Note on filtered colimits of Hilbert spaces

TL;DR

This paper investigates the existence and properties of filtered colimits in the category of Hilbert spaces with contractions, highlighting how tensoring preserves these colimits and discussing issues related to bounded maps.

Contribution

It establishes that the category of Hilbert spaces with contractions admits filtered colimits and that tensoring preserves these colimits, also addressing problems with bounded maps.

Findings

Filtered colimits exist in the category of Hilbert spaces with contractions.

Tensoring preserves filtered colimits in this category.

There are issues with bounded maps that are discussed.

Abstract

The category of Hilbert spaces and contractions has filtered colimits, and tensoring preserves them. We also discuss (problems with) bounded maps.