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  "body": "\n## Definition\n\nLet $A$ be a complex Banach algebra with an involution $a \\mapsto a^*$, that is, a map with $(a + b)^* = a^* + b^*$, $(\\lambda a)^* = \\overline{\\lambda} a^*$, $(ab)^* = b^* a^*$ and $a^{**} = a$ for $a, b \\in A$ and $\\lambda \\in \\C$.\nThe norm of $A$ satisfies the *C\\* identity* if\n$$\\norm{a^* a} = \\norm{a}^2 \\quad \\text{for all } a \\in A.$$\nA Banach algebra with involution whose norm satisfies the C\\* identity is a C*-algebra.\n\n## Remarks\n\nThe identity implies $\\norm{a^*} = \\norm{a}$: from $\\norm{a}^2 = \\norm{a^* a} \\le \\norm{a^*} \\norm{a}$ one gets $\\norm{a} \\le \\norm{a^*}$, and the same with $a^*$ in place of $a$.\n\nExample: let $X$ be a compact Hausdorff space and $A = C(X)$ with the sup norm and $f^* = \\overline{f}$.\nThen $\\norm{\\overline{f} f}_\\infty = \\max_x \\abs{f(x)}^2 = \\norm{f}_\\infty^2$.\nThe algebra of bounded operators on a Hilbert space with the adjoint also satisfies it: $\\norm{Tx}^2 = \\ip{T^* T x}{x} \\le \\norm{T^* T} \\norm{x}^2$ gives $\\norm{T}^2 \\le \\norm{T^* T} \\le \\norm{T}^2$.\n\nNon-example: $A = \\C^2$ with coordinatewise operations, $(x, y)^* = (\\overline{x}, \\overline{y})$, and the norm $\\norm{(x, y)} = \\abs{x} + \\abs{y}$.\nIt is a Banach algebra with involution, but for $a = (1, 1)$ one has $\\norm{a^* a} = \\norm{(1, 1)} = 2$, while $\\norm{a}^2 = 4$.\n",
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