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

Granirer, Edmond E.

doi:10.5802/aif.765

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.

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.

EuDML MR Zbl

DOI : 10.5802/aif.765

@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},
     pages = {37--52},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {29},
     number = {4},
     year = {1979},
     doi = {10.5802/aif.765},
     mrnumber = {81b:22007},
     zbl = {0403.46048},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.765/}
}

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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) no. 4, pp. 37-52. doi : 10.5802/aif.765. http://www.numdam.org/articles/10.5802/aif.765/

Bibliographie
Cité par

[1] C. A. Akeman, G. K. Pedersen, J. Tomiyama, Multipliers of C*-Algebras, J. of Functional Analysis, 13 (1973), 277-301. | MR | Zbl

[2] C. A. Akeman, M. E. Walter, Nonabelian Pontriagin duality, Duke Math. J., 39 (1972), 451-463. | Zbl

[3] G. Arsac, Sur l'espace de Banach engendré par les coefficients d'une représentation unitaire, Thèse, Université Claude-Bernard, Lyon I, 1973.

[4] M. C. F. Berglund, Ideal C*-algebras, Duke Math. J., 40 (1973), 241-257. | MR | Zbl

[5] R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach, 1970. | MR | Zbl

[6] J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, Paris, 2e edition, 1969. | Zbl

[7] P. Eymard, l'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France, 92 (1964), 181-236. | Numdam | MR | Zbl

[8] P. Eymard, Moyennes Invariantes et Représentations Unitaires, Lecture notes in mathematics, Springer Verlag, 1972, n° 300. | MR | Zbl

[9] Steven A. Gaal, Linear Analysis and Representation Theory, Springer Verlag, 1973. | Zbl

[10] E. Granirer, Density theorems for some linear subspaces and some C*-subalgebras of VN(G), Proc. of Symp. on Harmonic Analysis and Function Spaces, Inst. Nazionale di Alta Matematica Rome March 1976, pp. 61-70. | MR | Zbl

[11] G. D. Mostow, Homogeneous spaces with finite invariant measure, Annals of Math., 75 (1962), 17-37. | MR | Zbl

[12] G. D. Mostow, Factor spaces of solvable groups, Ann. of Math., 60 (1954), 1-27. | MR | Zbl

[13] S. Sakai, On topological properties of W*-algebras, Proc. Japan Acad., 33 (1957), 439-444. | MR | Zbl

[14] I. Schochetman, Topology and the duals of certain locally compact groups, Trans. AMS, 150 (1970), 477-489. | MR | Zbl

[15] M. E. Walter, W*-algebras and nonabelian harmonic analysis, J. of Functional Analysis, 11 (1972), 17-38. | MR | Zbl

[16] S. P. Wang, On S-subgroups of solvable Lie groups, Amer. J. Math., 92 (1970), 389-397. | MR | Zbl

[17] S. P. Wang, On isolated points in the dual spaces of locally compact groups, Math. Annalen, 218 (1975), 19-34. | MR | Zbl

[18] S. P. Wang, Compactness properties of topological groups III, Trans AMS, 209 (1975), 399-418. | MR | Zbl

[19] S. P. Wang, Homogeneous spaces with invariant measure, Amer. J. Math., 98 (1976), 311-324. | MR | Zbl

[20] L. Baggett, A separable group having a discrete dual space is compact, J. Functional Analysis, 10 (1972), 131-148. | MR | Zbl

[21] J. M. G. Fell, A new look at Mackey's imprimitivity theorem, Proc. Conference on Harmonic Analysis Maryland 1971, Springer Lecture Notes, Vol. 266, 42-58. | Zbl

[22] G.W. Mackey, Induced representations of locally compact groups I, Annals of Math., 55 (1952), 101-139. | MR | Zbl

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