Murray-von Neumann dimension for strictly semifinite weights

TL;DR

This paper extends the Murray-von Neumann dimension concept to modules over von Neumann algebras with strictly semifinite weights, providing new tools for analyzing subfactor indices in type III factors.

Contribution

It introduces a generalized dimension for modules over $(M, )$ in the semifinite setting, connecting it to subfactor index theory and spectral properties of modular operators.

Findings

Dimension depends only on the weight up to scaling for certain type III factors.

The dimension bounds the index of inclusions of diffuse factors with expectation.

Such inclusions always admit Pimsner-Popa orthogonal bases.

Abstract

Given a von Neumann algebra $M$ equipped with a faithful normal strictly semifinite weight $\varphi$, we develop a notion of Murray-von Neumann dimension over $(M,\varphi)$ that is defined for modules over the basic construction associated to the inclusion $M^\varphi \subset M$. For $\varphi=\tau$ a faithful normal tracial state, this recovers the usual Murray-von Neumann dimension for finite von Neumann algebras. If $M$ is either a type $\mathrm{III}_\lambda$ factor with $0<\lambda <1$ or a full type $\mathrm{III}_1$ factor with $\text{Sd}(M)\neq \mathbb{R}$, then amongst extremal almost periodic weights the dimension function depends on $\varphi$ 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 $\varphi$, then this dimension quantity bounds the index $\text{Ind}{\mathcal{E}}$, and in fact equals it when the modular operators $\Delta_\varphi$ and $\Delta_{\varphi|_N}$ have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner-Popa orthogonal bases.