Let $x$ be a positive operator on the Hilbert space direct sum $H:= \bigoplus_{i \in I} H_i$ and let $\pi_i: H \to H_i$ be the projections. Do we have $\|x\| = \sum_{i \in I} \|\pi_i^* x \pi_i\|?$
Attempt: Put $x_i:= \pi_i^*x\pi_i \in B(H)$. Then $x_i$ is positive, $x_ix_j = 0$ when $x\ne j$ and we have to show $$\|x\| = \sum_{i \in I} \|x_i\|.$$
I feel like I'm missing something basic here.
This is not true. Consider $H_1=H_2=\mathbb C$, $H=H_1\oplus H_2$, and the operator $$x = \begin{pmatrix}2&1\\1&2\end{pmatrix}\in M_2(\mathbb C)\cong B(H).$$ Then $x$ is positive, and $\|\pi_1x\pi_1^*\|+\|\pi_2x\pi_2^*\|=2+2=4$ while $\|x\|=3$.