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The trace

$\begin{prop} There is an unique trace $\tau \: \bCl(V) \to \bC$ such that $\tau(1) = 1$ and $\tau(a) = 0$ for $a$ odd. \end{prop}$

$\begin{proof} If $\{e_1, \dots, e_n\}$ is an orthonormal basis for $(V, g)$, th... ...hus $\tau(a) = a_\emptyset$ does not depend on $\{e_1,\dots,e_n\}$. \end{proof}$

$\begin{remark} % latex2html id marker 246At this point, it was remarked that f... ...ace} is useful in that it establishes the uniqueness of the trace. \end{remark}$

Now $\bCl(V)$ is a Hilbert space with scalar product

$\begin{displaymath} \braket{a}{b} := \tau(a^* b). \end{displaymath}$

Pawel Witkowski 2006-03-14