nondegenerate representation of a $C^*$ algebra

Question

Every representation $(\pi,H)$ of a $C^*$ algebra $A$ can be reduced to the case of a non-degenerate representation.Usually,we take $K=[\pi(A) H]$,then we get a non-degenerate representation $(\pi_K,K)$.

Can we let $K_0$ be the orthogonal complementary of $K$ to construct a non-degenerate representation $(\pi_{K_0},K_0)$

Answer 1

No. Take $A=M_2(\mathbb C)$ and $H=\mathbb C^3$, and $\pi(a)[x,y,z]^\perp=a[x,y]^\perp$. Then $K=\mathbb C^2$ (the first two coordinates of $H$), and the orthogonal complement is one-dimensional. There is no representation $M_2(\mathbb C)\to\mathbb C$.