巴拿赫代数的张量积的高维顺从性
The higher-dimensional amenability of tensor products of Banach algebras
摘要：
研究了巴拿赫代数$\A$和$\B$的张量积$\A \ptp \B$的高维相容性.证明了巴拿赫代数$\A$和$\B$的张量积$\A \ptp \B$的弱二维$db_w$满足\[ db_w \A \ptp \B = db_w \A + db_w \B. \]，并证明了它不能推广到任意巴拿赫代数。例如，对于一个双平坦Banach代数$\A$，如果它有一个左或右的，但不是双侧的有界逼近恒等式，我们有$db_w \A \ptp \A \le 1$和$db_w \A + db_w \A =2.$。我们明确地描述了具有有界逼近恒等式的巴拿赫代数$\A$和$\B$的某些张量积$\A \ptp \B$的连续Hochschild上同调$\H^n(\A \ptp \B, (X \ptp Y)^*)$和循环上同调$\H\C^n(\A \ptp \B)$;这里$(X \ptp Y)^*$是本质Banach双模$X$和$Y$分别在$\A$和$\B$上的张量积的对偶双模。
英文摘要：
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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