STUDIA MATHEMATICA 181 (3) (2007) Arens regularity of module actions by M. Eshaghi Gordji (Semnan and Tehran) and M. Filali (Oulu) Abstract. We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if A has a brai (blai), then the right (left) module action of A on A * is Arens regular if and only if A is reflexive. We find that Arens regularity is implied by the factorization of A * or A ** when A is a left or a right ideal in A ** . The Arens regularity and strong irregularity of A are related to those of the module actions of A on the nth dual A (n) of A. Banach algebras A for which Z(A ** )= A but A  Z t (A ** ) are found (here Z(A ** ) and Z t (A ** ) are the topo- logical centres of A ** with respect to the first and second Arens product, respectively). This also gives examples of Banach algebras such that A  Z(A ** )  A ** . Finally, the triangular Banach algebras T are used to find Banach algebras having the following prop- erties: (i) T * T = TT * but Z(T ** ) = Z t (T ** ); (ii) Z(T ** )= Z t (T ** ) and T * T = T * but TT * = T * ; (iii) Z(T ** )= T but T is not weakly sequentially complete. The results (ii) and (iii) are new examples answering questions asked by Lau and ¨ Ulger. 1. Introduction. The extension of bilinear maps on normed spaces and the concept of regularity of bilinear maps were introduced by Richard Arens in 1951 (see [1] and [2]). We start by recalling these definitions. Throughout the paper, we shall identify any Banach space with its natural image in the second dual. Let X , Y and Z be normed spaces and let f : X × Y → Z be a continuous bilinear map. Then the adjoint of f is defined by f ∗ : Z ∗ × X → Y ∗ , 〈f ∗ (z ′ ,x),y〉 = 〈z ′ ,f (x, y)〉 (z ′ ∈ Z ∗ ,x ∈ X, y ∈ Y ). Clearly, for each x ∈ X , the map z ′ → f ∗ (z ′ ,x): Z ∗ → Y ∗ is weak ∗ - weak ∗ continuous. Since f ∗ is a continuous bilinear map, this process may be repeated to define f ∗∗ =(f ∗ ) ∗ : Y ∗∗ × Z ∗ → X ∗ , and then f ∗∗∗ =(f ∗∗ ) ∗ : X ∗∗ × Y ∗∗ → Z ∗∗ . The map f ∗∗∗ is the unique extension of f such that • x ′′ → f ∗∗∗ (x ′′ ,y ′′ ): X ∗∗ → Z ∗∗ is weak ∗ -weak ∗ continuous for each y ′′ ∈ Y ∗∗ , 2000 Mathematics Subject Classification : Primary 46H25; Secondary 46H20, 43A20. Key words and phrases : Arens product, topological centres, strongly Arens irregular, module actions. [237] c  Instytut Matematyczny PAN, 2007