Range‐inclusive maps on C‐algebras

Let L be a linear ^*‐subspace in a C^*‐algebra A and let T : L → A be a linear home map: T maps any x = x^* ∈ L into the C^*‐subalgebra generated by x and 1. It is proved that if T(L) ⊆ L, or if L is rich, then there is λ ∈ C such that Tx – λx ∈ C1 for x ∈ L. This is used for the study of range‐inclusive linear maps T from L into A—the maps such that, for any x = x^* ∈ L and y ∈ A, there is z ∈ A such that [Tx, y] = [x, z]. It is shown that if T(L) ⊆ L then there is a function λ: Prim(A) → C such that Tx – λ(I)x ∈ C1 + I for x ∈ L and I ∈ Prim(A). If A is primitive, there is λ ∈ C such that Tx – λx ∈ C1. For finite‐dimensional L, there is a in the centre Z(A) of A and a linear map ξ : L → Z(A) such that Tx = ax + ξ(x), x ∈ L.