# state

A state $\mathrm{\Psi}$ on a ${C}^{*}$-algebra $A$ is a positive linear functional^{}
$\mathrm{\Psi}:A\to \u2102$, $\mathrm{\Psi}({a}^{*}a)\ge 0$ for all $a\in A$, with unit norm.
The norm of a positive linear functional is defined by

$$\parallel \mathrm{\Psi}\parallel =\underset{a\in A}{sup}\{|\mathrm{\Psi}(a)|:\parallel a\parallel \le 1\}.$$ | (1) |

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

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

$$\lambda {\mathrm{\Psi}}_{1}+(1-\lambda ){\mathrm{\Psi}}_{2},\lambda \in [0,1],$$ | (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^{} $\mathscr{H}$,
every unit vector^{} $\psi \in \mathscr{H}$ determines a (not necessarily pure) state in the form of an expectation value,

$$\mathrm{\Psi}(a)=\u27e8\psi ,a\psi \u27e9.$$ | (3) |

In physics, it is common to refer to such states by their vector $\psi $ rather than the linear functional^{} $\mathrm{\Psi}$.
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 |
---|---|

Canonical name | State |

Date of creation | 2013-03-22 13:50:18 |

Last modified on | 2013-03-22 13:50:18 |

Owner | mhale (572) |

Last modified by | mhale (572) |

Numerical id | 8 |

Author | mhale (572) |

Entry type | Definition |

Classification | msc 46L05 |

Related topic | ExtensionAndRestrictionOfStates |

Related topic | AlgebraicQuantumFieldTheoriesAQFT |

Defines | pure state |

Defines | tracial state |