Values of the Pukanszky Invariant in McDuff Factors

Abstract

In 1960 Pukánszky 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 Pukánszky invariant of some masa in a separable McDuff ${\mathrm{II}}_1$ factor which contains a masa with Pukánszky 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 Pukánskzy invariant of some masa.

26 pages, minor typos corrected

References in corpus (1)

• On a class of ${\mathrm{II}}_1$ factors with at most one Cartan subalgebra