homomorphism maps into dual Banach algebras

Let A be a Banach algebra and (B,B_*) be a dual Banach algebra. A linear map φ:A⟶B is said to be a δ - homomorphism map if ‖‖φ(a_1 a_2)-φ(a_1)φ(a_2)‖‖≤δ‖‖a_1‖‖ ‖‖a_2 ‖‖ for every a_1,a_2∈A. In this paper, we study the δ - homomorphism maps from A into B. Among other things, we prove that if φ:A⟶B is a δ - homomorphism map and B_* is multiplicative on the algebra generated by φ(A), then φ is bounded and ‖‖φ‖‖≤1+δ. Let A be a Banach algebra and (B,B_*) be a dual Banach algebra. A linear map φ:A⟶B is said to be a δ - homomorphism map if ‖‖φ(a_1 a_2)-φ(a_1)φ(a_2)‖‖≤δ‖‖a_1‖‖ ‖‖a_2 ‖‖ for every a_1,a_2∈A. In this paper, we study the δ - homomorphism maps from A into B. Among other things, we prove that if φ:A⟶B is a δ - homomorphism map and B_* is multiplicative on the algebra generated by φ(A), then φ is bounded and ‖‖φ‖‖≤1+δ.