Values of the Pukanszky Invariant in McDuff Factors

Details Values of the Pukanszky Invariant in McDuff Factors Values of the Pukanszky Invariant in McDuff Factors

2006-09-10

arxiv.org/abs/math/0609269v2

J. Funct. Anal. 254 (2008), no. 3, 612--631.

Mathematics

Operator Algebras

26 pages, minor typos corrected

Scientific paper

10.1016/j.jfa.2007.10.011

In 1960 Puk\'anszky introduced an invariant associating to every masa in a separable $\mathrm{II}_1$ factor a non-empty subset of $\mathbb N\cup\{\infty\}$. This invariant examines the multiplicity structure of the von Neumann algebra generated by the left-right action of the masa. In this paper it is shown that every non-empty subset of $\mathbb N\cup\{\infty\}$ arises as the Puk\'anszky invariant of some masa in a separable McDuff $\mathrm{II}_1$ factor which contains a masa with Puk\'anszky invariant $\{1\}$. In particular the hyperfinite $\mathrm{II}_1$ factor and all separable McDuff $\mathrm{II}_1$ factors with a Cartan masa satisfy this hypothesis. In a general separable McDuff factor we show that every subset of $\mathbb N\cup\{\infty\}$ containing $\infty$ is obtained as a Puk\'anskzy invariant of some masa.

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