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

Edmond E. Granirer

Annales de l'institut Fourier (1979)

  • Volume: 29, Issue: 4, page 37-52
  • ISSN: 0373-0956

Abstract

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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 ∈ V N τ ; φ C τ * + C τ * φ ⊂ 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.

How to cite

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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 29.4 (1979): 37-52. <http://eudml.org/doc/74432>.

@article{Granirer1979,
abstract = {Let $\tau $ be a continuous unitary representation of the locally compact group $G$ on the Hilbert space $H_\tau $. Let $C^*_\tau [VN_\tau ]$ be the $C^*[W^*]$ algebra generated by\begin\{\}(L^1(G)) \text\{and\} M\_\tau (C^*\_\tau ) = \big \lbrace \varphi \in VN\_\tau ;~\varphi C^*\_\tau + C^*\_\tau \varphi \subset C^*\_\tau \big \rbrace .\end\{\}The main result obtained in this paper is Theorem 1:If $G$ is $\sigma $-compact and $M_\tau (C^*_\tau )=VN_\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 $M_\{\pi _H\}(C^*_\{\pi _H\})=VN_\{\pi _H\}$ implies in many cases that $G/H$ is a compact coset space.},
author = {Granirer, Edmond E.},
journal = {Annales de l'institut Fourier},
keywords = {Unitary Representation; Locally Compact Group; Hilbert Space; Compact Coset Space},
language = {eng},
number = {4},
pages = {37-52},
publisher = {Association des Annales de l'Institut Fourier},
title = {On group representations whose $C^*$ algebra is an ideal in its von Neumann algebra},
url = {http://eudml.org/doc/74432},
volume = {29},
year = {1979},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 29
IS - 4
SP - 37
EP - 52
AB - Let $\tau $ be a continuous unitary representation of the locally compact group $G$ on the Hilbert space $H_\tau $. Let $C^*_\tau [VN_\tau ]$ be the $C^*[W^*]$ algebra generated by\begin{}(L^1(G)) \text{and} M_\tau (C^*_\tau ) = \big \lbrace \varphi \in VN_\tau ;~\varphi C^*_\tau + C^*_\tau \varphi \subset C^*_\tau \big \rbrace .\end{}The main result obtained in this paper is Theorem 1:If $G$ is $\sigma $-compact and $M_\tau (C^*_\tau )=VN_\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 $M_{\pi _H}(C^*_{\pi _H})=VN_{\pi _H}$ implies in many cases that $G/H$ is a compact coset space.
LA - eng
KW - Unitary Representation; Locally Compact Group; Hilbert Space; Compact Coset Space
UR - http://eudml.org/doc/74432
ER -

References

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