Murray-von Neumann dimension for strictly semifinite weights

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.
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