巴拿赫代数的相对上同调
Relative cohomology of Banach algebras
摘要：
设$A$是一个Banach代数，不一定是有单位的，且$B$是$A$的闭子代数。建立了$A$的Banach循环上同调群$ {\cal{HC}}^n(A)$与$A$的Banach $B$ -相对循环上同调群$ {\cal{HC}}^n_B(A) $之间的联系.本文证明了：对于具有有界近似恒等式的Banach代数$A$和$A$的顺从闭子代数$B$，在拓扑同构下，对所有$n \ge 0$都有${\cal{HC}}^n(A) = {\cal{HC}}^n_B(A) $.我们还通过一个顺从的闭双理想$I$建立了$A$的Banach单上同调群或循环上同调群与商代数$A/I$的Banach单上同调群或循环上同调群之间的联系。结果适用于计算这些群体的某些运营商代数，包括冯诺依曼代数。
英文摘要：
Let $A$ be a Banach algebra, not necessarily unital, and let $B$ be a closed subalgebra of $A$. We establish a connection between the Banach cyclic cohomology group $ {\cal{HC}}^n(A)$ of $A$ and the Banach $B$-relative cyclic cohomology group $ {\cal{HC}}^n_B(A) $ of $A$. We prove that, for a Banach algebra $A$ with a bounded approximate identity and an amenable closed subalgebra $B$ of $A$, up to topological isomorphism, ${\cal{HC}}^n(A) = {\cal{HC}}^n_B(A) $ for all $n \ge 0$. We also establish a connection between the Banach simplicial or cyclic cohomology groups of $A$ and those of the quotient algebra $A/I$ by an amenable closed bi-ideal $I$. The results are applied to the calculation of these groups for certain operator algebras, including von Neumann algebras.
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