Character amenability of real Banach algebras

Let (A,∥⋅∥) be a real Banach algebra. In this paper we first introduce left and right φ-amenability of A and discuss the relation between left (right, respectively) φ-menability and φ¯¯¯-amenability of A for φ∈△(A)∪{0} where φ¯¯¯∈△(A) is the conjugate of φ. Next we show that A is left (right, respectively) φ-amenable if and only if AC is left (right, respectively) φC-amenable, where AC is a suitable complexification of A and φC∈△(AC) is the induced character by φ on AC. In continue, we give a hereditary property for 0-amenability of A. We also study relations between the injectivity of Banach left A-modules and right φ-amenability of A. Finally, we characterize the left character amenability of certain real Banach algebras.