@prefix psr: . @prefix skos: . @prefix dc: . @prefix xsd: . psr:-V40NLNFZ-K skos:prefLabel "C*-algebra"@en, "C*-algèbre"@fr ; a skos:Concept ; skos:narrower psr:-NSPWMCHC-R . psr:-DWGDCN0G-R skos:prefLabel "operator algebra"@en, "algèbre d'opérateurs"@fr ; a skos:Concept ; skos:narrower psr:-NSPWMCHC-R . psr:-NSPWMCHC-R skos:exactMatch , ; skos:prefLabel "algèbre de Toeplitz"@fr, "Toeplitz algebra"@en ; skos:inScheme psr: ; a skos:Concept ; skos:definition """En théorie des algèbres d'opérateurs, l'algèbre de Toeplitz

{\\\\displaystyle {\\\\mathcal {T}}}

est la C*-algèbre universelle engendrée par une isométrie

{\\\\displaystyle S}

non unitaire. En clair, ce générateur vérifie :

{\\\\displaystyle S^{*}S=1\\\\qquad {\\ ext{mais}}\\\\qquad SS^{*}\\ eq 1.}

Si on définit l'élément

{\\\\displaystyle P}

de cette algèbre par

{\\\\displaystyle P:=1-SS^{*}}

, on obtient, comme pour toute isométrie, les relations :

{\\\\displaystyle PS=0\\\\qquad {\\ ext{et}}\\\\qquad S^{*}P=0.}

(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Toeplitz)"""@fr, """In operator algebras, the Toeplitz algebra is the C*-algebra generated by the unilateral shift on the Hilbert space l²(N). Taking l²(N) to be the Hardy space H², the Toeplitz algebra consists of elements of the form

{\\\\displaystyle T_{f}+K\\\\;}

where T_f is a Toeplitz operator with continuous symbol and K is a compact operator. Toeplitz operators with continuous symbols commute modulo the compact operators. So the Toeplitz algebra can be viewed as the C*-algebra extension of continuous functions on the circle by the compact operators. This extension is called the Toeplitz extension. By Atkinson's theorem, an element of the Toeplitz algebra T_f + K is a Fredholm operator if and only if the symbol f of T_f is invertible. In that case, the Fredholm index of T_f + K is precisely the winding number of f, the equivalence class of f in the fundamental group of the circle. This is a special case of the Atiyah-Singer index theorem. Wold decomposition characterizes proper isometries acting on a Hilbert space. From this, together with properties of Toeplitz operators, one can conclude that the Toeplitz algebra is the universal C*-algebra generated by a proper isometry; this is Coburn's theorem.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Toeplitz_algebra)"""@en ; skos:broader psr:-V40NLNFZ-K, psr:-DWGDCN0G-R, psr:-PTRWRXTF-0 ; dc:created "2023-08-04"^^xsd:date ; dc:modified "2024-10-18"^^xsd:date . psr:-PTRWRXTF-0 skos:prefLabel "K-théorie"@fr, "K-theory"@en ; a skos:Concept ; skos:narrower psr:-NSPWMCHC-R . psr: a skos:ConceptScheme .