a certain class of character module homomorphisms on normed algebras

For two normed algebras a and b with the character space △(b)≠∅ and a left b−module x, a certain class of bounded linear maps from a into x is introduced. we set cmhb(a,x) as the set of all non-zero b−character module homomorphisms from a into x. in the case where △(b)={φ} then cmhb(a,x)⋃{0} is a closed subspace of l(a,x) of all bounded linear operators from a into x. we define an equivalence relation on cmhb(a,x) and use it to show that cmhb(a,x)⋃{0} is a union of closed subspaces of l(a,x). also some basic results and some hereditary properties are presented. finally some relations between φ−amenable banach algebras and character module homomorphisms are examined.