CHARACTERIZATIONS OF (JORDAN) DERIVATIONS ON BANACH ALGEBRAS WITH LOCAL ACTIONS

Let A be a unital Banach *-algebra and M be a unital *-A-bimodule. If W is a left separating point of M, we show that every *-derivable mapping at W is a Jordan derivation, and every *-left derivable mapping at W is a Jordan left derivation under the condition WA = AW. Moreover we give a complete description of linear mappings & delta; and & tau; from A into M satisfying & delta;(A)B* +A & tau; (B)* = 0 for any A, B & ISIN; A with AB* = 0 or & delta;(A) o B* + A o & tau; (B)* = 0 for any A, B & ISIN; A with A o B* = 0, where A o B = AB + BA is the Jordan product.