Murray-von Neumann dimension for strictly semifinite weights

Abstract

Given a von Neumann algebra $M$ equipped with a faithful normal strictly semifinite weight $\ddot{I}†$, we develop a notion of Murray-von Neumann dimension over $(M,\ddot{I}†)$ that is defined for modules over the basic construction associated to the inclusion $M^{\ddot{I}}†\subset M$. For $\ddot{I}†=\ddot{I}„$ a faithful normal tracial state, this recovers the usual Murray-von Neumann dimension for finite von Neumann algebras. If $M$ is either a type ${\mathrm{III}}_{\hat{I}}»$ factor with $0<\hat{I}»<1$ or a full type ${\mathrm{III}}_1$ factor with $\text{Sd}(M)\ne \mathbb{R}$, then amongst extremal almost periodic weights the dimension function depends on $\ddot{I}†$ only up to scaling. As an application, we show that if an inclusion of diffuse factors with separable preduals $N\subset M$ is with expectation $\mathcal{E}$ and admits a compatible extremal almost periodic state $\ddot{I}†$, then this dimension quantity bounds the index $\text{Ind}\mathcal{E}$, and in fact equals it when the modular operators $\hat{I}{”}_{\ddot{I}}†$ and $\hat{I}{”}_{\ddot{I}†{|}_N}$ have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner-Popa orthogonal bases.

minor changes, to appear in the Journal of Functional Analysis