Values of the Pukánszky invariant in free group factors and the hyperfinite factor Academic Article uri icon

abstract

  • 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.

author list (cited authors)

  • Dykema, K. J., Sinclair, A. M., & Smith, R. R.

citation count

  • 9

complete list of authors

  • Dykema, Kenneth J||Sinclair, Allan M||Smith, Roger R

publication date

  • November 2006