The Pukánszky invariant for masas in group von Neumann factors

Abstract

The Pukanszky invariant associates to each maximal abelian self-adjoint subalgebra (masa) $A$ in a type $\operatorname{II}_1$ factor $M$ a certain subset ot $\mathbb N\cup\{\infty\}$, denoted by $\operatorname{Puk}(A)$. We study this invariant in the context of factors generated by infinite conjugacy class discrete countable groups $G$ with masas arising from abelian subgroups $H$. Our main result is that we are able to describe $\operatorname{Puk}(VN(H))$ in terms of the algebraic structure of $H\subseteq G$, specifically by examining the double cosets of $H$ in $G$. We illustrate our characterization by generating many new values for the invariant, mainly for masas in the hyperfinite type $\operatorname{II}_1$ factor $R$.