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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.

Abstract. Let A be a Banach algebra. We introduce the notions of approximate left φ-biprojective and approximate left character biprojective Banach algebras, where φ is a non-zero multiplicative linear functional on A. We show that for a SIN group G, the Segal algebra S(G) is approximate left φ 1 -biprojective if and only if G is amenable, where φ 1 is the augmentation character on S(G). Also we show that the measure algebra M (G) is approximate left character biprojective if and only if G is discrete and amenable. For a Clifford semigroup S, we show that ℓ 1 (S) is approximate left character biprojective if and only if ℓ 1 (S) is pseudo-amenable. We study the hereditary property of these notions. Finally we give some examples to show the differences of these notions and the classical ones.

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