States separate positive elements

Statement

Let $A\neq0$ be a complex $C^*$ -algebra, not necessarily unital, and let $S(A)$ be its state space. The statement that states separate positive elements is

\|a\|=\sup_{\varphi\in S(A)}\varphi(a) \qquad(a\in A_+).

In fact the supremum is attained: for every positive element $a$ , some state $\varphi$ satisfies $\varphi(a)=\|a\|$ . Consequently, if $a,b\in A_+$ and $\varphi(a)=\varphi(b)$ for every state, then $a=b$ . Equivalently, a self-adjoint $x\in A$ is positive exactly when $\varphi(x)\geq0$ for every state $\varphi$ .

Proof mechanism

Apply the continuous functional calculus to the commutative algebra generated by $a$ . Evaluation at the largest point of the spectrum gives a norm-one positive functional whose value at $a$ is $\|a\|$ . Extending that functional to $A$ without increasing its norm produces a state. Thus states attain the norm of positive elements.

To obtain the order criterion, if a self-adjoint $x$ is not positive, its negative spectral part yields a state with $\varphi(x)<0$ . Applying this to $a-b$ also shows that states separate arbitrary self-adjoint elements, not only elements of $A_+$ .

Norm consequences

For every $x\in A$ ,

\|x\|^2=\|x^*x\| =\sup_{\varphi\in S(A)}\varphi(x^*x).

For self-adjoint $x$ , one similarly has

\|x\|=\sup_{\varphi\in S(A)}|\varphi(x)|.

Thus states recover the norm even when $A$ is originally presented abstractly. These formulas underlie the isometry in the GNS construction and the universal representation.

Example and near-miss

For $A=M_n(\mathbb C)$ and $a\geq0$ , a unit eigenvector for the largest eigenvalue defines a vector state attaining $\|a\|$ .

A single faithful state need not recover the norm of every positive element. For example, normalized matrix trace assigns $1/n$ to a rank-one projection, although that projection has norm $1$ . The theorem requires the whole state space.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §3.3, especially Theorem 3.3.6 on states attaining the norm of normal elements.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. Publisher record. Relevant: §3.3 on positive functionals, states, and order separation.