Type II$_1$ von Neumann Algebras with Property $Gamma$

Wenhua Qian, University of New Hampshire, Durham

Abstract

In this dissertation, we give a definition of Property $\Gamma$ for type II$_1$ von Neumann algebras as a generalization of Property $\Gamma$ for type II$_1$ factors. We first look at a type II$_1$ von Neumann algebra $\mathcal{M}$ with separable predual and Property $\Gamma$. By applying the direct integral technique to $\mathcal{M}$, we give a description of $\mathcal{M}$ from the structures of its direct integral components. Then we prove that, if $\mathcal{M}$ is a countably decomposable type II$_1$ von Neumann algebra with Property $\Gamma$, then every finite subset of $\mathcal{M}$ is contained in a von Neumann subalgebra with separable predual and Property $\Gamma$. Combining with the fact that every II$_1$ von Neumann algebra is a direct sum of countably decomposable type II$_1$ von Neumann algebras, we give a clear characterization of general type II$_1$ von Neumann algebras with Property $\Gamma$. We apply these results to obtain that, if $\mathcal{M}$ is a countably decomposable type II$_1$ von Neumann algbera with Property $\Gamma$, then the Hochschild cohomology $H^k(\mathcal{M}, \mathcal{M}) =0$ for any $k \ge 2$. Another application arises in Kadison's Similarity Problem. We obtain that, if $\mathcal{M}$ is a type II$_1$ von Neumann algebra with Property $\Gamma$, then the similarity degree $d(\mathcal{M})=3$. As a corollary, we get that if $\mathcal{A}$ is a unital, separable, nonnuclear $\mathcal Z$-stable C$^*$-algebra, then the similarity degree $d(\mathcal{A}) = 3$.