Normal state - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra. A state $\varphi:M\to\mathbb C$ is normal if it is a normal functional, equivalently if $\varphi$ is a positive norm-one element of the predual $M_*$ . For a positive functional this means that whenever an increasing bounded net $(x_i)$ in $M_+$ has supremum $x$ ,

\varphi(x)=\sup_i\varphi(x_i).

Normality is therefore an order-continuity, or equivalently ultraweak continuity, requirement. It does not assert that $\varphi$ is faithful or pure.

Equivalent characterizations

For a positive linear functional on $M$ , membership in $M_*$ , ultraweak continuity, and preservation of suprema of bounded increasing nets in $M_+$ are equivalent. Normality can also be characterized by complete additivity on orthogonal families of projections. These equivalences are part of the standard predual theory of von Neumann algebras Takesaki, Chapter III, §2.

Concrete form on

B(H)

Every normal state on $B(H)$ is represented by a positive trace-class operator $\rho$ with $\operatorname{Tr}(\rho)=1$ :

\varphi(x)=\operatorname{Tr}(\rho x).

Conversely, every such density operator defines a normal state. Vector states $x\mapsto\langle x\xi,\xi\rangle$ , with $\|\xi\|=1$ , correspond to rank-one density operators and are normal Kadison–Ringrose, vol. II, §7.1.

Distinctions

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, AMS, 1997. DOI record. Relevant: §7.1 on normal functionals and states.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III, §2 on the predual and normal functionals.