invariant subspace of a representation

Question

Suppose $\pi:A \to B(H)$ is a representation of $A$ such that $\pi(A)K_1\subset K_,\pi(A)K_2\subset K_2$,where $H=K_1\oplus K_2$,can we conclude that there exist a projection $p\in \pi(A)^{'}$ such that $K_2=pH$?

Answer 1

Yes. If $a\in A$ and $p$ is the orthogonal projection onto $K_2$, for any $x\in H$ you have $px\in K_2$ and so $\pi(a)px\in K_2$. This gives you $p\pi(a)px=\pi(a)px$. As $x$ is arbitrary, you get $p\pi(a)p=\pi(a)p$. If $a=a^*$, the left-hand-side is selfadjoint, so $\pi(a)p=p\pi(a)$. As the selfadjoint elements span $A$, you get $p\in\pi(A)'$.