# Vigier's theorem for the spectral order and its applications

## TL;DR

This paper extends Vigier's theorem to the spectral order in von Neumann algebras, linking suprema and infima of self-adjoint operators to strong operator limits and exploring the order topology's properties.

## Contribution

It establishes a spectral order version of Vigier's theorem and applies it to describe bounds and topological structures in operator algebras.

## Key findings

- Decreasing nets with lower bounds have infima as strong operator limits.
- Similar results hold for increasing nets bounded above.
- The order topology is finer than the Mackey topology on self-adjoint operators.

## Abstract

The paper mainly deals with suprema and infima of self-adjoint operators in a von Neumann algebra $\mathcal{M}$ with respect to the spectral order. Let $\mathcal{M}_{sa}$ be the self-adjoint part of $\mathcal{M}$ and let $\preceq$ be the spectral order on $\mathcal{M}_{sa}$. We show that a decreasing net in $(\mathcal{M}_{sa},\preceq)$ with a lower bound has the infimum equal to the strong operator limit. The similar statement is proved for increasing net bounded above in $(\mathcal{M}_{sa},\preceq)$. This version of Vigier's theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self-adjoint operators in terms of the strong operator limit and operator means. As an application of our results on suprema and infima, we study the order topology on $\mathcal{M}_{sa}$ with respect to the spectral order. We show that it is finer than the restriction of the Mackey topology.