# invertible elements in a Banach algebra form an open set

Theorem - Let $\mathcal{A}$ be a Banach algebra with identity element^{} $e$ and $G(\mathcal{A})$ be the set of invertible elements in $\mathcal{A}$. Let ${B}_{r}(x)$ denote the open ball of radius $r$ centered in $x$.

Then, for all $x\in G(\mathcal{A})$ we have that

$${B}_{{\parallel {x}^{-1}\parallel}^{-1}}(x)\subseteq G(\mathcal{A})$$ |

and therefore $G(\mathcal{A})$ is open in $\mathcal{A}$.

Proof : Let $x\in G(\mathcal{A})$ and $y\in {B}_{{\parallel {x}^{-1}\parallel}^{-1}}(x)$. We have that

$$ |

So, by the Neumann series (http://planetmath.org/NeumannSeriesInBanachAlgebras) we conclude that $e-(e-{x}^{-1}y)$ is invertible^{},
i.e. ${x}^{-1}y\in G(\mathcal{A})$.

As $G(\mathcal{A})$ is a group we must have $y\in G(\mathcal{A})$.

So ${B}_{{\parallel {x}^{-1}\parallel}^{-1}}(x)\subseteq G(\mathcal{A})$ and the theorem follows. $\mathrm{\square}$

Title | invertible elements in a Banach algebra form an open set |
---|---|

Canonical name | InvertibleElementsInABanachAlgebraFormAnOpenSet |

Date of creation | 2013-03-22 17:23:22 |

Last modified on | 2013-03-22 17:23:22 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 6 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 46H05 |