This question can be formulated for general ($\sigma$-finite) von Neumann algebras, but for me it is enough to consider matrix algebras.
So let $M$ be a matrix algebra and $\rho$ a faithful state (density operator) on $M$. For $1\le p<\infty$, we define the $L_p$-norm on $M$ with respect to $\rho$ as $$ \|a\|_{p,\rho}^p=Tr(\rho^{1/2p}a\rho^{1/2p})^p. $$ Let us also introduce the inner product $\langle a,b\rangle_\rho=Tr a^*\rho^{1/2}b\rho^{1/2}$, for $a,b\in M$, then $L_p$ and $L_q$ are dual with respect to $\langle\cdot,\cdot\rangle_\rho$ for $1/p+1/q=1$.
Let $N$ be another matrix algebra and $\phi:N\to M$ be a unital completely positive (ucp) map, such that the state $\rho_0=\phi^*(\rho)$, given by $$ Tr (\rho_0 a)=Tr (\rho\phi(a)), \ a\in N $$ is faithful. Then $\phi$ is a contraction with respect to the $L_p$-norms, $$ \|\phi(a)\|_{p,\rho}\le \|a\|_{p,\phi^*(\rho)},\ a\in N. $$ I am interested in a characterization of the set of elements where equality is attained in the above inequality. Namely, let $1<p<\infty$ and let $$ \mathcal I_p:=\{a\in N, \|\phi(a)\|_{p,\rho}= \|a\|_{p,\phi^*(\rho)}\} $$
For $p=2$, this set can be determined easily. In this case, $M$ with the $L_2$-norm is a Hilbert space with the inner product $\langle\cdot,\cdot\rangle_\rho$ and $\mathcal I_2$ is equal to the set $\mathcal F$ of fixed points of the map $\phi_\rho^*\circ\phi$, where $\phi^*_\rho$ is the adjoint of $\phi$ with respect to this inner product. It is easy to see that $\phi^*_\rho$ is an ucp map $M\to N$, so that $\phi^*_\rho\circ\phi$ is an ucp map on $N$ that preserves the faithful state $\rho_0$. Consequently, $\mathcal F$ is a subalgebra invariant under the modular group $\sigma^{\rho_0}$ of $\rho_0$.
For $p\ne 2$, the inclusion $\mathcal F\subseteq \mathcal I_p$ is easily obtained by the fact that $\phi^*_\rho$ is an ucp map and therefore also a contraction. So my questions are:
Is the opposite inclusion true, that is, is it true that $\mathcal I_p=\mathcal F$ also for $p\ne 2$?
If the answer to 1. is negative, what is the structure of $\mathcal I_p$? Is it a subalgebra, is it invariant under $\sigma^{\rho_0}$, or maybe both?
