Marrakchi's relative bicentralizer conjecture

Theorem 1.1: for every unital inclusion $N\subset M$ with separable preduals and a faithful normal conditional expectation there are an amenable $P\subset N$ and an expectation $N\to P$ with $P'\cap c(M)=N'\cap c(M)$; with Marrakchi's Theorem D this is the relative bicentralizer conjecture, including arbitrary centers and types, and gives an expected MASA corollary relevant to Kadison's problem for subfactors. Along the way it reproves Connes' bicentralizer conjecture for separable type $\mathrm{III}_1$ factors; priority for that absolute case belongs to Houdayer and Marrakchi (arXiv 2609.11462, 10 Sep 2026), whose statement has no separability assumption. The companion gives a second absolute proof via bounded recovery. Non-separable preduals are not treated.

Theorem 1.1: for every unital inclusion $N\subset M$ with separable preduals and a faithful normal conditional expectation there are an amenable $P\subset N$ and an expectation $N\to P$ with $P'\cap c(M)=N'\cap c(M)$; with Marrakchi's Theorem D this is the relative bicentralizer conjecture, including arbitrary centers and types, and gives an expected MASA corollary relevant to Kadison's problem for subfactors. Along the way it reproves Connes' bicentralizer conjecture for separable type $\mathrm{III}_1$ factors; priority for that absolute case belongs to Houdayer and Marrakchi (arXiv 2609.11462, 10 Sep 2026), whose statement has no separability assumption. The companion gives a second absolute proof via bounded recovery. Non-separable preduals are not treated.