On Left φ-biflat and Left φ-biprojectivity of θ-lau Product Algebras

𝑀onfared defined θ-Lau product structure 𝐴 ×θ 𝐵 for two 𝐵anach algebras 𝐴 and 𝐵, where θ : 𝐵 → 𝐶 is a multiplicative linear functional. In this paper, we study the notion of left φ-biflatness and left φ-biprojectivity for the θ Lau product structure 𝐴×θ 𝐵. For a locally compact group 𝐺, we show that 𝑀(𝐺) ×θ 𝑀(𝐺) is left character biflat (left character biprojective) if and only if 𝐺 is discrete and amenable (𝐺 is finite), respectively. 𝐴lso we prove that ℓ^1 (𝑁∨) ×θ ℓ^1 (𝑁∨) is neither (φ𝑁∨ , θ)-biprojective nor (0, φ𝑁∨ )-biprojective, where φ𝑁∨ is the augmentation character on ℓ^1 (𝑁∨). Finally, we give an ex ample among the Lau product structure of matrix algebras which is not left φ-biflat.