Cyclic representation of a C*-algebra

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

\overline{\operatorname{span}}\{\pi(a)\xi:a\in A\}=H.

The representation is cyclic if it has at least one cyclic vector. For a nonunital algebra the closure in this formula is essential. The existence of a cyclic vector forces $\pi$ to be nondegenerate: the closed span of $\pi(A)H$ is then all of $H$ . Cyclicity is a property of the represented action together with its Hilbert space, not merely of $A$ .

Equivalent viewpoint

A vector is cyclic for $\pi(A)$ if and only if it is separating for the commutant $\pi(A)'$ : an operator $T\in\pi(A)'$ satisfying $T\xi=0$ must be zero. This converts density of one orbit into uniqueness detected by the commuting operators.

Sources of cyclic representations

The GNS construction associated with a positive linear functional produces a cyclic representation whose distinguished cyclic vector recovers the functional as a vector functional. Conversely, every cyclic representation with a chosen unit cyclic vector yields a state $a\mapsto\langle\pi(a)\xi,\xi\rangle$ .

Examples and limits

The identity representation of $C(X)$ on $L^2(X,\mu)$ is cyclic when the constant function $1$ belongs to the space and bounded continuous functions are dense there. A direct sum of cyclic representations need not be cyclic: one vector must simultaneously generate every summand with enough independence.

References

Gerald J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. Publisher record. Relevant: section 3.3 on cyclic representations and the GNS construction.