Murray–von Neumann dimension for strictly semifinite weights

Abstract

Given a von Neumann algebra M equipped with a faithful normal strictly semifinite weight φ, we develop a notion of Murray–von Neumann dimension over $(M,\phi )$ that is defined for modules over the basic construction associated to the inclusion $M^{\phi }\subset M$. For $\phi =\tau$ 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 $\mathrm{Sd}\left(M\right)\ne R$, then amongst extremal almost periodic weights the dimension function depends on φ 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 $E$ and admits a compatible extremal almost periodic state φ, then this dimension quantity bounds the index $\mathrm{Ind}E$, and in fact equals it when the modular operators ${\Delta }_{\phi }$ and ${\Delta }_{\phi {|}_N}$ have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner–Popa orthogonal bases.