On group representations whose $C^*$ algebra is an ideal in its von Neumann algebra

Granirer, Edmond E.

On group representations whose

C^{*}

algebra is an ideal in its von Neumann algebra

Granirer, Edmond E.

Annales de l'Institut Fourier, Tome 29 (1979), p. 37-52 / Harvested from Numdam

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Résumé
Abstract

Soit $τ$ une représentation unitaire continue d’un groupe $G$ localement compact sur l’espace de Hilbert $H_{τ}$ . Soit $C_{τ}^{*} [V N_{τ}]$ la $C^{*} [W^{*}]$ algèbre engendrée par

(L^{1} (G)) et M_{τ} (C_{τ}^{*}) = \{ϕ \in V N_{τ}; ϕ C_{τ}^{*} + C_{τ}^{*} ϕ \subset C_{τ}^{*}\} .

On obtient le théorème 1 :

Si $G$ est $σ$ -compact et $M_{τ} (C_{τ}^{*}) = V N_{τ}$ , alors le support de $τ$ est discret et chaque $π$ dans sup $τ$ est CCR.

Nous utilisons ce résultat dans le cas de la représentation quasi-régulière $τ = π_{H}$ . Cela nous permet d’obtenir, entre autres résultats, que $M_{π_{H}} (C_{π_{H}}^{*}) = V N_{π_{H}}$ impliquerait dans plusieurs cas que $G / H$ est compact.

Let $τ$ be a continuous unitary representation of the locally compact group $G$ on the Hilbert space $H_{τ}$ . Let $C_{τ}^{*} [V N_{τ}]$ be the $C^{*} [W^{*}]$ algebra generated by

(L^{1} (G)) and M_{τ} (C_{τ}^{*}) = \{ϕ \in V N_{τ}; ϕ C_{τ}^{*} + C_{τ}^{*} ϕ \subset C_{τ}^{*}\} .

The main result obtained in this paper is Theorem 1:

If $G$ is $σ$ -compact and $M_{τ} (C_{τ}^{*}) = V N_{τ}$ then supp $τ$ is discrete and each $π$ in supp $τ$ in CCR.

We apply this theorem to the quasiregular representation $τ = π_{H}$ and obtain among other results that $M_{π_{H}} (C_{π_{H}}^{*}) = V N_{π_{H}}$ implies in many cases that $G / H$ is a compact coset space.

Publié le : 1979-01-01

MR 81b:22007 | Zbl 0403.46048

DOI : https://doi.org/10.5802/aif.765

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     author = {Granirer, Edmond E.},
     title = {On group representations whose $C^*$ algebra is an ideal in its von Neumann algebra},
     journal = {Annales de l'Institut Fourier},
     volume = {29},
     year = {1979},
     pages = {37-52},
     doi = {10.5802/aif.765},
     mrnumber = {81b:22007},
     zbl = {0403.46048},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1979__29_4_37_0}
}

Granirer, Edmond E. On group representations whose $C^*$ algebra is an ideal in its von Neumann algebra. Annales de l'Institut Fourier, Tome 29 (1979) pp. 37-52. doi : 10.5802/aif.765. http://gdmltest.u-ga.fr/item/AIF_1979__29_4_37_0/

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