Jones’ representations of R. Thompson’s groups not induced by finite-dimensional ones
[Représentations de Jones des groupes de R. Thompson qui ne sont pas induites par celles de dimension finie]
Brothier, Arnaud1 ; Wijesena, Dilshan1
1 School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052 (Australia)
Annales de l'Institut Fourier, Online first, 40 p.

Étant donnée une isométrie linéaire entre un espace de Hilbert et sa somme directe avec lui-même on peut construire explicitement une représentation unitaire du groupe F de Richard Thompson. Nous définissons une condition sur l’isométrie qui assure que la représentation associée ne contienne pas de représentations induites par une représentation de dimension finie. Il s’agit du premier résultat de ce type. Nous illustrons ce théorème à l’aide d’une famille de représentations avec la propriété susdite qui est indexée par la sphère réelle de dimension 3 à laquelle on a retiré deux cercles.

Given any linear isometry from a Hilbert space to its square one can explicitly construct a so-called Pythagorean unitary representation of Richard Thompson’s group F. We introduce a condition on the isometry implying that the associated representation does not contain any representation induced by finite-dimensional ones. This provides the first result of this kind. We illustrate this theorem via a family of representations parameterized by the real 3-sphere for which all of them have this property except on two sub-circles.

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DOI : 10.5802/aif.3689
Classification : 20C99
Keywords: Thompson’s groups, unitary representations, Jones’ representations, mixing, fraction groups, Pythagorean C*-algebras
Mots-clés : Groupes de Thompson, représentations unitaires, représentations de Jones, mélange, groupes de fractions, C*-algèbres pythagoriciennes

Brothier, Arnaud 1 ; Wijesena, Dilshan 1

1 School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052 (Australia) 
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Brothier, Arnaud; Wijesena, Dilshan. Jones’ representations of R. Thompson’s groups not induced by finite-dimensional ones. Annales de l'Institut Fourier, Online first, 40 p.

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