http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Normalizers of finite von Neumann algebras
For an inclusion N [subset of or equal to] 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 [subset of or equal to] M is an inclusion of separable II₁ factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. An analyzing the bimodule structure of certain subalgebras of N_M(B)", then yieds 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 [subset of or equal to] 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 [subset of or equal to] M is a regular inclusion of II₁ factors, then N norms M: These new results and techniques develop further the study of normalizers of subfactors of II₁ factors. The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II₁ factors. We obtain a characterization of masas in separable II₁ subfactors of nonseparable II₁ factors, with a view toward computing cohomology groups. We prove that for a type II₁ factor N with a Cartan masa, the Hochschild cohomology groups H^n(N,N)=0, for all n [greater than or equal to] 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.The techniques and results in this part of the thesis represent new progress on the Hochschild cohomology problem for von Neumann algebras.
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Doctor of Philosophy Texas A&M University 2009 http://hdl.handle.net/1969.1/ETD-TAMU-2009-08-7141