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.

Keywords: Hochschild cohomology, cyclic cohomology, $C^*$-algebra, semigroup algebra

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