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Induced $*$-representations and $C^*$-envelopes of some quantum $*$-algebras

Philip Dowerk and Yurii Savchuk


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.

MSC Codes:
20G42, 47L60, 17B37
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 algebra

Related publications

2013 Repository Open Access
Philip Dowerk and Yurii Savchuk

Induced \(\ast\)-representations and \(C\)*-envelopes of some quantum \(\ast\)-algebras

In: Journal of lie theory, 23 (2013) 1, pp. 229-250