Index of the conditional expectation onto the almost periodic part

Determine the possible values of the index Ind(E_ap) of the φ-preserving faithful normal conditional expectation E_ap: M → M^{(φ,ap)} when φ is an extremal faithful normal strictly semifinite weight and M is a factor. Specifically, ascertain whether Ind(E_ap) is always either 1 or +∞ under these hypotheses.

Background

For a von Neumann algebra M with a faithful normal strictly semifinite weight φ, the authors define the almost periodic part M{(φ,ap)} as the von Neumann subalgebra generated by the eigenoperators of the modular automorphism group σφ. There is a unique φ-preserving faithful normal conditional expectation E_ap from M onto M{(φ,ap)}.

They note that (M{(φ,ap)}){φ_ap} = Mφ, so φ_ap is extremal if and only if φ is extremal. The index Ind(E) of a conditional expectation E is a central quantity in subfactor theory, here considered in the general (non-tracial) setting via Pimsner–Popa bases and related constructions developed earlier in the paper.

The authors express a specific unresolved claim about Ind(E_ap): in the case where φ is extremal and M is a factor, they suspect that Ind(E_ap) can take only two values (1 or +∞), but they do not have a proof. This dichotomy is illustrated in a subsequent example, but no general argument is known.

References

Additionally, one always has (M{(,\text{ap})}){_{\text{ap}} = M, so that {\text{ap}} is extremal if and only if is extremal. We suspect that in this case and when M is also a factor, one has that Ind{E{\text{ap}} is either 1 or +\infty, but we are unable to prove it at this time.

— Murray-von Neumann dimension for strictly semifinite weights  (2405.15725 - Guinto et al., 2024) in Section 5 (Constructing Extremal Almost Periodic Inclusions), paragraph following the definition of the almost periodic part