# Fock structure of complete Boolean algebras of type I factors and of unital factorizations

## TL;DR

This paper proves that factorizable vectors form a total set in complete Boolean algebras of type I factors, and develops a noncommutative spectral theory for noise-type Boolean algebras, providing new characterizations.

## Contribution

It resolves Araki and Woods' conjecture and introduces a noncommutative framework for spectral theory of Boolean algebras of factors.

## Key findings

- Factorizable vectors are total in the algebra
- Spectral theory of noise-type Boolean algebras is cast in noncommutative language
- Characterization of 'Fock type' factorizations provided

## Abstract

The factorizable vectors of a complete Boolean algebra of type I factors, acting on a separable Hilbert space, are shown to be total, resolving a conjecture of Araki and Woods. En route, the spectral theory of noise-type Boolean algebras of Tsirelson is cast in the noncommutative language of "factorizations with unit" for which a muti-layered characterization of being "of Fock type" is provided.