Noncommutative complex analytic geometry of a contractive quantum plane
Abstract
In the paper we investigate the Banach space representations of Manin's quantum q-plane for |q| is not 1. 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.
Cite
@article{arxiv.2412.04823,
title = {Noncommutative complex analytic geometry of a contractive quantum plane},
author = {Anar Dosi},
journal= {arXiv preprint arXiv:2412.04823},
year = {2024}
}
Comments
The quantum plane, Banach quantum plane, noncommutative Frechet algebra presheaf, Harte spectrum, Taylor spectrum, noncommutative holomorphic functional calculus