# Banach algebra

###### Definition 1.

A $\mathcal{A}$ 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.

Title Banach algebra BanachAlgebra 2013-03-22 12:57:52 2013-03-22 12:57:52 rspuzio (6075) rspuzio (6075) 12 rspuzio (6075) Definition msc 46H05 B-algebra Banach *-algebra B*-algebra $B^{*}$-algebra ExampleOfLinearInvolution GelfandTornheimTheorem MultiplicativeLinearFunctional TopologicalAlgebra