Masas and Bimodule Decompositions of II_1 Factors

Abstract

The measure-multiplicity-invariant for masas in II_1 factors was introduced by Dykema, Smith and Sinclair to distinguish masas that have the same Pukanszky invariant. In this dissertation, the measure class (left-right-measure) in the measuremultiplicity- invariant is studied, which equivalent to studying the structure of the standard Hilbert space as an associated bimodule. The focal point of this analysis is: To what extent the associated bimodule remembers properties of the masa. The structure of normaliser of any masa is characterized depending on this measure class, by using Baire category methods (Selection principle of Jankov and von Neumann). Measure theoretic proofs of Chifan's normaliser formula and the equivalence of weak asymptotic homomorphism property (WAHP) and singularity is presented. Stronger notions of singularity is also investigated. Analytical conditions based on Fourier coefficients of certain measures are discussed, that partially characterize strongly mixing masas and masas with nontrivial centralizing sequences. The analysis also provide conditions in terms of operators and L2 vectors that characterize masas whose left-right-measure belongs to the class of product measure. An example of a simple masa in the hyperfinite II1 factor whose left-right-measure is the class of product measure is exhibited. An example of a masa in the hyperfinite II1 factor whose leftright- measure is singular to the product measure is also presented. Unitary conjugacy of masas is studied by providing examples of non unitary conjugate masas. Finally, it is shown that for k greater than/equal to 2 and for each subset S \subseteq N, there exist uncountably many non conjugate singular masas in L(Fk) whose Pukanszky invariant is S u {1}.