# extension and restriction of states

## 0.1 Restriction of States

Let $\mathcal{A}$ be a ${C}^{*}$-algebra (http://planetmath.org/CAlgebra) and $\mathcal{B}\subset \mathcal{A}$ a ${C}^{*}$-subalgebra, both having the same identity element^{}.

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- Given a state $\varphi $ of $\mathcal{A}$, its restriction (http://planetmath.org/RestrictionOfAFunction) ${\varphi |}_{\mathcal{B}}$ to $\mathcal{B}$ is also a state of $\mathcal{B}$.

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Remark - Note that the requirement that the ${C}^{*}$-algebras $\mathcal{A}$ and $\mathcal{B}$ have a (common) identity element is necessary.

For example, let $X$ be a compact space and $C(X)$ the ${C}^{*}$-algebra of continuous functions^{} $X\to \u2102$. Pick a point ${x}_{0}\in X$ and consider the ${C}^{*}$-subalgebra of continuous functions $X\to \u2102$ which vanish at ${x}_{0}$. Notice that this subalgebra never has the same identity element of $C(X)$ (the constant function that equals $1$). In fact, this subalgebra may not have an identity^{} at all.

## 0.2 Extension of States

Let $\mathcal{A}$ be a ${C}^{*}$-algebra and $\mathcal{B}\subset \mathcal{A}$ a ${C}^{*}$-subalgebra (not necessarily unital).

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Theorem 1 - Every state $\varphi $ of $\mathcal{B}$ admits an extension^{} to a state $\stackrel{~}{\varphi}$ of $\mathcal{A}$. Moreover, every pure state $\varphi $ of $\mathcal{B}$ admits an extension to a pure state $\stackrel{~}{\varphi}$ of $\mathcal{A}$.

Theorem 2 - The set of extensions of a state $\varphi $ of $\mathcal{B}$ is a compact^{} and convex subset of ${S}_{\mathcal{A}}$, the of $\mathcal{A}$ endowed with the weak-* topology^{}.

Title | extension and restriction of states |
---|---|

Canonical name | ExtensionAndRestrictionOfStates |

Date of creation | 2013-03-22 18:09:35 |

Last modified on | 2013-03-22 18:09:35 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 10 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46L30 |

Related topic | State |