Active lattices determine AW*-algebras

TL;DR

This paper demonstrates that operator algebras with sufficient projections, including AW*-algebras, are fully characterized by their projections, symmetries, and the action of these symmetries, through the concept of active lattices.

Contribution

It introduces active lattices as a new framework to completely determine AW*-algebras from their projections and symmetries, establishing an equivalence of categories.

Findings

AW*-algebras are determined by their active lattices.

Category of AW*-algebras is equivalent to a subcategory of active lattices.

Established an equivalence between piecewise AW*-algebras and piecewise complete Boolean algebras.

Abstract

We prove that operator algebras that have enough projections are completely determined by those projections, their symmetries, and the action of the latter on the former. This includes all von Neumann algebras and all AW*-algebras. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*-algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*-algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*-algebra that enables recovering its total structure.