Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
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.
