Journal of Operator Theory

Volume 95, Issue 1, Winter 2026 pp. 189-222.

Noncommutative complex analytic geometry of a contractive quantum plane

Authors: Anar Dosi
Author institution: College of Mathematical Sciences, Harbin Engineering University, Nangang District, Harbin, 150001, China

Summary: In the paper we investigate the Banach space representations of Manin's quantum $q$-plane for $\vert q\vert \neq1$. 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.

DOI: http://dx.doi.org/10.7900/jot.2024apr27.2485
Keywords: the quantum plane, Banach quantum plane, noncommutative Fréchet algebra presheaf, Harte spectrum, Taylor spectrum, noncommutative holomorphic functional calculus

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