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.

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.

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.

@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},
     zbl = {0403.46048},
     mrnumber = {81b:22007},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.765/}
}
TY  - JOUR
AU  - Granirer, Edmond E.
TI  - On group representations whose $C^*$ algebra is an ideal in its von Neumann algebra
JO  - Annales de l'Institut Fourier
PY  - 1979
SP  - 37
EP  - 52
VL  - 29
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.765/
DO  - 10.5802/aif.765
LA  - en
ID  - AIF_1979__29_4_37_0
ER  - 
%0 Journal Article
%A Granirer, Edmond E.
%T On group representations whose $C^*$ algebra is an ideal in its von Neumann algebra
%J Annales de l'Institut Fourier
%D 1979
%P 37-52
%V 29
%N 4
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.765/
%R 10.5802/aif.765
%G en
%F 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) no. 4, pp. 37-52. doi : 10.5802/aif.765. https://aif.centre-mersenne.org/articles/10.5802/aif.765/

[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

Cité par Sources :