Example 41.6.5 (Conditional Expectation).label Let $\phi: M_{n}(\complex) \to \complex^{n}$ be defined by $\phi(A)_{k} = a_{kk}$, then:
- (1)
$\phi$ is unital and completely positive.
- (2)
The multiplicative domain of $\phi$ is the diagonal matrices on $M_{n}(\complex)$.
Proof. (1): Identify $\complex^{n}$ as the diagonal matrices in $M_{n}(\complex)$, then for each $A \in M_{n}(\complex)$,
\[\phi(A) = \sum_{j = 1}^{n} E_{jj}AE_{jj}\]
By Proposition 41.5.3, $\phi$ is completely positive.
(2): ???$\square$
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