Murray-von Neumann dimension for strictly semifinite weights

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,φ) that is defined for modules over the basic construction associated to the inclusion M^φ ⊂ M. For φ=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 III_λ factor with 0<λ <1 or a full type III₁ factor with Sd(M)≠ 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⊂ M is with expectation E and admits a compatible extremal almost periodic state φ, then this dimension quantity bounds the index IndE, and in fact equals it when the modular operators Δ_φ and Δφ|N have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner-Popa orthogonal bases.