Equality of analytic and algebraic bicentralizers for amenable groups

Prove that the analytic and algebraic relative bicentralizers coincide for every trace-preserving action of an amenable locally compact group on a tracial inclusion $(M,\tau)\subset(N,\tau)$, namely establish $(M\subset N,\alpha)_{\mathrm{analytic}}=(M\subset N,\alpha)_{\mathrm{algebraic}}$.

Background

The paper introduces two versions of the bicentralizer for a trace-preserving action: an analytic bicentralizer defined through asymptotic commutation with almost invariant unitaries, and an algebraic bicentralizer defined using commutants after amplification by $L^2(G)$. It proves the inclusion of the algebraic bicentralizer in the analytic one.

The reverse inclusion is conjectured for amenable groups. The result would generalize the earlier bicentralizer conjecture for actions on a single II_1 factor and would identify the analytic asymptotic formulation with the algebraic operator-algebraic formulation for arbitrary tracial inclusions.