In this paper we prove that for a commutative character amenable Banach algebra A, if T : A → A is a multiplier then T has closed range if and only if T = BP = PB, where B ∈ M(A) is invertible and p ∈ M(A) is idempotent. By this result we characterize each multiplier with closed range on such Banach algebra (proposition 3.7), and so we get a necessary condition for character amenability of alge...