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