On I-biflat and I-biprojective Banach algebras

In this paper, we introduce new notions of I-biflatness and I-biprojectivity, for a Banach algebra A, where I  is a closed ideal of A. We show that M(G) is L^{۱}(G)-biprojective (I-biflat) if and only if G is a compact group (an amenable group), respectively. Also, we show that, for a non-zero ideal I, if the Fourier algebra  A(G) is I-biprojective, then G is a discrete group. Some examples are given to show the differences between these new notions and the classical ones.