= Involution alone does not imply an isometric Gelfand transform
The complex <dual numbers> $\mathbb C[\varepsilon]/(\varepsilon^2)$, with norm $\|a+b\varepsilon\|=|a|+|b|$ and involution $a+b\varepsilon\mapsto\bar a+\bar b\varepsilon$, is a commutative unital <Banach algebra> with an isometric involution. Its sole <character of an algebra> kills $\varepsilon$, so the <Gelfand transform> is neither injective nor isometric. The <C-star identity> excludes this example and gives the <Commutative Gelfand--Naimark theorem>. Even preservation of the involution on characters can fail without that identity: on $\mathbb C^2$, the involution $(a,b)^*=(\bar b,\bar a)$ exchanges the two evaluation characters.
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