Values of the Pukánszky invariant in free group factors and the hyperfinite factor

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 Pukánszky 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 Størmer 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 Pukánszky invariant {n,∞}, for arbitrary n ∈ N. © 2006 Elsevier Inc. All rights reserved.