Noncommutative complex analytic geometry of a contractive quantum plane

In the present talk we discuss the Banach space representations of Manin's quantum $q$-plane for $|q|\ne 1$. The Arens-Michael envelope of the quantum plane is extended up to a Fréchet algebra presheaf over its spectrum. The obtained ringed space represents the geometry of the quantum plane as a union of two irreducible components being copies of the complex plane equipped with the $q$-topology and the disk topology, respectively. It turns out that the Fréchet algebra presheaf is commutative modulo its Jacobson radical, which is decomposed into a topological direct sum. The related noncommutative functional calculus problem and the spectral mapping property are solved in terms of the noncommutative Harte spectrum.