THE HIGHER-DIMENSIONAL AMENABILITY OF TENSOR PRODUCTS OF BANACH ALGEBRAS

We investigate the higher-dimensional amenability of tensor products A (circle times) over cap B of Banach algebras A and B. We prove that the weak bidimension db(w) of the tensor product A (circle times) over cap B of Banach algebras A and B with bounded approximate identities satisfies db(w)A (circle times) over cap B = db(w)A + db(w)B. 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 db(w)A (circle times) over cap A <= 1 and db(w)A + db(w)A = 2. We describe explicitly the continuous Hochschild cohomology H-n(A (circle times) over cap B, (X (circle times) over cap Y)*) and the cyclic cohomology HCn (A (circle times) over cap B) of certain tensor products A (circle times) over cap B of Banach algebras A and B with bounded approximate identities; here (X (circle times) over cap Y)* is the dual bimodule of the tensor product of essential Banach bimodules X and Y over A and B respectively.