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Noncommutative localizations of a contractive quantum plane

Anar Dosi

TL;DR

<3-5 sentence high-level summary> This paper analyzes noncommutative localizations for the contractive quantum plane through the Taylor framework on Arens-Michael-Fréchet algebras. It shows that all Fréchet structure-sheaf localizations over Runge $q$-open subsets are genuine localizations, enabling exact diagonal-cohomology computations via a Decomposition Theorem and diagonal $q$-complexes. By connecting localization to joint spectra, it relates the Taylor and Putinar spectra in the $q$-setting and highlights a notable discrepancy due to the weak $q$-topology, with implications for noncommutative spectral theory. Overall, the work provides a complete localization picture for the formal geometry of the contractive quantum plane and its spectral calculus.

Abstract

In the present paper we investigate the localizations in the sense of J. L. Taylor of the Arens-Michael-Fréchet algebras associated with noncommutative analytic spaces of a contractive q-plane representing its formal geometry. It turns out that all noncommutative Fréchet algebras obtained by the Fréchet algebra structure sheaves over open subsets from the topology bases are indeed localizations. That topological homology property of the structure sheaves results in the key properties of Taylor and Putinar spectra of the left Banach q-modules over the algebras of global sections.

Noncommutative localizations of a contractive quantum plane

TL;DR

<3-5 sentence high-level summary> This paper analyzes noncommutative localizations for the contractive quantum plane through the Taylor framework on Arens-Michael-Fréchet algebras. It shows that all Fréchet structure-sheaf localizations over Runge -open subsets are genuine localizations, enabling exact diagonal-cohomology computations via a Decomposition Theorem and diagonal -complexes. By connecting localization to joint spectra, it relates the Taylor and Putinar spectra in the -setting and highlights a notable discrepancy due to the weak -topology, with implications for noncommutative spectral theory. Overall, the work provides a complete localization picture for the formal geometry of the contractive quantum plane and its spectral calculus.

Abstract

In the present paper we investigate the localizations in the sense of J. L. Taylor of the Arens-Michael-Fréchet algebras associated with noncommutative analytic spaces of a contractive q-plane representing its formal geometry. It turns out that all noncommutative Fréchet algebras obtained by the Fréchet algebra structure sheaves over open subsets from the topology bases are indeed localizations. That topological homology property of the structure sheaves results in the key properties of Taylor and Putinar spectra of the left Banach q-modules over the algebras of global sections.

Paper Structure

This paper contains 39 sections, 32 theorems, 273 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Triangular (co)chains
  4. Topological homology and localizations
  5. Arens-Michael envelope
  6. The $q$-topology of the complex plane
  7. The Fréchet space sheaves $\mathcal{O}\left( d\right)$ on $\left( \mathbb{C},\mathfrak{q}\right)$
  8. Arens-Michael envelope of $\mathfrak{A}_{q}$ and the character space $\mathbb{C}_{xy}$
  9. The $x$-diagonal and $y$-diagonal Fréchet space complexes
  10. The $y$-diagonal formal $q$-complex
  11. The $x$-diagonal formal $q$-complex
  12. The exactness of the diagonal formal complexes
  13. The $y$-diagonal holomorphic $q$-complex
  14. The localizations of the quantum plane
  15. The bimodule resolution of the quantum plane
  16. ...and 24 more sections

Key Result

Lemma 2.1

(Aristov-Dosi) Let $X\overset{S}{\longrightarrow}Y\overset{T}{\longrightarrow}Z$ be a triangular cochain of Fréchet spaces. If all diagonal cochains $X_{m}\overset{S_{mm}}{\longrightarrow}Y_{m}\overset{T_{mm}}{\longrightarrow}Z_{m}$, $m\in\mathbb{N}$ are exact then so is the original cochain. Moreov

Theorems & Definitions (67)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • ...and 57 more