---
tag: PCX8R81H96
kind: definition
name: State on a C*-algebra
msc: [46L30, 46L05]
defines: F7MB2WREBJ
uses: [NA8DF5X3M4, 4A82S5FVRT]
sources:
  - url: https://en.wikipedia.org/wiki/State_(functional_analysis)
---

## Definition

Let $A$ be a C*-algebra.
A linear functional $\varphi \colon A \to \C$ is *positive* if $\varphi(a) \ge 0$ for every [[positive element|NA8DF5X3M4]] $a \in A$.
A *state* on $A$ is a positive linear functional $\varphi$ with $\norm{\varphi} = 1$.

## Remarks

A positive linear functional is automatically bounded, so it lies in the [[dual space|4A82S5FVRT]] $A^*$.
If $A$ has a unit, then $\norm{\varphi} = \varphi(1)$ for positive $\varphi$, so a state is a positive linear functional with $\varphi(1) = 1$.
The set $S(A)$ of states of a unital $A$ is convex and weak-* compact; its [[extreme points|3MSGFJM2NX]] are the *pure states*.

Example: let $H$ be a Hilbert space, $A$ the bounded operators on $H$, and $\xi \in H$ a unit vector.
The vector state $\varphi(T) = \ip{T \xi}{\xi}$ is linear, satisfies $\varphi(T^* T) = \norm{T \xi}^2 \ge 0$, and $\varphi(I) = \norm{\xi}^2 = 1$; so it is a state.
On $M_n(\C)$, the normalized trace $\tfrac{1}{n} \tr$ is a state.

Non-example: the trace $\tr$ on $M_2(\C)$ is positive, but $\tr(I) = 2$, so its norm is $2$ and it is not a state.