# Cameron, Jan Michael (2009-08). Normalizers of Finite von Neumann Algebras. Doctoral Dissertation. Thesis

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### abstract

• For an inclusion N \subseteq M of finite von Neumann algebras, we study the group of normalizers N_M(B) = {u: uBu^* = B} and the von Neumann algebra it generates. In the first part of the dissertation, we focus on the special case in which N \subseteq M is an inclusion of separable II_1 factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of N_M(B)'', this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of N_M(B)'' in terms of a unique countable subgroup of N_M(B). We then apply these general techniques to obtain results for inclusions B \subseteq M arising from the crossed product, group von Neumann algebra, and tensor product constructions. Our work also leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N \subseteq M is a regular inclusion of II_1 factors, then N norms M: These new results and techniques develop further the study of normalizers of subfactors of II_1 factors. The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II_1 factors. We obtain a characterization of masas in separable II_1 subfactors of nonseparable II_1 factors, with a view toward computing cohomology groups. We prove that for a type II_1 factor N with a Cartan masa, the Hochschild cohomology groups H^n(N,N)=0, for all n \geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.
• For an inclusion N subseteq M of finite von Neumann algebras, we study the group of normalizers

N_M(B) = {u: uBu^* = B}

and the von Neumann algebra it generates. In the first part of the dissertation, we focus on the special case in which N subseteq M is an inclusion of separable II_1 factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of N_M(B)'', this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of N_M(B)'' in terms of a unique countable subgroup of N_M(B). We then apply these general techniques to obtain results for inclusions B subseteq M arising from the crossed product, group von Neumann algebra, and tensor product constructions. Our work also leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N subseteq M is a regular inclusion

of II_1 factors, then N norms M: These new results and techniques develop further

the study of normalizers of subfactors of II_1 factors.

The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II_1 factors. We obtain a characterization of masas in separable II_1 subfactors of nonseparable II_1 factors, with a view toward computing cohomology groups. We prove that for a type II_1 factor N with a Cartan masa, the Hochschild cohomology groups H^n(N,N)=0, for all n geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.

• August 2009