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Revision difference : Banach algebra |
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Version 1 |
| \begin{definition} |
\begin{definition} |
| A \defn{Banach algebra} is a Banach space with a multiplication law |
A \defn{Banach algebra} is a Banach space with a multiplication law |
| compatible with the norm, i.e.\ $||ab|| \leq ||a||\,||b||$ (product inequality). |
compatible with the norm, i.e.\ $||ab|| \leq ||a||\,||b||$ (product inequality). |
| \end{definition} |
\end{definition} |
| \begin{definition} |
\begin{definition} |
| A \defn{Banach *-algebra} is a Banach algebra with an involution ${}^*$ |
A \defn{Banach *-algebra} is a Banach algebra with an involution ${}^*$ |
| satisfying the following properties: |
satisfying the following properties: |
| \begin{eqnarray} |
\begin{eqnarray} |
| a^{**} & = & a, \\ |
a^{**} & = & a, \\ |
| (ab)^* & = & b^* a^*, \\ |
(ab)^* & = & b^* a^*, \\ |
| (\lambda a+\mu b)^* & = & \bar{\lambda} a^*+\bar{\mu} b^* \quad\forall\lambda,\mu\in\Cset, \\ |
(\lambda a+\mu b)^* & = & \bar{\lambda} a^*+\bar{\mu} b^* \quad\forall\lambda,\mu\in\Cset, \\ |
| ||a^*|| & = & ||a||. |
||a^*|| & = & ||a||. |
| \end{eqnarray} |
\end{eqnarray} |
| \end{definition} |
\end{definition} |
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