Cohomology for operator algebras: The Mayer-Vietoris sequence (Q1365223)

The authors consider bounded (standard) and completely bounded Hochschild cohomologies of operator algebras. For any couple \(\mathcal{A} \subset \mathcal{B}\) of operator algebras they introduce the so-called \(\mathcal{B}\)-nill cohomology \(H^n(\mathcal{A}|\mathcal{B})\) and derive the long exact sequence \[ \to H^1(\mathcal{A}|\mathcal{B}) \to \ldots \to H^n(\mathcal{A}) \to H^n(\mathcal{B}) \to H^{n+1}(\mathcal{A}|\mathcal{B}) \to \ldots \] Also they obtain a kind of Mayer-Vietoris sequence. The first term of this sequence is \(H^1\). By this reason the authors discuss in details \(H^0\) and \(H^1\) for tensor products. At the end of the paper several applications are considered.