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# Banach algebra

###### Definition 1.

A Banach algebra $\mathcal{A}$ is a Banach space (over $\mathbb{C}$) 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:

$\|ab\|\leq\|a\|\,\|b\|$ |

###### Definition 2.

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

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

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

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

$\displaystyle(\lambda a)^{*}$ | $\displaystyle=$ | $\displaystyle\bar{\lambda}a^{*}\quad\forall\lambda\in\mathbb{C},$ | (4) | ||

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

where $\bar{\lambda}$ 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.

Related:

ExampleOfLinearInvolution, GelfandTornheimTheorem, MultiplicativeLinearFunctional, TopologicalAlgebra

Synonym:

B-algebra, Banach *-algebra, B*-algebra, $B^*$-algebra

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

46H05*no label found*

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