The higher-dimensional amenability of tensor products of Banach algebras

Details The higher-dimensional amenability of tensor products of Banach algebras The higher-dimensional amenability of tensor products of Banach algebras

2009-04-29

arxiv.org/abs/0904.4548v1

Mathematics

K-Theory and Homology

Scientific paper

We investigate the higher-dimensional amenability of tensor products $\A \ptp \B$ of Banach algebras $\A$ and $\B$. We prove that the weak bidimension $db_w$ of the tensor product $\A \ptp \B$ of Banach algebras $\A$ and $\B$ with bounded approximate identities satisfies \[ db_w \A \ptp \B = db_w \A + db_w \B. \] 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 $db_w \A \ptp \A \le 1$ and $db_w \A + db_w \A =2.$ We describe explicitly the continuous Hochschild cohomology $\H^n(\A \ptp \B, (X \ptp Y)^*)$ and the cyclic cohomology $\H\C^n(\A \ptp \B)$ of certain tensor products $\A \ptp \B$ of Banach algebras $\A$ and $\B$ with bounded approximate identities; here $(X \ptp Y)^*$ is the dual bimodule of the tensor product of essential Banach bimodules $X$ and $Y$ over $\A$ and $\B$ respectively.

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