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Murray-von Neumann dimension for strictly semifinite weights

Aldo Garcia Guinto, Matthew Lorentz, Brent Nelson

TL;DR

The paper extends Murray–von Neumann dimension to von Neumann algebras equipped with faithful normal strictly semifinite weights by employing the basic construction $\langle M,e_\varphi\rangle$ and the associated $(M,\varphi)$-modules, recovering the classical dimension in the tracial case. It analyzes the discrete core, Connes invariants, and the interplay between modular data and dimension, establishing that for extremal almost periodic weights the dimension is unique up to a scaling constant in type $\mathrm{III}_\lambda$ factors ($0<\lambda<1$) or full factors, with a concrete isomorphism between basic constructions yielding a precise scaling relation. The main application bounds (and in some cases determines) the index of subfactor inclusions in the type III setting via Murray–von Neumann dimension, and the authors prove the universal existence of Pimsner–Popa bases in these AP contexts. They also develop constructive methods to produce extremal almost periodic inclusions by passing to the AP part and adjoining eigenoperators, providing tools to generate new examples and study index–dimension phenomena in broader von Neumann algebra settings.

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 $\varphi=τ$ a faithful normal tracial state, this recovers the usual Murray-von Neumann dimension for finite von Neumann algebras. If $M$ is either a type $\mathrm{III}_λ$ factor with $0<λ<1$ 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 $Δ_\varphi$ and $Δ_{\varphi|_N}$ have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner-Popa orthogonal bases.

Murray-von Neumann dimension for strictly semifinite weights

TL;DR

The paper extends Murray–von Neumann dimension to von Neumann algebras equipped with faithful normal strictly semifinite weights by employing the basic construction and the associated -modules, recovering the classical dimension in the tracial case. It analyzes the discrete core, Connes invariants, and the interplay between modular data and dimension, establishing that for extremal almost periodic weights the dimension is unique up to a scaling constant in type factors () or full factors, with a concrete isomorphism between basic constructions yielding a precise scaling relation. The main application bounds (and in some cases determines) the index of subfactor inclusions in the type III setting via Murray–von Neumann dimension, and the authors prove the universal existence of Pimsner–Popa bases in these AP contexts. They also develop constructive methods to produce extremal almost periodic inclusions by passing to the AP part and adjoining eigenoperators, providing tools to generate new examples and study index–dimension phenomena in broader von Neumann algebra settings.

Abstract

Given a von Neumann algebra equipped with a faithful normal strictly semifinite weight , we develop a notion of Murray-von Neumann dimension over that is defined for modules over the basic construction associated to the inclusion . For a faithful normal tracial state, this recovers the usual Murray-von Neumann dimension for finite von Neumann algebras. If is either a type factor with or a full type factor with , then amongst extremal almost periodic weights the dimension function depends on only up to scaling. As an application, we show that if an inclusion of diffuse factors with separable preduals is with expectation and admits a compatible extremal almost periodic state , then this dimension quantity bounds the index , and in fact equals it when the modular operators and have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner-Popa orthogonal bases.
Paper Structure (14 sections, 34 theorems, 222 equations)

This paper contains 14 sections, 34 theorems, 222 equations.

Table of Contents

  1. Preliminaries
  2. Weights
  3. Strictly semifinite weights
  4. Almost periodic weights
  5. Discrete cores
  6. Connes invariants
  7. The basic construction, Pimsner--Popa orthogonal bases, and index
  8. Modules and Dimension
  9. Centralizer basic construction
  10. Definitions and general properties
  11. Connection to semifinite Murray--von Neumann dimension
  12. Extremal almost periodic advantages
  13. Application to the Index for Subfactors
  14. Constructing Extremal Almost Periodic Inclusions

Key Result

Theorem A

Let $M$ be a factor with separable predual, and let $\varphi$ and $\psi$ be extremal almost periodic weights. Suppose that either: Then there exists a $*$-isomorphism $\theta\colon \left\langle M,e_\psi\right\rangle \to \left\langle M,e_\varphi\right\rangle$ and a constant $c_{\psi,\varphi}>0$ so that for all left $(M,\varphi)$-modules $(\mathcal{H},\pi)$ one has

Theorems & Definitions (89)

  • Theorem A: Theorem \ref{['thm:dim_scaling_constant']}
  • Theorem B: Theorem \ref{['thm:dimension_vs_index']}
  • Remark 1.1
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Remark 1.5
  • Lemma 1.6
  • ...and 79 more