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  "body": "\n## Summary\n\nLet $A$ be a commutative unital Banach algebra.\nIts Shilov boundary is the smallest closed subset of the [[character space|M7JVFJ3MMX]] on which every [[Gelfand transform|VWNSHKJT92]] attains its maximum modulus.\nIt is an abstract version of the maximum modulus principle of complex analysis.\nFor the disc algebra it is the unit circle, and for $C(X)$ it is all of $X$.\n\n## Explanations\n\n### L1 human\n\nA holomorphic function on the closed unit disc takes its largest absolute value on the boundary circle.\nSo to know how large such a function can be, it suffices to look at the circle.\nThe Shilov boundary is the smallest closed set with this property, defined for any commutative Banach algebra with unit.\nFor the disc algebra the circle cannot be made smaller: for each point of the circle there is a function in the algebra that peaks exactly there.\n",
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