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×θ B. 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.

In this paper, we study the notion of Johnson pseudo-contractibility for certain Banach algebras. For a bicyclic semigroup S, we show that ℓ 1 (S) is not Johnson pseudo-contractible. Also for a Johnson pseudo-contractible Banach algebra A, we show that A has no non-zero complemented closed nilpotent ideal.

We investigate Johnson pseudo-contractibility and pseudo-contractibility of Clifford semigroup algebras. We show that, for a Clifford semigroup S, if 1 (S) has a central approximate identity in c 00 (S), then 1 (S) is (Johnson) pseudo-contractible if and only if E(S) is locally finite and each maximal subgroup of S is (amenable) finite, respectively. As an application, we characterize Johnson pseudo-contractibility and pseudo-contractibility of 1 (S), where S is a commutative semigroup, a band semigroup, or an inverse semigroup with totally ordered idempotents set.

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