# Journal of Operator Theory

Volume 44, Issue 2, Fall 2000 pp. 335-345.

Cohomology for finite index inclusions of factors**Authors**: Allan M. Sinclair (1), and Roger R. Smith (2)

**Author institution:**(1) Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland

(2) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

**Summary:**If $\caln \subseteq \calm$ is an inclusion of type ${\rm II}_1$ factors of finite index on a separable Hilbert space, and if $\caln$ has a Cartan subalgebra then we show that $H^n(\caln, \calm) = 0$ for $n\ge 1$. We also show that $H^n_{\rm cb}(\caln, \calm)= 0$, $n\ge 1$, for an arbitrary finite index inclusion $\caln \subseteq \calm$ of von~Neumann algebras.

**Keywords:**von Neumann algebra, factor, Jones index, cohomology, Cartan subagebra , completely bounded, $C^*$-algebra

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