# Banach algebra

###### Definition 1.

A Banach algebra^{} $\mathcal{A}$ is a Banach space^{} (over $\u2102$) with an multiplication law compatible with the norm which turns $\mathcal{A}$ into an algebra. Compatibility with the norm means that, for all $a,b\in \mathcal{A}$, it is the case that the following product inequality holds:

$$\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel $$ |

###### Definition 2.

A Banach *-algebra is a Banach algebra $\mathcal{A}$ with a map ${}^{*}:\mathcal{A}\to \mathcal{A}$ which satisfies the following properties:

${a}^{**}$ | $=$ | $a,$ | (1) | ||

${(ab)}^{*}$ | $=$ | ${b}^{*}{a}^{*},$ | (2) | ||

${(a+b)}^{*}$ | $=$ | ${a}^{*}+{b}^{*},$ | (3) | ||

${(\lambda a)}^{*}$ | $=$ | $\overline{\lambda}{a}^{*}\mathit{\hspace{1em}}\forall \lambda \in \u2102,$ | (4) | ||

$\parallel {a}^{*}\parallel $ | $=$ | $\parallel a\parallel ,$ | (5) |

where $\overline{\lambda}$ is the complex conjugation of $\lambda $. In other words, the operator ${}^{*}$ is an involution.

###### Example 1

The algebra of bounded operators^{} on a Banach space is a Banach algebra
for the operator norm.

Title | Banach algebra |
---|---|

Canonical name | BanachAlgebra |

Date of creation | 2013-03-22 12:57:52 |

Last modified on | 2013-03-22 12:57:52 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 12 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 46H05 |

Synonym | B-algebra |

Synonym | Banach *-algebra |

Synonym | B*-algebra |

Synonym | ${B}^{*}$-algebra |

Related topic | ExampleOfLinearInvolution |

Related topic | GelfandTornheimTheorem |

Related topic | MultiplicativeLinearFunctional |

Related topic | TopologicalAlgebra |