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 locally compact space $X$. In this paper, we investigate the relation between Arens regularity of $cL$ and the topology of $X$. We find conditions under which the Arens regularity of $fL$ implies the compactness of $X$. We show that these conditions are necessary. We also present some examples in showing that the new conditions are different from Theorem 3.1 of cite{7}.