Values of the Pukanszky invariant in free group factors and the hyperfinite factor
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Let A M B (L2 (M)) be a maximal abelian self-adjoint subalgebra (masa) in a type II1 factor M in its standard representation. The abelian von Neumann algebra A generated by A and JAJ has a type I commutant which contains the projection eA A onto L2 (A). Then A (1 - eA) decomposes into a direct sum of type In algebras for n {1, 2, ..., }, and those n's which occur in the direct sum form a set called the Puknszky invariant, Puk (A), also denoted PukM (A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S {} when M is both a free group factor and the hyperfinite factor, and where S is an arbitrary subset of N. The only previously known values for masas in free group factors were {} and {1, }, and some values of the form S {} 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 Strmer 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 Puknszky invariant {n, }, for arbitrary n N. 2006 Elsevier Inc. All rights reserved.
