Induced *-representations and C*-envelopes of some quantum *-algebras
Philip Dowerk and Yurii Savchuk
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Submission date: 15. Jun. 2012
published in: Journl of Lie theory, 23 (2013) 1, p. 229-250
MSC-Numbers: 20G42, 47L60, 17B37
Keywords and phrases: 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
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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 (su(2)) we show the existence of “bad” representations.