Banach algebra

Definition 1.

A Banach algebra $𝒜$ is a Banach space (over $ℂ$) with an multiplication law compatible with the norm which turns $𝒜$ into an algebra. Compatibility with the norm means that, for all $a,b\in 𝒜$, 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 $𝒜$ with a map ${}^{*}:𝒜\to 𝒜$ which satisfies the following properties:

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

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

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

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