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Banach algebra
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(Definition)
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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:$$ \norm{ab} \leq \norm{a}\,\norm{b}$$
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: \begin{eqnarray} a^{**} & = & a, \\ (ab)^* & = & b^* a^*, \\ (a+b)^* & = & a^* + b^*, \\ (\lambda a)^* & = & \bar{\lambda} a^* \quad\forall\lambda\in\Cset, \\ \norm{a^*} & = & \norm{a}, \end{eqnarray}
where $\bar{\lambda}$ is the complex conjugation of $\lambda$ . In other words, the operator $^*$ is an involution.
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"Banach algebra" is owned by rspuzio. [ full author list (3) | owner history (1) ]
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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 41 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 14312 times total.
Classification:
| AMS MSC: | 46H05 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: General theory of topological algebras) |
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Pending Errata and Addenda
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