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I have a question about the Gelfand and norm topologies on the character space of $L^{1} (\mathbb Z)$.

Are the Gelfand and norm topologies equal, on the character space of

$L^{1} (\mathbb Z)$?

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Based on Proposition 3.10, pp. 67 of the following Book

“Functional Analysis and infinite-dimensional geometry By Vaclav Zizler and …Springer 2001”,

if $X$ is an infinite-dimensional normed space, then weak and norm topologies do not coincide.

Because $L^{1}(\mathbb{Z})^{*}$ is an infinite-dimensional normed space by using the mentioned theorem the Gelfand and norm topologies are not equal.

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