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.