Conditional expectation, pinching

Question

Let $\mathfrak{C}$ be a unital $*$-subalgebra of the full matrix algebra $M_n(\mathbb{C}).$ Let $\mathbb{E}_\mathfrak{C}$ be the orthogonal projection from $M_n(\mathbb{C}),$ endowed with the Hilbert-Schmidt inner product $\langle A,B \rangle = {\rm Tr} \: B^*A$, onto the $*$-subalgebra $\mathfrak{C}.$ Then it seems to me that the inequality $${\rm Tr } \: |\mathbb{E}_\mathfrak{C}(A)|^p \leq {\rm Tr} \: |A|^p$$ holds, for every $1 \leq p < \infty,$ where $|X|=(X^*X)^{1/2}$ denote the absolute value of $X$. I guess this must be written somewhere in literature. Please, could you give me a reference for the result?

Answer 1

I will prove the case when $p=2$. I hope it will be helpful for you.

Note that $E_\mathfrak{C}$ is a completely positive, unital, trace-preserving map. Using the Schwarz type inequality, we have \begin{eqnarray} E_\mathfrak{C}(A)^*E_\mathfrak{C}(A)\leq E_\mathfrak{C}(A^*A). \end{eqnarray} Since $\textrm{Tr}E_\mathfrak{C}(A^*A)=\textrm{Tr}A^*A$, we have the desired inequality.