In this paper, we introduce the new notions of -bi atness, -biprojectivity, -Johnson amenability and -Johnson contractibility for Banach algebras, where is a non-zero homomorphism from a Banach algebra A into C. We show that a Banach algebra A is -Johnson amenable if and only if it is -inner amenable and -bi at. Also we show that -Johnson amenability is equivalent with the existence of left and right -means for A. We give some examples to show di erences between these new notions and the classical ones. Finally, we show that L1(G) is -bi at if and only if G is an amenable group and A(G) is -biprojective if and only if G is a discrete group.
