Positive element

Summary

A positive element of a C*-algebra is a self-adjoint element whose spectrum lies in $[0, \infty)$. Equivalently, it has the form $b^* b$. Positive elements generalize nonnegative functions and positive semidefinite matrices, and they define the order $a \le b$ on self-adjoint elements. They are the basis of the notions of states and completely positive maps.