Values of the Pukanszky invariant in free group factors and the hyperfinite factor
Abstract
Let A be a maximal abelian self-adjoint subalgebra (masa) in a type II_1 factor M acting via standard representation on L^2(M). The abelian von Neumann algebra A generated by A and JAJ has a type I commutant which contains the projection e_A from L^2(M) onto L^2(A). Then A'(1-e_A) decomposes into a direct sum of type I_n algebras for n in {1,2,...,\infty}, and those n's which occur in the direct sum form a set called the Pukanszky invariant, Puk(A), also denoted Puk_M(A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S\cup {\infty} when M is both a free group factor and the hyperfinite factor, and where S is an arbitrary subset of the natural numbers. The only previously known values for masas in free group factors were {\infty} and {1,\infty}, and some values of the form S\cup {\infty} are new also for the hyperfinite factor. We also consider a more refined invariant (that we will call the measure-multiplicity invariant), which was considered recently by Neshveyev and Stormer and has been known to experts for a long time. We use the measure-multiplicity invariant to distinguish two masas in a free group factor, both having Pukanszky invariant {n,\infty}, for an arbitrary natural number n.
Keywords
Cite
@article{arxiv.math/0512405,
title = {Values of the Pukanszky invariant in free group factors and the hyperfinite factor},
author = {Ken Dykema and Allan Sinclair and Roger Smith},
journal= {arXiv preprint arXiv:math/0512405},
year = {2007}
}