We consider three quantum algebras: the $q$-oscillator algebra, the Podle\'s sphere and the $q$-deformed enveloping algebra of $su(2)$. To each of these $*$-algebras we associate a certain partial dynamical system and perform the ``Mackey analysis'' of $*$-representations developed by Yu. Savchuk and K. Schm\"udgen [``Unbounded induced representations of $*$-algebras'', Algebr. Represent. Theory, DOI: 10.1007/s10468-011-9310-6]. As a result we get the description of ``standard'' irreducible $*$-representations. Further, for each of these examples we show the existence of a ``$C^*$-envelope'' which is canonically isomorphic to the covariance $C^*$-algebra of the partial dynamical system. Finally, for the $q$-oscillator algebra and the $q$-deformed ${\cal U}(su(2))$ we show the existence of ``bad'' representations.
1
MPI für Mathematik, Inselstrasse 22, 04103 Leipzig, Germany
2
Universität Leipzig, Mathematisches Institut, Johannisgasse 26, 04103 Leipzig, Germany
Philip A. Dowerk; Yurii Savchuk. Induced *-Representations and C*-Envelopes of Some Quantum *-Algebras. Journal of Lie Theory, Tome 23 (2013) no. 1, pp. 229-250. http://geodesic.mathdoc.fr/item/JOLT_2013_23_1_a12/
@article{JOLT_2013_23_1_a12,
author = {Philip A. Dowerk and Yurii Savchuk},
title = {Induced {*-Representations} and {C\protect\textsuperscript{*}-Envelopes} of {Some} {Quantum} {*-Algebras}},
journal = {Journal of Lie Theory},
pages = {229--250},
year = {2013},
volume = {23},
number = {1},
url = {http://geodesic.mathdoc.fr/item/JOLT_2013_23_1_a12/}
}
TY - JOUR
AU - Philip A. Dowerk
AU - Yurii Savchuk
TI - Induced *-Representations and C*-Envelopes of Some Quantum *-Algebras
JO - Journal of Lie Theory
PY - 2013
SP - 229
EP - 250
VL - 23
IS - 1
UR - http://geodesic.mathdoc.fr/item/JOLT_2013_23_1_a12/
ID - JOLT_2013_23_1_a12
ER -
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%A Yurii Savchuk
%T Induced *-Representations and C*-Envelopes of Some Quantum *-Algebras
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%D 2013
%P 229-250
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%U http://geodesic.mathdoc.fr/item/JOLT_2013_23_1_a12/
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