Definition 34.11.1 (Cyclic Representation).label Let $A$ be a $C^{*}$-algebra, $(H, \pi)$ be a representation of $A$, and $\xi \in H$, then $\xi$ is a cyclic vector for $(H, \pi)$ if $\bracsn{\pi(x)(\xi)|x \in A}$ is dense in $H$. The representation $(H, \pi)$ is cyclic if it admits a cyclic vector.