Noncommutative complex analytic geometry of a contractive quantum plane

TL;DR

This paper explores the noncommutative complex geometry of the quantum q-plane, extending its algebraic structure to a Frechet algebra presheaf and analyzing its spectral and topological properties.

Contribution

It introduces a novel geometric framework for the quantum q-plane using Frechet algebra presheaves and solves related spectral calculus problems.

Findings

The quantum plane's geometry is represented as a union of two irreducible components.

The Frechet algebra presheaf is shown to be commutative modulo its Jacobson radical.

Spectral mapping properties are established using the noncommutative Harte spectrum.

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.