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  "body": "\n## Description\n\nLet $H$ be an infinite-dimensional separable complex [[Hilbert space|HM2HHE3J51]].\nLet $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$, and let $K(H)$ be the [[ideal of compact operators|2WAZF1RV2Y]] in it.\nThe *Calkin algebra* of $H$ is the [[quotient|9240K052CS]] $Q(H) = B(H) / K(H)$.\nIt carries the quotient norm and the induced product.\nIts involution is $\\pi(T)^* = \\pi(T^*)$, where $\\pi \\colon B(H) \\to Q(H)$ is the quotient map.\n\n## Remarks\n\nSince $K(H)$ is a closed two-sided ideal of $B(H)$ that is closed under adjoints, $Q(H)$ is a unital C*-algebra.\nCalkin showed that for separable $H$, $K(H)$ is the only closed two-sided ideal of $B(H)$ other than $\\set{0}$ and $B(H)$, so $Q(H)$ is a simple C*-algebra.\nAn operator $T \\in B(H)$ is Fredholm exactly when $\\pi(T)$ is invertible in $Q(H)$.\nThe spectrum of $\\pi(T)$ in $Q(H)$ is the essential spectrum of $T$.\nThe algebra $Q(H)$ is infinite-dimensional and not separable as a normed space.\nThe construction makes sense for any Hilbert space, but for non-separable $H$ the ideal $K(H)$ is no longer the only nontrivial closed ideal.\n\n## Explanations\n\n### L1 human\n\nThe Calkin algebra is what remains of the bounded operators when compact operators are treated as zero.\nTwo operators are the same element of it when they differ by a compact operator, a small error in a precise sense.\nIn this algebra, the right shift on sequences becomes invertible, since it has an inverse up to a rank-one error.\nIt is the natural home of the Fredholm index and of the essential spectrum.\n",
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