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.