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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.

Abstract. In this paper, we investigate the higher simplicial cohomology groups of the convolution algebra ℓ 1 (S) for various semigroups S. The classes of semigroups considered are semilattices, Clifford semigroups, regular Rees semigroups and the additive semigroups of integers greater than a for some integer a. Our results are of two types: in some cases, we show that some cohomology groups are 0, while in some other cases, we show that some cohomology groups are Banach spaces.

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