MODULE (φ,φ) - BIPROJECTIVITY AND MODULE (φ,φ) - BIFLATNESS OF BANACH ALGEBRAS

Let A be a Banach algebra. In this paper for a Banach algebra Awhich is also an A-bimodule we introduce the notions of module (phi,phi)-biprojectivity and module (phi,phi)-biflatness of A, where phi epsilon Delta (A)boolean OR {0} and phi epsilon Omega(A), the space consisting of all linear maps phi : Lambda -> A such that phi(ab) = phi(a)phi(b), phi(alpha.alpha) = phi(alpha)phi(a) (a, b epsilon A, alpha epsilon A). We investigate relations between module (phi,phi)-biprojectivity and phi degrees phi-biprojectivity of A and we show that under some conditions A is module (phi,phi)-biflat if and only if A is module (phi,phi)-amenable. Finally, for an inverse semigroup S with the set of idempotents E, we show that the semigroup algebra l(1)(S), as an l(1)(E)-module, is module (phi,phi)-biflat if and only if S is amenable.