Ambient factor choice for approximately factorial perturbations (Theorem E)

Determine whether, under the hypotheses that (M, τ) is a tracial von Neumann algebra and M1, M2 ⊂ M satisfy M1 ≠ C1, M2 ≠ C1, dim(M1) + dim(M2) ≥ 5, and e(M1) + e(M2) ≤ 1, one can choose the ambient II1 factor for the perturbation conclusion of Theorem E as follows: (a) if M is itself a II1 factor, take the ambient factor to be M and find, for every ε > 0, a unitary v ∈ U(M) with ∥v − 1∥2 ≤ ε such that M ∨ v M1 v∗ ∨ M2 is a II1 factor; (b) for general M, take the ambient factor to be M ∗ L(Z) and find, for every ε > 0, a unitary v ∈ U(M ∗ L(Z)) with v ∈ L(Z) and ∥v − 1∥2 ≤ ε such that (M ∗ L(Z)) ∨ v M1 v∗ ∨ M2 is a II1 factor.

Background

Theorem E provides an approximate factoriality result: given a tracial von Neumann algebra (M, τ) with subalgebras M1, M2 satisfying basic size and minimal-projection constraints, there exists a larger II1 factor (Mf, τf) containing M, and for every ε > 0 a unitary vε ∈ U(Mf) with ∥vε − 1∥2 < ε such that the von Neumann algebra generated by M and the conjugate vε M1 vε∗ together with M2 is a II1 factor.

The open question asks whether the ambient II1 factor Mf can be taken in a particularly canonical or minimal form. Part (a) asks if one can avoid enlarging M at all when M is already a II1 factor. Part (b) asks whether, in general, it suffices to take Mf to be the free product M ∗ L(Z) and to choose the perturbing unitary inside the L(Z) leg, while still achieving factoriality after conjugating M1 by a near-identity unitary.