Geodesic
Induced *-Representations and C*-Envelopes of Some Quantum *-Algebras
Journal of Lie theory, Tome 23 (2013) no. 1, pp. 229-25.

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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.
Classification : 20G42, 47L60, 17B37, 16G99, 22D30, 16W50, 47L65
Mots-clés : Induced representations, group graded algebras, well-behaved representations, partial action of a group, Mackey analysis, C*-envelope, q-deformed enveloping algebra, Podles sphere, q-oscillator 
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     author = {P. A. Dowerk and Y. Savchuk },
     title = {Induced {*-Representations} and {C\protect\textsuperscript{*}-Envelopes} of {Some} {Quantum} {*-Algebras}},
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P. A. Dowerk; Y. Savchuk . Induced *-Representations and C*-Envelopes of Some Quantum *-Algebras. Journal of Lie theory, Tome 23 (2013) no. 1, pp. 229-25. http://geodesic.mathdoc.fr/item/JLT_2013_23_1_JLT_2013_23_1_a12/