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  "body": "\n## Description\n\nLet $\\ell^2(\\N)$ be the Hilbert space of square-summable sequences, with orthonormal basis $e_0, e_1, e_2, \\ldots$.\nThe unilateral shift is the operator $S$ with $S e_n = e_{n+1}$ for all $n$.\nThe *Toeplitz algebra* $\\mathcal{T}$ is the C*-subalgebra of the [[algebra of bounded operators|BMDG4JQPEH]] $B(\\ell^2(\\N))$ generated by $S$.\nIt is the norm closure of the set of polynomials in $S$ and $S^*$.\nIt is a C*-algebra by construction.\nIdentify $\\ell^2(\\N)$ with the [[Hardy space|PNNNFJVF6W]] $H^2$ of the unit circle.\nThen $\\mathcal{T}$ is the C*-algebra generated by the [[Toeplitz operators|GQ0X3R3WM2]] $T_f$ with continuous symbol $f$.\n\nCheck that $\\mathcal{T}$ contains the [[compact operators|M2NEP1K5AE]]: $S^* S = 1$, and $P = 1 - S S^*$ is the projection onto $\\C e_0$.\nThen $S^m P (S^*)^n$ is the rank-one operator sending $e_n$ to $e_m$ and the other basis vectors to $0$.\nThese matrix units lie in $\\mathcal{T}$, and the norm closure of their span is the [[ideal of compact operators|2WAZF1RV2Y]] $K$.\n\n## Remarks\n\nThe quotient $\\mathcal{T} / K$ is isomorphic to the algebra $C(\\mathbb{T})$ of continuous functions on the unit circle, by a map that sends the class of $T_f$ to $f$.\nThis gives the exact sequence $0 \\to K \\to \\mathcal{T} \\to C(\\mathbb{T}) \\to 0$, often called the Toeplitz extension.\nCoburn's theorem states that the C*-algebra generated by any isometry that is not unitary is isomorphic to $\\mathcal{T}$, by an isomorphism that sends the isometry to $S$.\nThe index of the Fredholm operator $T_f$, for a continuous nowhere-vanishing symbol $f$, is minus the winding number of $f$ around $0$.\n\n## Explanations\n\n### L1 human\n\nThe shift operator moves each entry of a sequence one place to the right and puts a zero in front.\nIt almost has an inverse, the backward shift, but shifting back and then forward loses the first entry.\nThe Toeplitz algebra is everything one can build from these two operators with sums, products and limits.\nModulo the compact operators, the shift becomes multiplication by $z$ on the circle, so the algebra links operators with functions on the circle and with winding numbers.\n",
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