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

TL;DR

This paper analyzes *-representations of three quantum algebras using partial dynamical systems, describing their irreducible representations, $C^*$-envelopes, and identifying some 'bad' representations.

Contribution

It introduces a Mackey analysis approach for these quantum algebras and establishes the existence of $C^*$-envelopes and standard irreducible representations.

Findings

Description of standard irreducible *-representations

Existence of $C^*$-envelopes isomorphic to covariance $C^*$-algebras

Identification of 'bad' representations for certain algebras

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.