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MiS Preprint
34/2012
Induced $*$-representations and $C^*$-envelopes of some quantum $*$-algebras
Philip Dowerk and Yurii Savchuk
Abstract
We consider three quantum algebras: the $q$-oscillator algebra, the Podleś sphere and the $q$-deformed enveloping algebra of $su(2).$ To each of these $*$-algebras we associate certain partial dynamical system and perform the "Mackey analysis" of $*$-representations developed in the paper "Unbounded induced representations of $*$-algebras" written by Y. Savchuk and K. Schmüdgen. 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 $\mathcal{U}(su(2))$ we show the existence of "bad" representations.
