The Higher-dimensional Amenability of Tensor Products of Banach Algebras

We investigate the higher-dimensional amenability of tensor products A ⊗̂ B of Banach algebras A and B. We prove that the weak bidimension dbw of the tensor product A⊗̂B of Banach algebras A and B with bounded approximate identities satisfies dbwA ⊗̂ B = dbwA+ dbwB. We show that it cannot be extended to arbitrary Banach algebras. For example, for a biflat Banach algebra A which has a left or right, but not two-sided, bounded approximate identity, we have dbwA ⊗̂A ≤ 1 and dbwA+ dbwA = 2. We describe explicitly the continuous Hochschild cohomology H(A ⊗̂ B, (X ⊗̂ Y )∗) and the cyclic cohomologyHC(A⊗̂B) of certain tensor products A⊗̂B of Banach algebras A and B with bounded approximate identities; here (X ⊗̂Y )∗ is the dual bimodule of the tensor product of essential Banach bimodules X and Y over A and B respectively. 2000 Mathematics Subject Classification: Primary 55U25, 19D55, 43A20.