Cyclic vector - Knowlpedia

Let $\pi:A\to B(H)$ be a representation of a $C^*$ -algebra on a Hilbert space. A vector $\xi\in H$ is cyclic for $\pi$ if

\overline{\pi(A)\xi}=H.

Equivalently, every vector of $H$ can be approximated by vectors $\pi(a)\xi$ with $a\in A$ . If $M\subseteq B(H)$ is a concrete operator algebra, one likewise calls $\xi$ cyclic for $M$ when $\overline{M\xi}=H$ . A representation possessing a cyclic vector is called cyclic; cyclicity is always relative to the represented algebra.

Generated cyclic subspace

For any $\xi\in H$ , the closed subspace

H_\xi=\overline{\pi(A)\xi}

is the cyclic subspace generated by $\xi$ . If $\pi$ is a $*$ -representation, $H_\xi$ is reducing, and the restriction of $\pi$ to $H_\xi$ is cyclic. A cyclic nonzero representation is nondegenerate. Direct sums need not remain cyclic unless a single vector simultaneously generates every summand with the required approximation compatibility.

GNS construction

Given a positive functional $\varphi$ on $A$ , the GNS construction produces a representation $(\pi_\varphi,H_\varphi)$ with a distinguished cyclic vector $\xi_\varphi$ satisfying

\varphi(a)=\langle\pi_\varphi(a)\xi_\varphi,\xi_\varphi\rangle.

Conversely, every vector in a representation defines a positive vector functional by this formula. Cyclicity says that no proper closed reducing subspace containing $\xi_\varphi$ is needed to realize the functional.

Relation to the commutant

If $M$ is a unital self-adjoint operator algebra, then $\xi$ is cyclic for $M$ exactly when it is separating for the commutant $M'$ . Indeed, an operator in $M'$ that kills $\xi$ kills the dense set $M\xi$ ; the converse uses the projection onto $\overline{M\xi}$ , which belongs to $M'$ .

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997. AMS DOI record. Relevant: §5.5 on cyclic and separating vectors.
Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: Chapter 3 on cyclic representations and the GNS construction.