Uniform spectral gap property (P1) for non-full factors

Ascertain whether the following uniform spectral gap property holds for non-full factors M: for every automorphism θ ∈ Aut(M) that is not approximately inner (θ ∉ \overline{Inn}(M)), there exists a neighborhood U of the class \underline{p}(θ) in \underline{Out}(M) = Aut(M)/\overline{Inn}(M) such that the standard M–M bimodule 2(id) is not weakly contained in the direct sum \bigoplus_{β ∈ \underline{p}^{-1}(U)} 2(β), where 2(β) denotes the M–M bimodule associated to β.

Background

Jones proved a uniform spectral gap property (P2) for full II1 factors, and Marrakchi–Vaes extended such uniformity to full σ-finite factors. The weaker property (P1) would suffice, together with topological outerness of an action, to deduce strict outerness via a general mechanism outlined by the authors.

However, the existence of a uniform spectral gap at the automorphism-group level for non-full factors remains unresolved. The authors explicitly state that even the weaker property (P1) is unknown in this generality, underscoring a key open direction in the extension of rigidity phenomena beyond the full factor case.