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 Frechet 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 Frechet 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 Fr\'{e}chet algebra
presheaf, Harte spectrum, Taylor spectrum, noncommutative holomorphic
functional calculus
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