State on a C*-algebra

Builds on Positive element Continuous dual space

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 $a \in A$ . 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 $A^*$ . 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 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.

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Tag	`PCX8R81H96`
URI	`m2://PCX8R81H96`
Kind	definition
Writing	W0 stub
MSC	46L30, 46L05
Defines	State on a C*-algebra
Uses	Positive element, Continuous dual space
Changed	2 November 2026

State on a C*-algebra

Definition

Remarks

Reading path

Further reading

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