NON-REGULARITY OF MULTIPLICATIONS FOR GENERAL MEASURE ALGEBRAS

Let M(X) be the space of all finite regular Borel measures on X. A general measure algebra is a subspace of M(X), which is an L-space and has a multiplication preserving the probability measures. Let L subseteq M(X) be a general measure algebra on a locally compact space X. In this paper, we investigate the relation between Arens regularity of L and the topology of X. We find conditions under which the Arens regularity of L 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 [7].