Journal of Operator Theory

Volume 67, Issue 1, Winter 2012 pp. 73-100.

The higher-dimensional amenability of tensor products of Banach algebras

Authors: Zinaida A. Lykova
Author institution: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, U.K.

Summary: We investigate the higher-dimensional amenability of tensor\mathcal{B}reak products

$\mathcal{A} \widehat{\otimes} \mathcal{B}$ of Banach algebras

$\mathcal{A}$ and

$\mathcal{B}$ . We prove that the weak bidimension

$db_{\mathrm w}$ of the tensor product

$\mathcal{A} \widehat{\otimes} \mathcal{B}$ of Banach algebras

$\mathcal{A}$ and

$\mathcal{B}$ with bounded approximate identities satisfies

$db_{\mathrm w} \mathcal{A} \widehat{\otimes} \mathcal{B} = db_{\mathrm w} \mathcal{A} + db_{\mathrm w} \mathcal{B}.$ We show that it cannot be extended to arbitrary Banach algebras. For example, for a biflat Banach algebra

$\mathcal{A}$ which has a left or right, but not two-sided, bounded approximate identity, we have

$db_{\mathrm w} \mathcal{A} \widehat{\otimes} \mathcal{A} \leqslant 1$ and

$db_{\mathrm w} \mathcal{A} + db_{\mathrm w} \mathcal{A} =2.$ We describe explicitly the continuous Hochschild cohomology

$\H^n(\mathcal{A} \widehat{\otimes} \mathcal{B}, (X \widehat{\otimes} Y)^*)$ and the cyclic cohomology

$\H\C^n(\mathcal{A} \widehat{\otimes} \mathcal{B})$ of certain tensor products

$\mathcal{A} \widehat{\otimes} \mathcal{B}$ of Banach algebras

$\mathcal{A}$ and

$\mathcal{B}$ with bounded approximate identities; here

$(X \widehat{\otimes} Y)^*$ is the dual bimodule of the tensor product of essential Banach bimodules

$X$ and

$Y$ over

$\mathcal{A}$ and

$\mathcal{B}$ respectively.

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