Some Cohomological Notions in Banach Algebras Based on Maximal Ideal Space

In this paper, we investigate the homological notion of left phi-biprojectivity for certain Banach algebras, where phi is a nonzero multiplicative linear functional. Our initial result states that this notion is equivalent to left phi-contractibility, provided that A has a left approximate identity. As an application, we study left phi-biprojectivity of Banach algebras related to locally compact groups. For instance, we show that for a locally compact group G, the Segal algebra S(G) is left phi is compact and the Fourier algebra A(G) is left phi-biprojective if and only if G is discrete. Finally, we give some examples which show the differences between our new notion and the classical ones.