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Banach algebra (Definition)
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:
$\displaystyle \Vert ab\Vert \leq \Vert a\Vert \,\Vert b\Vert $
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)
$\displaystyle \Vert a^*\Vert$ $\displaystyle =$ $\displaystyle \Vert a\Vert ,$ (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.



"Banach algebra" is owned by rspuzio. [ full author list (3) | owner history (1) ]
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See Also: example of linear involution, Gelfand-Tornheim theorem, multiplicative linear functional, topological $*$-algebra

Other names:  B-algebra, Banach *-algebra, B*-algebra, $B^*$-algebra

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invertible elements in a Banach algebra form an open set (Theorem) by asteroid
$L^1(G)$ is a Banach *-algebra (Example) by asteroid
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Cross-references: operator norm, bounded operators, involution, operator, complex conjugation, properties, map, inequality, product, algebra, norm, compatible, multiplication, Banach space
There are 39 references to this entry.

This is version 9 of Banach algebra, born on 2002-08-23, modified 2007-08-29.
Object id is 3333, canonical name is BanachAlgebra.
Accessed 10852 times total.

Classification:
AMS MSC: 46H05 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: General theory of topological algebras)
Pending Errata and Addenda
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