| Version 6 |
Version 5 |
| \begin{definition} |
\begin{definition} |
|
A \textbf{Banach algebra} $\mathcal{A}$ is a Banach space with an multiplication law compatible with the norm which turns $A$ into an algebra. Compatibility with the norm means that, for all $a,b \in \mathcal{B}$, it is the case that the following product inequality holds:
|
A \textbf{Banach algebra} is a Banach space with a multiplication law
|
| \[ \norm{ab} \leq \norm{a}\,\norm{b} \] |
compatible with the norm, i.e.\ $\norm{ab} \leq \norm{a}\,\norm{b}$ (product inequality). |
| \end{definition} |
\end{definition} |
|
|
| \begin{definition} |
\begin{definition} |
| A \textbf{Banach *-algebra} is a Banach algebra with a map ${}^* \colon A \to A$ which satisfies the following properties: |
A \textbf{Banach *-algebra} is a Banach algebra with an involution ${}^*$ |
|
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, \\ |
| \norm{a^*} & = & \norm{a}. |
\norm{a^*} & = & \norm{a}. |
| \end{eqnarray} |
\end{eqnarray} |
| \end{definition} |
\end{definition} |
|
|
| \begin{example} |
\begin{example} |
| The algebra of bounded operators on a Banach space is a Banach algebra |
The algebra of bounded operators on a Banach space is a Banach algebra |
| for the operator norm. |
for the operator norm. |
| \end{example} |
\end{example} |
