Pure state of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A state $\varphi\in S(A)$ is pure if it is an extreme point of the state space: whenever

\varphi=t\psi+(1-t)\chi

for states $\psi,\chi\in S(A)$ and $0<t<1$ , one must have $\psi=\chi=\varphi$ . Thus purity means that $\varphi$ admits no nontrivial convex decomposition into other states. It is a convex-geometric condition, not a continuity or nondegeneracy condition; in particular, purity does not mean normality or faithfulness.

GNS characterization

A state is pure if and only if its GNS representation is irreducible. This connects extreme points of $S(A)$ with irreducible representations of $A$ . The equivalence is a theorem, not an alternative convention for the definition Murphy, §3.3.

Standard examples

For $A=C_0(X)$ , pure states are evaluations $f\mapsto f(x)$ at points $x\in X$ ; equivalently, they are the multiplicative states. On $M_n(\mathbb C)$ , pure states are vector states $\varphi(a)=\langle a\xi,\xi\rangle$ for unit vectors $\xi$ , or density matrices of rank one. A density matrix of rank greater than one gives a mixed, not pure, state.

Distinctions

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §3.3 on pure states and irreducible GNS representations.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: §3.3 on states and purity.