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Thank you Sam.
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Bokuan Li
2026-10-06 19:04:41 -04:00
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src/op/system/complete-upgrade.tex
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@@ -239,7 +239,7 @@ The following is a slight generalisation of one form of the above example, but s
\begin{align*} \begin{align*}
\sum_{k = 1}^n S_k (E_{ij} \otimes 1_B) S_k^* &= \sum_{k = 1}^n \sum_{s, t = 1}^n (E_{ks} \otimes b_{k, s}^*) (E_{ij} \otimes 1_B) (E_{kt} \otimes b_{k, t}^*)^* \\ \sum_{k = 1}^n S_k (E_{ij} \otimes 1_B) S_k^* &= \sum_{k = 1}^n \sum_{s, t = 1}^n (E_{ks} \otimes b_{k, s}^*) (E_{ij} \otimes 1_B) (E_{kt} \otimes b_{k, t}^*)^* \\
&= \sum_{k = 1}^n \sum_{s, t = 1}^n (E_{ks} \otimes b_{k, s}^*) (E_{ij} \otimes 1_B) (E_{tk} \otimes b_{k, t}) \\ &= \sum_{k = 1}^n \sum_{s, t = 1}^n (E_{ks} \otimes b_{k, s}^*) (E_{ij} \otimes 1_B) (E_{tk} \otimes b_{k, t}) \\
&= \sum_{k = 1}^n E_{kk} \otimes b_{k, i}^* b_{k, j}^* = \sum_{k = 1}^n E_{kk} \otimes \phi(E_{ij}) &= \sum_{k = 1}^n E_{kk} \otimes b_{k, i}^* b_{k, j} = \sum_{k = 1}^n E_{kk} \otimes \phi(E_{ij})
\end{align*} \end{align*}
By linearity, for each $T \in M_n(\complex)$, By linearity, for each $T \in M_n(\complex)$,