# Fibre-wise linear Poisson structures related to W*-algebras

## TL;DR

This paper explores fiber-wise linear Poisson structures derived from W*-algebras, revealing their organization within a short exact sequence of Banach sub-Poisson VB-groupoids linked to the algebra's invertible elements.

## Contribution

It introduces a canonical fiber-wise linear sub-Poisson structure associated with W*-algebras and describes its organization within a short exact sequence of Banach sub-Poisson VB-groupoids.

## Key findings

- Structures form a short exact sequence of Banach sub-Poisson VB-groupoids
- The groupoid of partially invertible elements plays a key role
- Canonical fiber-wise linear Poisson structures are characterized

## Abstract

In this paper we investigate fiber-wise linear complex Banach sub-Poisson structures defined canonically by the structure of a W*-algebra M. In particular we show that these structures are arranged in the short exact sequence of complex Banach sub-Poisson VB-groupoids with the groupoid of partially invertible elements of M as the side groupoid.