---
tag: NA8DF5X3M4
kind: concept
name: Positive element
msc: [46L05, 47B65]
sources:
  - url: https://en.wikipedia.org/wiki/Positive_element
same_as:
  wikidata: Q124257255
  wikipedia: Positive element
---

## Summary

A positive element of a C*-algebra is a self-adjoint element whose [[spectrum|EH0CZ19RKP]] lies in $[0, \infty)$.
Equivalently, it has the form $b^* b$.
Positive elements generalize nonnegative functions and [[positive semidefinite matrices|01MMVW8CKC]], and they define the order $a \le b$ on self-adjoint elements.
They are the basis of the notions of [[states|F7MB2WREBJ]] and [[completely positive maps|CGY299QMC3]].

## Explanations

### L1 human

In the algebra of continuous functions, a positive element is a function that never takes a negative value.
In the algebra of matrices, it is a Hermitian matrix with no negative eigenvalue, like $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, whose eigenvalues are $1$ and $3$.
Every element of the form $b^* b$ is positive, just as $\abs{z}^2 = \overline{z} z \ge 0$ for a complex number $z$.