Proposition 30.5.2.label The Gelfand transform of $\ell^{1}(\integer)$ is not isometric.
Proof. Let $f = \one_{\bracs{n = 1}}- \one_{\bracs{2 \le n \le 3}}$, then
\[f^{2}(n) = \begin{cases}-1 &n \in \bracs{1, 5} \\ -2 &n = 2 \\ -1 &n = 3 \\ 2 &n = 4 \\ 0 &n \not\in [1, 5]\end{cases}\]
so $\normn{f^2}_{\ell^1(\integer)}= 7 < \normn{f}_{\ell^1(\integer)}^{2}$. By Proposition 29.8.3, the Gelfand transform is not isometric.$\square$
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