On group representations whose C * algebra is an ideal in its von Neumann algebra
Annales de l'Institut Fourier, Volume 29 (1979) no. 4, p. 37-52

Let τ be a continuous unitary representation of the locally compact group G on the Hilbert space H τ . Let C τ * [VN τ ] be the C * [W * ] algebra generated by

(L1(G))andMτ(Cτ*)=ϕVNτ;ϕCτ*+Cτ*ϕCτ*.

The main result obtained in this paper is Theorem 1:

If G is σ-compact and M τ (C τ * )=VN 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 * )=VN π H implies in many cases that G/H is a compact coset space.

Soit τ une représentation unitaire continue d’un groupe G localement compact sur l’espace de Hilbert H τ . Soit C τ * [VN τ ] la C * [W * ] algèbre engendrée par

(L1(G))etMτ(Cτ*)=ϕVNτ;ϕCτ*+Cτ*ϕCτ*.

On obtient le théorème 1 :

Si G est σ-compact et M τ (C τ * )=VN 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 * )=VN π H impliquerait dans plusieurs cas que G/H est compact.

@article{AIF_1979__29_4_37_0,
     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},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {29},
     number = {4},
     year = {1979},
     pages = {37-52},
     doi = {10.5802/aif.765},
     zbl = {0403.46048},
     mrnumber = {81b:22007},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1979__29_4_37_0}
}
On group representations whose $C^*$ algebra is an ideal in its von Neumann algebra. Annales de l'Institut Fourier, Volume 29 (1979) no. 4, pp. 37-52. doi : 10.5802/aif.765. http://www.numdam.org/item/AIF_1979__29_4_37_0/

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