JOHNSON PSEUDO-CONTRACTIBILITY AND PSEUDO-AMENABILITY OF θ-LAU PRODUCT

Given Banach algebras A and B and θ ∈ ∆(B). We shall study the Johnson pseudo-contractibility and pseudo-amenability of the θ-Lau product A×θB. We show that if A ×θ B is Johnson pseudo-contractible, then both A and B are Johnson pseudo-contractible and A has a bounded approximate identity. In some particular cases, a complete characterization of Johnson pseudo-contractibility of A ×θ B is given. Also, we show that pseudo-amenability of A ×θ B implies the approximate amenability of A and pseudo-amenability of B.

a Banach algebra A is called amenable if there is an element M ∈ (A ⊗ p A) * * such that a · M = M · a and π * * A (M )a = a for every a ∈ A, where π : A ⊗ p A → A is the product morphism and A ⊗ p A is the projective tensor product of A. Motivated by this construction of Johnson, some authors introduce several modifications of this notion by relaxing some conditions in different versions of definitions of amenability. The notion of pseudo-amenability was introduced by F. Ghahramani and Y. Zhang [13]. A Banach algebra A is called pseudo-amenable if there is a net (m α ) ⊆ A ⊗ p A such that a · m α − m α · a → 0 and π A (m α )a → a for every a ∈ A. The concept of approximately amenable Banach algebras was introduced by F. Ghahramani and R. J. Loy in [11], see also [12]. A Banach algebra A is called approximately amenable if there are nets (M α Recently the second and third authors [19] have defined a new concept related to amenability called Johnson pseudo-contractibility. Indeed, a Banach algebra A is called Johnson pseudo-contractible if there is a not necessarily bounded net (M α In the Section 2 we deal with Johnson pseudo-contractible Banach algebras. We show that if A × θ B is Johnson pseudo-contractible, then A is Johnson pseudocontractible and has a bounded approximate identity and B is Johnson pseudocontractible. Moreover, we show that in particular cases, for example when A is Arens regular and weakly sequentially complete or when A is a dual Banach algebra, Johnson pseudo-contractibility of A × θ B is equivalent with amenability of A and Johnson pseudo-contractibility of B. Some example are given at the end of the section.
In the Section 3 we focus on pseudo-amenability of A × θ B. Pseudo-amenability of A × θ B was studied by E. Ghaderi et al. [10]. They showed that pseudo-amenability of A × θ B implies pseudo-amenability of B, and implies pseudo-amenability of A whenever A has a bounded approximate identity. We show that the existence of bounded approximate identity in this result is not a necessary condition. Indeed, we show that if A × θ B is pseudo-amenable, then A is approximately amenable and B is pseudo-amenable.

Johnson Pseudo-Contractibility of A × θ B
We state a result from [2] that will be used frequently in this section. Proof. By hypothesis there is a net (M α ) ⊆ (A ⊗ p A) * * such that a · M α = M α · a and π * * A (M α )a − a → 0 for every a ∈ A. Let (e β ) be a bounded approximate identity for I and let E be a weak* cluster point of (e β ) in I * * . Then by setting (N α ) = (E · M α · E) ⊆ (I ⊗ p I) * * , we have x → x, for every x ∈ I. It follows that I is Johnson pseudo-contractible.

Theorem 2.2. Let
A and B be two Banach algebras and θ ∈ ∆(B). If A × θ B is Johnson pseudo-contractible, then the following statements hold.
(a) A is Johnson pseudo-contractible and has a bounded approximate identity.
, for every a ∈ A and b ∈ B. Then by Goldstine's theorem for every α there exists Note that θ has an extensionθ ∈ ∆(B * * ) given byθ(F ) = F (θ) for every F ∈ B * * . Since Φ and θ are bounded, Φ * * andθ are weak* continuous maps. Now we have Also, we have This shows that A has a bounded right approximate identity. A similar argument shows that A has a bounded left approximate identity. It follows that A has a bounded approximate identity. Since A is a two sided closed ideal of (A× θ B) and has a bounded approximate identity, by Lemma 2.1 it is Johnson pseudo-contractible.
It is well known that ( [19,Proposition 2.9] implies Johnson pseudocontractibility of B. We remark that the converse of the previous theorem does not hold in general. For example, A(H), the Fourier algebra on the integer Heisenberg group H, is Johnson pseudo-contractible and has a bounded approximate identity and M (H), the measure algebra over H, is Johnson pseudo-contractible (H is discrete and amenable). But It seems that Johnson pseudo-contractibility of A × θ B is related with amenability of A. We believe that Corollary 2.2 holds without the assumption that A has an identity. However, it remains as a conjecture. We left it as an open problem in the following questions.

Question 1. Does Johnson pseudo-contractibility of A × θ B implies the amenability of A?
Question 2. Suppose that A is an amenable Banach algebra and B is a Johnson pseudocontractible Banach algebra and θ ∈ ∆(B). Is A × θ B a Johnson pseudo-contractible Banach algebra?
We finish this section with some examples. First we recall some concepts and notations from semigroup theory. A semigroup S is called regular if for every s ∈ S there exists an element t ∈ S such that sts = s and tst = t. A semigroup S is an inverse semigroup if for every s ∈ S there exists a unique element t ∈ S such that sts = s and tst = t. The set of idempotents of a semigroup S is denoted by E(S), which is a partially ordered set with the following order  A linear subspace S 1 (G) of L 1 (G) is said to be a Segal algebra on G if it satisfies the following conditions: and y ∈ G) and the map y , for every f ∈ S 1 (G) and y ∈ G. Example 2.3. Suppose that B is a Banach algebra and θ ∈ ∆(B). Let S 1 (G) be a Segal algebra on G. If S 1 (G) × θ B is Johnson pseudo-contractible, then S 1 (G) = L 1 (G).

Pseudo-Amenability of
for every x ∈ A, y ∈ B. Particularly for every x ∈ A we have One can easily see that For an arbitrary element b in B, we have Similarly we have From (3.2), (3.3) and (3.1), by using Remark 3.1 we obtain for every x ∈ A. It follows that A is approximately amenable.
Example 3.1. Let S be a uniformly locally finite inverse semigroup and let B be a Banach algebra and θ ∈ ∆(B). If 1 (S)× θ B is pseudo-amenable, then by Theorem 3.1 1 (S) is approximately amenable. Theorem 4.3 of [17] shows that 1 (S) is amenable.
Example 3.3. Let G be an infinite abelian compact group and let B be a Banach algebra and θ ∈ ∆(B). We claim that L 2 (G) × θ B is not pseudo-amenable. To see this, suppose that L 2 (G) × θ B is pseudo-amenable. Then Theorem 3.1 implies that L 2 (G) is approximately amenable. But by the Plancherel theorem L 2 (G) is isometrically isomorphism to 2 (Ĝ), whereĜ is the dual group of G and 2 (Ĝ) is equipped with the pointwise product. So 2 (Ĝ) is approximately amenable which is a contradiction with the main result of [6].