2005/12/16 by Ken Dykema, Dykema, Ken, Allan Sinclair +5
Mathematics · #46L10 (46L09) #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.OA #msc:46L10
paper · pdf · doi:10.48550/arxiv.math/0512405
arxiv created 2005/12/16 · openalex publication_date 2005/12/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a maximal abelian self-adjoint subalgebra (masa) in a type II1 factor M acting via standard representation on L2(M). The abelian von Neumann algebra A generated by A and JAJ has a type I commutant which contains the projection eA from L2(M) onto L2(A). Then A'(1-eA) decomposes into a direct sum of type In algebras for n in 1,2,...,∞, and those n's which occur in the direct sum form a set called the Pukanszky 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 the natural numbers. 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 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,∞, for an arbitrary natural number n.