Vector state - Imported Knowlpedia Content

Definition

Let $\pi:A\to\mathcal B(H)$ be a nondegenerate representation of a $C^*$ -algebra, and let $\xi\in H$ be a unit vector. The vector state determined by $\xi$ is the state

\omega_\xi(a)=\langle\pi(a)\xi,\xi\rangle\qquad(a\in A).

The representation is part of the ambient data, even when suppressed from the notation. More generally, the same formula for an arbitrary vector gives a positive vector functional. For a degenerate representation and a unit vector, that functional can have norm less than one, so it need not be a state.

Positivity and normalization

Positivity follows from

\omega_\xi(a^*a)=\|\pi(a)\xi\|^2\geq0.

Nondegeneracy implies that an approximate identity converges strongly to $I_H$ , which yields $\|\omega_\xi\|=\|\xi\|^2=1$ . If $\xi$ is cyclic, the given pointed representation is unitarily equivalent to the GNS representation of $\omega_\xi$ Murphy, §3.3.

Concrete operator algebras

For $A=\mathcal B(H)$ with its identity representation, every unit vector defines a vector state. This state is normal, and its density operator is the rank-one projection onto $\mathbb C\xi$ . A general normal state on $\mathcal B(H)$ is instead represented by a positive trace-class operator of trace one, and need not be a single vector state.

Dependence on representation

The same abstract state can appear as a vector state in different representations. The GNS construction guarantees at least one cyclic-vector realization for every state. Purity is not automatic: a vector state is pure exactly when its cyclic GNS subrepresentation is irreducible, not merely because it is specified by one vector.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §§3.2–3.3 on vector functionals, states, and GNS representations.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I, American Mathematical Society, 1997. DOI record. Relevant: Chapter 4 on positive functionals and vector states.