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  "body": "\n## Summary\n\nThe Gelfand transform turns each element $a$ of a commutative complex Banach algebra into a continuous function $\\hat a$ on the [[character space|M7JVFJ3MMX]], by $\\hat a(\\varphi) = \\varphi(a)$.\nIt is an algebra homomorphism into the functions vanishing at infinity, and it does not increase norms.\nFor a unital algebra, the range of $\\hat a$ is the spectrum of $a$.\nFor commutative C*-algebras it is an isometric isomorphism onto all such functions, by the [[commutative Gelfand–Naimark theorem|NT564Y315M]].\n\n## Explanations\n\n### L1 human\n\nThe idea is to view the elements of an abstract commutative algebra as functions.\nThe points are the characters, and the value of $a$ at a character $\\varphi$ is the number $\\varphi(a)$.\nFor sequences $(a_n)_{n \\in \\Z}$ with $\\sum \\abs{a_n} \u003c \\infty$ and convolution as product, the characters are the points $z$ of the unit circle.\nThere the Gelfand transform is the Fourier series $\\sum_n a_n z^n$, which turns convolution into ordinary multiplication.\n",
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