Definition 38.5.8 (Invariant Subspace).label Let $A$ be a $C^{*}$-algebra, $(H, \pi)$ be a representation of $A$, $M \subset H$ be a closed subspace, and $P \in B(H)$ be the orthogonal projection onto $M$, then the following are equivalent:
- (1)
For each $x \in A$, $\pi(x)(M) \subset M$.
- (2)
$P \in \pi(A)'$.
If the above holds, then $M$ is an invariant subspace for $\pi$.
Proof. Since $\pi(A)$ is a self-adjoint subspace of $B(H)$, $M$ is invariant for $\pi(A)$ if and only if it is a reducing subspace for $\pi(A)$, if and only if $\pi(A)$ commutes with $P$.$\square$
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