and only if K is compact, where K is a hypergroup. We also study the approximate biflatness and left phi-biflatness of hypergroup algebras in terms of amenability of their related hypergroups.