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) no. 4, pp. 37-52.

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

$$(L^1\left(G\right))\text{et}{M}_{\tau }(C_{\tau }^{*})=\left\{\varphi \in V{N}_{\tau };\varphi C_{\tau }^{*}+C_{\tau }^{*}\varphi \subset C_{\tau }^{*}\right\}.$$

On obtient le théorème 1 :

Si $G$ est $\sigma$-compact et $M_{\tau }(C_{\tau }^{*})=V{N}_{\tau }$, alors le support de $\tau$ est discret et chaque $\pi$ dans sup $\tau$ est CCR.

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

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

$$(L^1\left(G\right))\text{and}{M}_{\tau }(C_{\tau }^{*})=\left\{\varphi \in V{N}_{\tau };\varphi C_{\tau }^{*}+C_{\tau }^{*}\varphi \subset C_{\tau }^{*}\right\}.$$

The main result obtained in this paper is Theorem 1:

If $G$ is $\sigma$-compact and $M_{\tau }(C_{\tau }^{*})=V{N}_{\tau }$ then supp $\tau$ is discrete and each $\pi$ in supp $\tau$ in CCR.

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

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