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  "body": "\n## Definition\n\nLet $A$ be a commutative complex Banach algebra.\nA *character* of $A$ is a nonzero linear map $\\varphi \\colon A \\to \\C$ with $\\varphi(ab) = \\varphi(a) \\varphi(b)$ for all $a, b \\in A$.\nThe *character space* $\\Delta(A)$ of $A$ is the set of characters of $A$.\nEach character is continuous with $\\norm{\\varphi} \\le 1$, so $\\Delta(A)$ is a subset of the [[continuous dual space|4A82S5FVRT]] $A^*$.\nThe topology on $\\Delta(A)$ is the relative weak-* topology, the [[weak topology|P6KHVSTPZV]] of the maps $\\varphi \\mapsto \\varphi(a)$, $a \\in A$.\n\n## Remarks\n\nOther names are the *maximal ideal space* and the *Gelfand spectrum*, written $\\sigma(A)$ or $\\mathfrak{M}(A)$.\nThe space $\\Delta(A)$ is locally compact and Hausdorff.\nIf $A$ has a unit, then $\\varphi(1) = 1$ for every character, $\\Delta(A)$ is weak-* closed in the unit ball of $A^*$, and so it is compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]].\nFor unital $A$, the map $\\varphi \\mapsto \\ker \\varphi$ is a bijection from $\\Delta(A)$ onto the set of [[maximal ideals|0PV74TR5FP]] of $A$; the [[Gelfand–Mazur theorem|3D6XH41A3B]] shows that each quotient $A / \\mathfrak{m}$ is $\\C$.\n\nExample: let $X$ be a compact Hausdorff space and $A = C(X)$ with the sup norm.\nFor $x \\in X$, the evaluation $\\varphi_x(f) = f(x)$ is linear and multiplicative, and $\\varphi_x(1) = 1$, so it is a character.\nEvery character of $C(X)$ has this form, and $x \\mapsto \\varphi_x$ is a homeomorphism from $X$ onto $\\Delta(A)$.\n\nNon-example: on $C([0,1])$, the map $\\varphi(f) = \\int_0^1 f(t) \\, dt$ is linear and nonzero but not multiplicative.\nFor $f(t) = t$ one has $\\varphi(f^2) = 1/3$, while $\\varphi(f)^2 = 1/4$.\n",
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