The higher-dimensional amenability of tensor products of Banach algebras

TL;DR

This paper studies the higher-dimensional amenability of tensor products of Banach algebras, establishing a formula for their weak bidimension when they have bounded approximate identities and exploring their Hochschild and cyclic cohomology.

Contribution

It proves a formula for the weak bidimension of tensor products of Banach algebras with bounded approximate identities and describes their Hochschild and cyclic cohomology explicitly.

Findings

Weak bidimension of tensor products equals the sum of individual bidimensions.

The bidimension formula does not extend to all Banach algebras.

Explicit descriptions of Hochschild and cyclic cohomology for certain tensor products.

Abstract

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.