Non-regularity of multiplications for general measure algebras

Let $fM(X)$ be the  space of  all finite regular Borel measures on $X$. A general measure algebra is a subspace  of$fM(X)$,which is an $L$-space and has a multiplication preserving the probability measures. Let $cLsubseteqfM(X)$ be a general measure algebra on a locallycompact space $X$. In this paper, we investigate the relation between Arensregularity of $cL$ and the topology of $X$. We  find conditionsunder which the Arens regularity of $fL$ implies the compactness of $X$.Weshow that these conditions are necessary.We also  present some examples in showing that the new conditions aredifferent from  Theorem 3.1 of cite{7}.