Faithful normal state - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra. A faithful normal state is a normal state $\varphi:M\to\mathbb C$ that is faithful:

x\in M_+,\quad \varphi(x)=0\quad\Longrightarrow\quad x=0.

Equivalently, $\varphi(x^*x)=0$ implies $x=0$ for every $x\in M$ . Thus the state is both ultraweakly continuous and nondegenerate on the positive cone. Neither adjective implies the other: normality is a continuity condition, whereas faithfulness says that no nonzero positive element is invisible to the state. In particular, a faithful normal state distinguishes zero from every positive element.

Equivalent characterizations

A normal state $\varphi$ is faithful exactly when its support projection is $1$ . Equivalently, in the GNS representation of $\varphi$ , the cyclic vector is separating for the represented von Neumann algebra. These characterizations connect positivity, support, and representation theory Takesaki, discussion of faithful normal states.

Existence and examples

A von Neumann algebra admits a faithful normal state exactly when it is sigma-finite, also called countably decomposable in this setting. If $H$ is separable, choose a positive trace-class operator $\rho$ with trivial kernel and $\operatorname{Tr}(\rho)=1$ ; then $\varphi(x)=\operatorname{Tr}(\rho x)$ is a faithful normal state on $B(H)$ . By contrast, $B(H)$ for nonseparable $H$ has no faithful normal state.

Distinctions

A faithful state on the underlying $C^*$ -algebra need not be normal. Likewise, a normal state can fail to be faithful when its support is a proper projection. Faithful normal states are bounded normalized functionals, not the possibly unbounded faithful normal semifinite weights used in modular theory.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: the chapters on normal positive functionals, sigma-finiteness, and standard representations.