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  "body": "\n## Summary\n\nThe C* identity is the condition $\\norm{a^* a} = \\norm{a}^2$ on a Banach algebra with an involution.\nA Banach $*$-algebra whose norm satisfies it is a C*-algebra.\nThis one equation ties the norm to the algebra so tightly that the algebraic structure determines the norm.\nIndeed, $\\norm{a}$ is the square root of the spectral radius of $a^* a$.\n\n## Explanations\n\n### L1 human\n\nFor a complex number, $\\abs{\\overline{z} z} = \\abs{z}^2$.\nThe C* identity asks the same of a whole algebra of objects, such as matrices or operators, where $a^*$ plays the role of the complex conjugate.\nFor a matrix with the operator norm and the conjugate transpose as $a^*$, the identity holds.\nAlgebras with this property behave like algebras of operators on a Hilbert space, and in fact every one of them is such an algebra.\n",
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