Proposition 41.8.9 (Multiplicative Domain of Stinespring Representations).label Let $A$ be a unital $C^{*}$-algebra, $H$ be a complex Hilbert space, $\phi: A \to B(H)$ be a unital completely positive map, and $(\pi, V, K)$ be a minimal Stinespring representation of $\phi$, then:
- (1)
For any $a \in A$, $\phi(a)^{*}\phi(a) = \phi(a^{*}a)$ if and only if $V(H)$ is invariant for $\pi(a)$[1]. In other words, $a$ is in the right multiplicative domain of $\phi$ if and only if $V(H)$ is invariant for $\pi(a)$.
- (2)
For any $a \in A$, $a$ is in the multiplicative domain of $\phi$ if and only if $V(H)$ is reducing for $\pi(a)$.
Proof. (1): Since $\phi$ is unital, $V$ is an isometry. Identify $H = V(H) \subset K$ and let $P = V^{*} \in B(K)$ be the orthogonal projection onto $H$.
Let $a \in A$ and $\xi, \eta \in H$, then
By Lemma 39.3.2, $\pi(a)^{*}P\pi(a)P = \pi(a)^{*}P\pi(a)P$ if and only if $P(K) = V(H)$ is an invariant subspace for $\pi(a)$. Therefore $\phi(a)^{*}\phi(a) = \phi(a^{*}a)$ if and only if $V(H)$ is invariant for $\pi(a)$.$\square$
- This also shows that multiplicative domains of $\phi$ are subalgebras of $A$. keyboard_return
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