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The Arens-Michael functor in noncommutative geometry is an analogue of theanalytification functor in algebraic geometry: out of the ring of "algebraicfunctions" on a noncommutative space it constructs the ring of "holomorphicfunctions" on it. In this paper, we explicitly compute the Arens-Michaelenvelopes of the Jordanian plane and the quantum enveloping algebra$U_q(\mathfrak{sl}(2))$ of $\mathfrak{sl}(2)$ for $|q|=1$.

The Arens-Michael envelopes of the Jordanian Plane and $U_q(\mathfrak{sl}(2))$