On the Arens Homomorphism

Let \(E\) be a unital \(f\)-module over an \(f\)-algebra \(A\). With the help of Arens extension theory, a \((A^{\sim})_{n}^{\sim}\) module structure on \(E^{\sim}\) can be defined. The paper deals mainly with properties of the Arens homomorphism \(\eta\colon(A^{\sim})_{n}^{\sim}\to \operatorname {Orth}(E^{\sim})\), which is defined by the \((A^{\sim})_{n}^{\sim}\) module structure on \(E^{\sim}\). Necessary and sufficient conditions for an \(A\) submodule of \(E\) to be an order ideal are obtained.