Flattening and asymptotic orthogonalization of completely positive maps

Let $M$ be a ${\mathrm{I}\mathrm{I}}_1$ factor, $N$ a tracial von Neumann algebra, and $\mathrm{\Phi }:M\to N$ a subtracial completely positive map. For an irreducible ${\mathrm{I}\mathrm{I}}_1$ subfactor $P\subseteq M$, we characterize when $\mathrm{\Phi }$ exhibits a flattening property under conjugation by unitaries in $P$. To be specific, we show that the failure of a Pimsner-Popa type inequality for $E_P\circ {\mathrm{\Phi }}^{\ast }\circ \mathrm{\Phi }$ is the precise obstruction, equivalently characterized by left weak mixing of a naturally associated $P$-$N$ bimodule. As an application, we obtain an asymptotic orthogonalization result generalizing a result of Popa.