LAUSR

The purpose of this article is to study the concepts Arens regularity for semigroup algebras $$M_a(S)$$Ma(S), $$\ell ^1(S)$$ℓ1(S), and $${ LUC}(S)^*$$LUC(S)∗. We give a necessary and sufficient condition for $$M_a(S)$$Ma(S) to be pointwise Arens regular. We then give some characterizations for Arens regularity of $$M_a(S)$$Ma(S) and $$\ell ^1(S)$$ℓ1(S). Also, we investigate weakly compact multipliers on $$M_a(S)$$Ma(S) and then for a compactly cancellative foundation semigroup S with identity, we show that S is compact if $${ LUC}(S)^*$$LUC(S)∗ is Arens regular.