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NC functions over the nc Grassmannian

Hyuga Ito

TL;DR

The paper develops a non‑affine extension of kvv14 nc function theory to nc Grassmannians by defining $Gr^{(d;m)}(\mathcal{A})$, a Grassmannian intertwining property, and a new nc $(u,v)$‑difference‑differential operator $\widetilde{\Delta}_{u,v}^{(d;m)}$. It introduces the $(d;m)$‑Grassmannian $\mathcal{B}$‑resolvent $\widetilde{\mathfrak{R}}^{(d;m)}(\pi;\mathcal{B}|v,u)$ and its resolvent set $\widetilde{\rho}^{(d;m)}(\pi;\mathcal{B})$, proving a Grassmannian generalization of Voiculescu’s resolvent equation. The work connects Grassmannian resolvents to classical resolvents of unbounded operators via Halmos dilation, providing explicit formulas and illustrating with examples, while laying groundwork for flag manifold extensions and nc manifold perspectives. It also outlines several natural open questions about the representation, spectral data, and geometric structure (e.g., Schubert decomposition) in this non‑commutative geometric setting, highlighting potential applications to non‑commutative spectral analysis and function theory on manifolds beyond the affine case.

Abstract

We explain how to formulate Voiculescu's non-commutative Riemann sphere framework for fully matricial functions \cite{v10} within the theory of nc functions developed by Vinnikov and Kaliuzhnyi-Verbovetskyi \cite{kvv14}. We then extend this framework from the Riemann sphere to Grassmannians (and flag manifolds). Moreover, as an example of nc functions in this setting, we introduce a generalization of Voiculescu's non-commutative resolvent on the Riemann sphere, study a corresponding generalization of the resolvent equation, and discuss aspects of the spectral analysis of unbounded operators in Voiculescu's framework \cite{v10}.

NC functions over the nc Grassmannian

TL;DR

The paper develops a non‑affine extension of kvv14 nc function theory to nc Grassmannians by defining , a Grassmannian intertwining property, and a new nc ‑difference‑differential operator . It introduces the ‑Grassmannian ‑resolvent and its resolvent set , proving a Grassmannian generalization of Voiculescu’s resolvent equation. The work connects Grassmannian resolvents to classical resolvents of unbounded operators via Halmos dilation, providing explicit formulas and illustrating with examples, while laying groundwork for flag manifold extensions and nc manifold perspectives. It also outlines several natural open questions about the representation, spectral data, and geometric structure (e.g., Schubert decomposition) in this non‑commutative geometric setting, highlighting potential applications to non‑commutative spectral analysis and function theory on manifolds beyond the affine case.

Abstract

We explain how to formulate Voiculescu's non-commutative Riemann sphere framework for fully matricial functions \cite{v10} within the theory of nc functions developed by Vinnikov and Kaliuzhnyi-Verbovetskyi \cite{kvv14}. We then extend this framework from the Riemann sphere to Grassmannians (and flag manifolds). Moreover, as an example of nc functions in this setting, we introduce a generalization of Voiculescu's non-commutative resolvent on the Riemann sphere, study a corresponding generalization of the resolvent equation, and discuss aspects of the spectral analysis of unbounded operators in Voiculescu's framework \cite{v10}.
Paper Structure (13 sections, 11 theorems, 49 equations)

This paper contains 13 sections, 11 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. The nc Grassmannian $Gr^{(d;m)}(\mathcal{A})$ over $\mathcal{R}$-algebra $\mathcal{A}$
  3. Nc sets and nc functions over $Gr^{(d;m)}(\mathcal{A})$
  4. The nc $(u,v)$-difference-differential operator $\widetilde{\Delta}_{u,v}^{(d;m)}$
  5. Construction
  6. Properties
  7. A generalization of the first order difference formulae
  8. The $(d;m)$-Grassmannian $\mathcal{B}$-resolvent $\widetilde{\mathfrak{R}}^{(d;m)}(\pi;\mathcal{B}|v,u)$ with respect to $(u,v)$
  9. More about the Grassmannian resolvent
  10. A generalization of Voiculescu's resolvent equation
  11. Aspects of the spectral analysis of unbounded operators
  12. Questions
  13. The nc flag manifold $F\ell^{(\mathbf{d};m)}(\mathcal{A})$ over $\mathcal{R}$-algebra $\mathcal{A}$

Key Result

Proposition 3.3

Let $\Omega$ be an nc set over $Gr^{(d;m)}(\mathcal{A})$, and let $f$ be a graded function from $\Omega$ to $M(\mathcal{B})$. Then, the following are equivalent:

Theorems & Definitions (34)

  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • Definition 3.6
  • Remark 3.7
  • ...and 24 more