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Classification of a Family of Free Bogoliubov Actions of R on L(F∞) up to Cocycle Conjugacy

  • Author(s): Keneda, Joshua Tyler
  • Advisor(s): Shlyakhtenko, Dimitri
  • et al.
Abstract

Much of the research into operator algebras concerns the classification problem for von Neumann algebras, where one hopes to find useful invariants to categorize von Neumann algebras up to isomorphism. Murray and von Neumann initiated the study of these objects and reduced the classification problem to that of classifying the so-called factors. And while Type I factors are fully understood, the Type II and Type III cases are still active areas of research. Relatedly, one can attempt to classify group actions on von Neumann algebras up to a suitable notion of equivalence, for example unitary or cocycle conjugacy. The aim of this dissertation is to classify a family of free Bogoliubov actions of R on the Type II1 free group factor L(F∞) up to cocycle conjugacy. We consider a certain collection C of measure classes on R, corresponding to the spectral measure classes of the infinitesimal generators for orthogonal representations α:R→ O(H), with H separable and infinite-dimensional. Such representations give rise to a free Bogoliubov action σα of R on L(F∞) and an associated von Neumann algebra: the crossed product L(F∞)oσαR. Note that the cocycle conjugacy of two actions gives an isomorphism between their crossed product von Neumann algebras. We show that our family of free Bogoliubov actions are completely classified up to cocycle conjugacy by their associated spectral measure class [α]∈C.Our main technical tool for this result mirrors a recent result of Houdayer, Shlyakhtenko,and Vaes and relates the equality of the spectral measure classes [α],[β] to the embeddability

(in the sense of Popa’s intertwining-by-bimodules) of the group algebra Lα(R) into Lβ(R)inside of their shared crossed product M'L(F∞)oσαR'L(F∞)oσβR. By restricting to representationsαofRwhich act trivially on a large subspace of H,we force Lα(R) to have large (non-amenable) relative commutant in L(F∞)oσαR. This allows us to use solidity arguments and a rigidity result of Houdayer and Ueda to “trap” Lα(R) inside ofLβ(R) within the crossed product (i.e.Lα(R)MLβ(R)). Applying the technical tool of the previous paragraph, we obtain that [α] = [β], which establishes the desired cocycle conjugacy invariant.

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