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  "body": "\n## Summary\n\nA uniform algebra is a closed subalgebra of the continuous complex functions on a compact Hausdorff space that contains the constants and separates points.\nIt is a commutative Banach algebra with the sup norm.\nThe main examples come from complex analysis, such as the disc algebra of functions holomorphic inside the disc and continuous up to the boundary.\nThe [[Stone–Weierstrass theorem|1P8D3MK79H]] says the only uniform algebra on $X$ closed under complex conjugation is $C(X)$ itself.\nSo the theory of uniform algebras studies the closed subalgebras that are not closed under conjugation.\n\n## Explanations\n\n### L1 human\n\nStart with all continuous functions on a compact space, measured by their largest absolute value.\nA uniform algebra is a smaller family that is still closed under sums, products and uniform limits, contains the constants, and can tell any two points apart.\nThe polynomials in $z$ on the closed unit disc, together with all their uniform limits, form such a family.\nIt is smaller than all continuous functions: $\\overline{z}$ is not a uniform limit of polynomials in $z$.\n",
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