From e99ca5818aa76f5e094e0b689ed67b9fbcf4ba60 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Tue, 6 Oct 2026 19:04:41 -0400 Subject: [PATCH] Thank you Sam. --- src/op/system/complete-upgrade.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/op/system/complete-upgrade.tex b/src/op/system/complete-upgrade.tex index 76e5568..9ac84d3 100644 --- a/src/op/system/complete-upgrade.tex +++ b/src/op/system/complete-upgrade.tex @@ -239,7 +239,7 @@ The following is a slight generalisation of one form of the above example, but s \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 \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*} By linearity, for each $T \in M_n(\complex)$,