Relative bicentralizers and modular spectral recovery

Proves Connes' bicentralizer conjecture for every type III₁ factor with separable predual and every faithful normal state. More generally, for every inclusion $N\subset M$ of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra $P\subset N$ with $P'\cap c(M)=N'\cap c(M)$, resolving the relative bicentralizer conjecture.