Relative bicentralizers and modular spectral recovery

We prove that every inclusion N\subset M of von Neumann algebras with separable preduals and a faithful normal conditional expectation contains an expected amenable subalgebra P\subset N such that P'\cap c(M)=N'\cap c(M), where c(M) denotes the continuous core of M. This resolves the relative bicentralizer conjecture in this setting.

We prove a bounded spectral recovery theorem for a von Neumann algebra with a faithful normal state whose centralizer consists only of scalars. A positive averaged squared norm for an arbitrary bounded Hilbert-space operator on shrinking modular spectral bands can be recovered on uniformly bounded algebra elements with shrinking spectral support. We apply this theorem to spectral intertwining rigidity and obtain an alternative proof of bicentralizer triviality for type III_(1) factors with separable predual.