Equivalence of two formulations of the Peterson–Thom property beyond the finite case

Determine whether, for an arbitrary (not necessarily finite) von Neumann algebra M, the following two properties are equivalent: (i) for every diffuse amenable von Neumann subalgebra Q ≤ M, the set {P ≤ M : P is with expectation in M, P ⊇ Q, and P is amenable} has a largest element; (ii) whenever {Q_j : j ∈ J} are diffuse, amenable, and with expectation in M and the intersection ⋂_{j∈J} Q_j contains a diffuse subalgebra with expectation in M, then every von Neumann subalgebra B ≤ W_j Q_j with expectation in M is amenable. In particular, ascertain whether (ii) implies (i) for general von Neumann algebras.

Background

Proposition 1.7 introduces two properties related to maximal amenable extensions with expectation. Property (i) is taken as the definition of the Peterson–Thom property in this paper, while property (ii) is a consequence of (i). In the finite (tracial) setting, these two properties are known to be equivalent.

Outside the finite setting, the equivalence is not established. A key obstruction noted by the authors is that intersections of subalgebras that are individually with expectation may fail to remain with expectation, complicating the implication from (ii) back to (i). Clarifying whether the equivalence persists in the general (possibly type III) case would refine the conceptual status of the Peterson–Thom property beyond the tracial framework.