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

2012/06/13 by Dowerk, Philip A., Savchuk, Yurii
#16W50 #17B37 #22D30 #47L60 #47L65 #FOS: Mathematics #Operator Algebras (math.OA) #Primary 20G42 #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary 16G99

paper · doi:10.48550/arxiv.1206.2806

Abstract

We consider three quantum algebras: the q-oscillator algebra, the Podles' 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 [SS]. 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 \cU(su(2)) we show the existence of "bad" representations.

Related

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• Introduction to Quantum Algebras
• Operator representations of cross product algebras of Podles' quantum spheres
• *-Representations of a Quantum Heisenberg Group Algebra