A Certain Class of Character Module Homomorphisms on Normed Algebras

For two normed algebras $A$ and $B$ with the character space   $bigtriangleup(B)neq emptyset$  and a left $B-$module $X,$  a certain class of bounded linear maps from $A$ into $X$ is introduced. We set $CMH_B(A, X)$  as the set of all non-zero $B-$character module homomorphisms from $A$ into $X$. In the case where $bigtriangleup(B)=lbrace varphirbrace$ then $CMH_B(A, X)bigcup lbrace 0rbrace$ is a closed subspace of $L(A, X)$  of all bounded linear operators from $A$ into $X$.   We  define an  equivalence  relation on  $CMH_B(A, X)$ and use it  to show that  $CMH_B(A, X)bigcuplbrace 0rbrace $ is  a union of closed subspaces of $L(A, X)$.  Also some basic results and some hereditary properties are presented. Finally some relations between $varphi-$amenable Banach algebras and character module homomorphisms are examined.