Homological framework of noncommutative complex analytic geometry and functional calculus

TL;DR

This paper develops a topological homology framework for noncommutative complex analytic geometries of Fréchet algebras, addressing functional calculus and spectral properties with applications to q-geometry and operator theory.

Contribution

It introduces a homological approach to noncommutative complex geometry of Fréchet algebras and solves related functional calculus problems using a ch categorical perspective.

Findings

Describes ideal analytic geometry via ch categories over Fréchet algebras.

Provides solutions to functional calculus problems using homological spectra.

Applies the framework to noncommutative q-geometry and spectra of operators.

Abstract

In the paper we propose topological homology framework of noncommutative complex analytic geometries of Fr\'echet algebras, and investigate the related functional calculus and spectral mapping properties. It turns out that an ideal analytic geometry of a Fr\'echet algebra A can be described in terms of a \v{C}ech category over A. The functional calculus problem within a particular \v{C}ech A-category, and a left Fr\'echet A-module X is solved in term of the homological spectrum of X with respect to that category. As an application, we use the formal q-geometry of a contractive operator q-plane, and solve the related noncommutative holomorphic functional calculus problem. The related spectrum is reduced to Putinar spectrum of a Fr\'echet q-module. In the case of a Banach q-module we come up with the closure of its Taylor spectrum.