Faithful state on a C*-algebra

Let $A$ be a $C^*$ -algebra. A state $\varphi$ on $A$ is faithful if, for every positive element $a\in A$ ,

\varphi(a)=0\quad\Longrightarrow\quad a=0.

Equivalently, $\varphi(a^*a)=0$ implies $a=0$ for every $a\in A$ . Thus faithfulness is a nondegeneracy condition on a state: no nonzero positive element is invisible to it. It is independent of normality, which is a continuity condition available when $A$ is a von Neumann algebra, and it is also independent of purity.

Equivalent tests and representations

The two tests in the core agree because every positive element has a positive square root. In the GNS construction, faithfulness says that the cyclic vector has zero expectation on $a^*a$ only when $a=0$ . It implies that the associated GNS representation is faithful, but the converse can fail: a faithful representation may have a cyclic vector whose vector state vanishes on a nonzero positive element.

Examples and existence

On $M_n(\mathbb C)$ , a state has the form $\varphi(a)=\operatorname{Tr}(\rho a)$ for a positive matrix $\rho$ of trace one; it is faithful exactly when $\rho$ is invertible. On $C_0(X)$ , states correspond to probability Radon measures, and the state is faithful exactly when the measure has full support. Not every $C^*$ -algebra has a faithful state, although every nonzero separable $C^*$ -algebra does.

Distinctions

References

Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §3.3 on positive functionals, states, and faithfulness.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter I, §9 on positive functionals and representations.