University of MaraghehSahand Communications in Mathematical Analysis2322-580712120181101A Certain Class of Character Module Homomorphisms on Normed Algebras1131203119910.22130/scma.2018.78500.364ENAli Reza KhoddamiFaculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran.Journal Article20180101For 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.http://scma.maragheh.ac.ir/article_31199_6d259fdf33f0dff36ef61b8685f930c1.pdf