A semilattice structure for Arens products

Abstract

Let (A,⋅) be a Banach algebra. Then for n∈ℕ, A (2n) has 2 n Arens products. In this paper we study the relations between the Arens products on A (2n). Moreover, if P n (A) denotes the set of all Arens products on A (2n), for n∈ℕ, we show that $P(A)=\bigcup_{n=1}^{\infty} P_{n}(A)$ is a ∧-semilattice. Also, we study P(A) as an infinite commutative semigroup and P(A)−{⋅} as a free semigroup generated by two elements. Then we investigate amenability and weak amenability for their semigroup Banach algebras.