Let $\cal A$ be a Banach algebra. Then $\cal A^{ **}$ the second dual of $\cal A$ is a Banach algebra with first (second) Arens product. We study the Arens products of $\cal A^{ 4}(=({ \cal A^{ **}})^{ **}).$ We found some conditions on $\cal A^{ **}$ to be a left ideal in $\cal A^{ 4}.$ We found the biggest two sided ideal $I$ of ${ \cal A},$ in which $I$ is a left (right) ideal of ${ \cal A}^{ **}$.
Categories: math.FA