^{}Department of Mathematics, University of Ayatollah Borujerdi, Borujerd, Iran

Abstract

Let $\mathcal{A}$ be a Banach algebra with BAI and $E$ be an introverted subspace of $\mathcal{A}^\prime$. In this paper we study the quotient Arens regularity of $\mathcal{A}$ with respect to $E$ and prove that the group algebra $L^1(G)$ for a locally compact group $G$, is quotient Arens regular with respect to certain introverted subspace $E$ of $L^\infty(G)$. Some related result are given as well.