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Banach Lie groupoid of partial isometries over restricted Grassmannian

Tomasz Goliński, Grzegorz Jakimowicz, Aneta Sliżewska

TL;DR

This work constructs a Banach Lie groupoid structure on the set of partial isometries over the restricted Grassmannian Gr_{res}, denoted U_{res} ⇢ Gr_{res}, by coupling the known groupoid on H with a differential framework on Gr_{res} built from local cross-sections. It proves that U_{res} is a transitive, pure Banach Lie groupoid modeled on $L^2$-type spaces and a Banach Lie algebra $U_+$, with smooth maps for source, target, multiplication, inversion, and identity. The construction clarifies that Gr_{res} is not a submanifold of the full Grassmannian, hence U_{res} is not a Banach Lie subgroupoid of U(H); it also discusses generalizations to Gr_{res}^p and the immersed but non-split case when p=0. The results enrich infinite-dimensional differential geometry and may inform integrable systems linked to the Sato Grassmannian and Banach Poisson–Lie structures.

Abstract

The set of partial isometries in a W*-algebra possesses a structure of Banach Lie groupoid. In this paper the differential structure on the set of partial isometries over the restricted Grassmannian is constructed, which makes it into a Banach Lie groupoid.

Banach Lie groupoid of partial isometries over restricted Grassmannian

TL;DR

This work constructs a Banach Lie groupoid structure on the set of partial isometries over the restricted Grassmannian Gr_{res}, denoted U_{res} ⇢ Gr_{res}, by coupling the known groupoid on H with a differential framework on Gr_{res} built from local cross-sections. It proves that U_{res} is a transitive, pure Banach Lie groupoid modeled on -type spaces and a Banach Lie algebra , with smooth maps for source, target, multiplication, inversion, and identity. The construction clarifies that Gr_{res} is not a submanifold of the full Grassmannian, hence U_{res} is not a Banach Lie subgroupoid of U(H); it also discusses generalizations to Gr_{res}^p and the immersed but non-split case when p=0. The results enrich infinite-dimensional differential geometry and may inform integrable systems linked to the Sato Grassmannian and Banach Poisson–Lie structures.

Abstract

The set of partial isometries in a W*-algebra possesses a structure of Banach Lie groupoid. In this paper the differential structure on the set of partial isometries over the restricted Grassmannian is constructed, which makes it into a Banach Lie groupoid.
Paper Structure (5 sections, 8 theorems, 38 equations)

This paper contains 5 sections, 8 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Banach Lie groupoids
  3. Banach Lie groupoid of partial isometries
  4. Restricted Grassmannian as a Hilbert manifold
  5. Banach Lie groupoid of partial isometries over restricted Grassmannian

Key Result

Proposition 4.2

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 4.1
  • Proposition 4.2
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Theorem 5.3
  • proof
  • Theorem 5.4
  • ...and 5 more