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  "body": "\n## Statement\n\nLet $A$ be a complex Banach algebra with unit $1 \\ne 0$.\nSuppose every nonzero element of $A$ is invertible, so that $A$ is a [[division ring|0WZXK05ME7]].\nThen every $a \\in A$ equals $\\lambda 1$ for a unique $\\lambda \\in \\C$, and the map $\\C \\to A$, $\\lambda \\mapsto \\lambda 1$, is an isomorphism of algebras.\n\n## Remarks\n\nProof idea: the spectrum $\\set{\\lambda \\in \\C : a - \\lambda 1 \\text{ is not invertible}}$ of each $a \\in A$ is nonempty.\nSuppose it were empty, and let $\\varphi$ be a continuous linear functional on $A$.\nThen $\\lambda \\mapsto \\varphi((a - \\lambda 1)^{-1})$ would be a bounded entire function that tends to $0$, hence zero by Liouville's theorem.\nAs this holds for every $\\varphi$, the Hahn–Banach theorem would give $(a - \\lambda 1)^{-1} = 0$, a contradiction.\nSo some $a - \\lambda 1$ is not invertible; in a division algebra this forces $a - \\lambda 1 = 0$.\nThe complex scalars are needed.\nThe quaternions $\\mathbb{H}$ with the Euclidean norm on $\\R^4$ form a real Banach algebra in which every nonzero element is invertible, but $\\mathbb{H}$ is not $\\R$ or $\\C$.\nConsequence: a field that contains $\\C$ and has an absolute value extending the usual one on $\\C$, complete for it, is $\\C$ itself.\n\n## Explanations\n\n### L1 human\n\nA Banach algebra is a space where one can add, multiply, scale by complex numbers and measure size, and where limits exist.\nThe theorem says that if such an algebra lets you divide by every nonzero element, it is nothing more than the complex numbers.\nSo there is no room for a bigger \"complex number system\" with a good notion of size.\n\n### L2 human\n\nThe theorem is a basic step in the theory of commutative Banach algebras.\nIf $A$ is a commutative unital complex Banach algebra and $\\mathfrak{m}$ a maximal ideal, then $\\mathfrak{m}$ is closed and $A/\\mathfrak{m}$ is a Banach division algebra, hence $\\C$.\nThis gives the characters $A \\to \\C$ that underlie the Gelfand transform.\nSome proofs of Ostrowski's theorem, that a field complete for an Archimedean absolute value is $\\R$ or $\\C$, use the Gelfand–Mazur theorem.\n",
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