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Journal of Lie Theory 23 (2013), No. 1, 229--250
Copyright Heldermann Verlag 2013

Induced *-Representations and C*-Envelopes of Some Quantum *-Algebras

Philip A. Dowerk
MPI für Mathematik, Inselstrasse 22, 04103 Leipzig, Germany

Yurii Savchuk
Universität Leipzig, Mathematisches Institut, Johannisgasse 26, 04103 Leipzig, Germany


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.

Keywords: 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.

MSC: 20G42, 47L60, 17B37; 16G99, 22D30, 16W50, 47L65

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