# Reference Request – Defining a multiplier on a Banach algebra

To let $$A$$ Be a Banach algebra. Some textbooks define a (left) multiplier as a map $$T: A rightarrow A$$ satisfying $$T (ab) = T (a) b$$ for all $$a, b in A$$ and assume $$A$$ must be a Banach algebra without an order, that is, if $$xA = (0)$$ for some $$x in A$$, the $$x = 0$$, (The right multiplier is defined in a similar way). However, some others define a (left) multiplier as a linear map on a (general) Banach algebra that satisfies the above condition. Am I missing something here? I know, the first definition does not assume $$T$$ is linear, but it follows $$A$$ is without order.