Positive element of a C*-algebra

An element $a$ of a $C^*$ -algebra $A$ is positive, written $a\geq0$ , if it is self-adjoint and its spectrum satisfies $\sigma_A(a)\subseteq[0,\infty)$ . Equivalently, $a=b^*b$ for some $b\in A$ . Also equivalently, there is a unique positive element $a^{1/2}\in A$ such that $(a^{1/2})^2=a$ . These characterizations agree in unital and nonunital $C^*$ -algebras; the square root is constructed by the continuous functional calculus. The positive elements form the cone $A_+$ , which defines the canonical order on the self-adjoint part: $x\leq y$ exactly when $y-x\in A_+$ .

The positive cone and order

The set $A_+$ of positive elements is a norm-closed convex cone, satisfies $A_+\cap(-A_+)=\{0\}$ , and linearly spans $A$ . It defines an order on the self-adjoint part $A_{\mathrm{sa}}$ by

x\leq y\quad\Longleftrightarrow\quad y-x\in A_+.

Positive functionals, states, weights, and positive maps are defined using this cone.

Concrete interpretation

Under any faithful representation $A\subseteq B(H)$ , abstract positivity agrees with operator positivity:

\langle a\xi,\xi\rangle\geq0 \qquad\text{for every }\xi\in H.

This criterion is representation-independent. Positivity is stronger than self-adjointness: a self-adjoint element with negative spectrum is not positive. It is also not defined by signs of arbitrary coefficients in a presentation of $A$ .

References

G. J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: Section 2.2 on positive elements and square roots.
G. K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Section 1.2 on positivity and order.