# norm and spectral radius in ${C}^{*}$-algebras

Let $\mathcal{A}$ be a ${C}^{*}$-algebra (http://planetmath.org/CAlgebra). Let ${R}_{\sigma}(a)$ denote the spectral radius of the element $a\in \mathcal{A}$.

Theorem - For every $a\in \mathcal{A}$ we have that $\parallel a\parallel =\sqrt{{R}_{\sigma}({a}^{*}a)}$.

This result shows that the norm in a ${C}^{*}$-algebra has a purely nature. Moreover, the norm in a ${C}^{*}$-algebra is unique (in the sense that there is no other norm for which the algebra is a ${C}^{*}$-algebra).

Title | norm and spectral radius in ${C}^{*}$-algebras |
---|---|

Canonical name | NormAndSpectralRadiusInCalgebras |

Date of creation | 2013-03-22 17:38:41 |

Last modified on | 2013-03-22 17:38:41 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 7 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46L05 |

Related topic | HomomorphismsOfCAlgebrasAreContinuous |

Related topic | CAlgebra |