Approximate biprojectivity and ϕ-biflatness of certain Banach algebras

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In this paper we are going to investigate the approximate biprojectivity and the ϕ-biflatness of some Banach algebras related to the locally compact groups. We show that a Segal algebra S(G) is approximate biprojective if and only if G is compact. Also for a continuous weight w≥ 1, we show that L¹(G,w) is a approximate biprojective if and only if G is compact. We study ϕ-biflatness of some Banach algebras, where ϕ:A→ C is a multiplicative linear functional. We show that if S(G) is ϕ-biflat, then G is amenable group. Also we show that the ϕ-biflatness of L¹(G)** implies the amenability of G.