Type Ⅱ₁ Von Neumann Algebras With Property Γ

In this dissertation, we give a definition of Property Γ for type Ⅱ1 von Neumann algebras as a generalization of Property Γ for type Ⅱ1 factors. We begin with type Ⅱ1 von Neumann algberas with separable predual and Property Γ. Without loss of generality, we may assume that a von Neumann algebra with separable predual is acting on a separable Hilbert space and thus we can apply the direct integral technique. The main result is that, if M is a type Ⅱ1 von Neumann algebra acting on a separable Hilbert space, then M has Property Γ if and only if almost every component in its direct integral decomposition (relative to its center) is a type Ⅱ1 factor with Property Γ. Then we look at countably decomposable type Ⅱ1 von Neumann algebras with Property Γ and we obtain that, if M is a countably decomposable type Ⅱ1 von Neumann algebra with Property Γ, then any finite subset of M is contained in a type Ⅱ1 von Neumann subalgebra with separable predual and Property Γ. Combining these results and the fact that every type Ⅱ1 von Neumann algebra is a direct sum of coutably decomposable type Ⅱ1 von Neumann algebras, we are able to give a nice characterization of general type Ⅱ1 von Neumann algebras with Property Γ in Theorem 2.1. We apply these results to obtain that, if M is a countably decomposable type Ⅱ1 von Neumann algbera with Property Γ, then the Hochschild cohomology Hk(M, M)= 0 for any k> 2. Another application arises in Kadison’s Similarity Problem. We obtain that, if M is a type Ⅱ1 von Neumann algebra with Property Γ, then the similarity degree d(M)=3. As a corollary, we get that if A is a unital, separable, nonnuclear Z-stable C*-algebra, then the similarity degree d(A)= 3.