# state

A state $\Psi$ on a $C^{*}$-algebra $A$ is a positive linear functional $\Psi\colon A\to\mathbb{C}$, $\Psi(a^{*}a)\geq 0$ for all $a\in A$, with unit norm. The norm of a positive linear functional is defined by

 $\|\Psi\|=\sup_{a\in A}\{|\Psi(a)|:\|a\|\leq 1\}.$ (1)

For a unital $C^{*}$-algebra, $\|\Psi\|=\Psi(\mathord{\mathrm{1\!\!\!\>I}})$.

The space of states is a convex set. Let $\Psi_{1}$ and $\Psi_{2}$ be states, then the convex combination

 $\lambda\Psi_{1}+(1-\lambda)\Psi_{2},\quad\lambda\in[0,1],$ (2)

is also a state.

A state is pure if it is not a convex combination of two other states. Pure states are the extreme points of the convex set of states. A pure state on a commutative $C^{*}$-algebra is equivalent to a character.

A state is called a tracial state if it is also a trace.

When a $C^{*}$-algebra is represented on a Hilbert space $\mathord{\mathcal{H}}$, every unit vector $\psi\in\mathord{\mathcal{H}}$ determines a (not necessarily pure) state in the form of an expectation value,

 $\Psi(a)=\langle\psi,a\psi\rangle.$ (3)

In physics, it is common to refer to such states by their vector $\psi$ rather than the linear functional $\Psi$. The converse is not always true; not every state need be given by an expectation value. For example, delta functions (which are distributions not functions) give pure states on $C_{0}(X)$, but they do not correspond to any vector in a Hilbert space (such a vector would not be square-integrable).

## References

• 1 G. Murphy, $C^{*}$-Algebras and Operator Theory. Academic Press, 1990.
Title state State 2013-03-22 13:50:18 2013-03-22 13:50:18 mhale (572) mhale (572) 8 mhale (572) Definition msc 46L05 ExtensionAndRestrictionOfStates AlgebraicQuantumFieldTheoriesAQFT pure state tracial state