The fourth duals of Banach algebras

Abstract

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

7 pages. To appear in Italian J. of Pure and Appl. Math