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In this paper, we study an approximate biflatness of l 1 S , where S is a Clifford semigroup. Indeed, we show that a Clifford semigroup algebra l 1 S is approximately biflat if and only if every maximal subgroup of S is amenable, E S is locally finite, and l 1 S has an approximate identity in c 00 S . Moreover, we prove that l 1 S is approximately biflat if and only if each maximal subgroup of S is amenable for an inverse semigroup S such that E S , the set of its idempotent elements, is totally ordered and locally finite.

Let T be a homomorphism from a Banach algebra B to a Banach algebra A. The Cartesian product space A × B with T -Lau multiplication and ℓ 1 -norm becomes a new Banach algebra A × T B.We investigate the notions such as approximate amenability, pseudo amenability, φ-pseudo amenability, φ-biflatness and φ-biprojectivity for Banach algebra A × T B. We also present an example to show that approximate amenability of A and B is not stable for A × T B. Finally we characterize the double centralizer algebra of A × T B and present an application of this characterization.

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