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In this paper, following [1], we develop the theory of globalpseudo-differential operators defined on the quantum group $SU_q(2)$, andprovide some spectral results concerning these operators. We define agraduation for this algebra of pseudo-differential operators in terms of itsnatural Fourier decomposition and, using the infinite-dimensionalrepresentations introduced by Woronowicz [15], we also provide a$*$-representation of $SU_q(2)$ as bounded pseudo-differential operators actingon $L^2(\mathbb{S}^1)$.

Global Pseudo-differential Operators on the Quantum Group $SU_q(2)$