An example of a generalized completely continuous representation of a locally compact group

Detlev Poguntke

Studia Mathematica (1993)

  • Volume: 105, Issue: 2, page 189-205
  • ISSN: 0039-3223

Abstract

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There is constructed a compactly generated, separable, locally compact group G and a continuous irreducible unitary representation π of G such that the image π(C*(G)) of the group C*-algebra contains the algebra of compact operators, while the image π ( L 1 ( G ) ) of the L 1 -group algebra does not containany nonzero compact operator. The group G is a semidirect product of a metabelian discrete group and a “generalized Heisenberg group”.

How to cite

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Poguntke, Detlev. "An example of a generalized completely continuous representation of a locally compact group." Studia Mathematica 105.2 (1993): 189-205. <http://eudml.org/doc/215994>.

@article{Poguntke1993,
abstract = {There is constructed a compactly generated, separable, locally compact group G and a continuous irreducible unitary representation π of G such that the image π(C*(G)) of the group C*-algebra contains the algebra of compact operators, while the image $π(L^1(G))$ of the $L^1$-group algebra does not containany nonzero compact operator. The group G is a semidirect product of a metabelian discrete group and a “generalized Heisenberg group”.},
author = {Poguntke, Detlev},
journal = {Studia Mathematica},
keywords = {generalized Heisenberg group; compactly generated, separable locally compact group; continuous irreducible unitary representation; group -algebra; algebra of compact operators; -group algebra; semidirect product},
language = {eng},
number = {2},
pages = {189-205},
title = {An example of a generalized completely continuous representation of a locally compact group},
url = {http://eudml.org/doc/215994},
volume = {105},
year = {1993},
}

TY - JOUR
AU - Poguntke, Detlev
TI - An example of a generalized completely continuous representation of a locally compact group
JO - Studia Mathematica
PY - 1993
VL - 105
IS - 2
SP - 189
EP - 205
AB - There is constructed a compactly generated, separable, locally compact group G and a continuous irreducible unitary representation π of G such that the image π(C*(G)) of the group C*-algebra contains the algebra of compact operators, while the image $π(L^1(G))$ of the $L^1$-group algebra does not containany nonzero compact operator. The group G is a semidirect product of a metabelian discrete group and a “generalized Heisenberg group”.
LA - eng
KW - generalized Heisenberg group; compactly generated, separable locally compact group; continuous irreducible unitary representation; group -algebra; algebra of compact operators; -group algebra; semidirect product
UR - http://eudml.org/doc/215994
ER -

References

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  1. [1] J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, Paris 1969. 
  2. [2] Ph. Green, The structure of imprimitivity algebras, J. Funct. Anal. 36 (1980), 88-104. Zbl0422.46048
  3. [3] A. Guichardet, Caractères des algèbres de Banach involutives, Ann. Inst. Fourier (Grenoble) 13 (1963), 1-81. Zbl0124.07003
  4. [4] H. Leptin, Verallgemeinerte L 1 -Algebren und projektive Darstellungen lokal kompakter Gruppen, Invent. Math. 3 (1967), 257-281, 4 (1967), 68-86. 
  5. [5] H. Leptin and D. Poguntke, Symmetry and nonsymmetry for locally compact groups, J. Funct. Anal. 33 (1979), 119-134. Zbl0414.43004
  6. [6] D. Poguntke, Unitary representations of Lie groups and operators of finite rank, Ann. of Math., to appear. Zbl0828.22013
  7. [7] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Clarendon, Oxford 1968. Zbl0165.15601

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