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  "body": "\n## Summary\n\nThe character space of a commutative complex Banach algebra $A$ is the set of nonzero algebra homomorphisms $A \\to \\C$, with the weak-* topology.\nIt is also called the maximal ideal space or the spectrum of $A$.\nFor unital $A$, the characters correspond to the [[maximal ideals|0PV74TR5FP]] of $A$.\nIt is a locally compact Hausdorff space, compact when $A$ has a unit.\nIt is the domain of the [[Gelfand transform|VWNSHKJT92]], which turns elements of $A$ into continuous functions.\n\n## Explanations\n\n### L1 human\n\nA character is a way to turn each element of the algebra into a single complex number while respecting sums and products.\nFor the algebra of continuous functions on a compact space $X$, the characters are exactly the evaluations $f \\mapsto f(x)$ at points $x \\in X$.\nSo the character space of $C(X)$ is $X$ itself.\nFor a general commutative Banach algebra, the character space is the space of \"points\" on which its elements become functions.\n",
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