For two Banach algebras A and B, an interesting product A × θ B, called the θ-Lau product, was recently introduced and studied for some nonzero characters θ on B. Here, we characterize some notions of amenability as approximate amenability, essential amenability, n-weak amenability and cyclic amenability between A and B and their θ-Lau product. Introduction. LetA and B be two Banach algebras and θ ∈ σ(B), the spectrum of B of all nonzero characters on B. Then the θ-Lau product of A and B, denoted by A × θ B, is defined as the space A × B equipped with the multiplication (a, b)(a , b ) = (aa + θ(b)a + θ(b )a, bb ), and the norm (a, b) = a + b , for all a, a ∈ A and b, b ∈ B. The θ-Lau product A × θ B is a Banach algebra.This product was first introduced by Lau [L1] for Lau algebras; recall that a Lau algebra is a Banach algebra which is the predual of a von Neumann algebra for which the identity of the dual is a multiplicative linear functional. The study of this large class of Banach algebras originated with a paper published in 1983 by Lau [L1] in which he referred to them as "F-algebras"; see also Lau [L2]. Later on, in his useful monograph Pier [Pi] introduced the name "Lau algebra". Examples of Lau algebras include the group algebra and the measure algebra of a locally compact group or hypergroup (see Lau [L1]), and also the Fourier algebra and

Let A and B be two Banach algebras and let θ be a nonzero character on B. In this paper, we deal with the θ-Lau product A × θ B that was first introduced by A. T. Lau for certain Banach algebras known as Lau algebras and recently by M. S. Monfared for all Banach algebras; we study pseudo-amenability, pseudo-contractibility and character pseudo-amenability of A × θ B and their relations with A and B.

Let A and U be Banach algebras and θ be a nonzero character on A. Then the Lau product Banach algebra A × θ U associated with the Banach algebras A and U is the l 1 -direct sum A ⊕ U equipped with the algebra multiplication (a, u)(a ′ , u ′ ) = (ab, θ(a)u ′ + θ(a ′ )u + uu ′ ) (a, a ′ ∈ A, u, u ′ ∈ U ) and l 1 -norm. In this paper we shall investigate the derivations and multipliers from this Banach algebras and study the automatic continuity of these mappings. We also study continuity of the derivations for some special cases of Banach algebra U and Banach A × θ U -bimodule X and establish various results on the continuity of derivations and give some examples.

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