# nLab localizable Boolean algebra

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

A Boolean algebra is localizable if it admits “sufficiently many” measures.

## Definition

A localizable Boolean algebra is a complete Boolean algebra $A$ such that $1\in A$ equals the supremum of all $a\in A$ such that the Boolean algebra $a A$ admits a faithful continuous valuation $\nu\colon A\to[0,1]$. Here a valuation $\nu\colon A\to[0,\infty]$ is faithful if $\nu(a)=0$ implies $a=0$.

A morphism of localizable Boolean algebras is a complete (i.e., suprema-preserving) homomorphism of Boolean algebras.

## Properties

The category of localizable Boolean algebras admits all small limits and small colimits.

It is equivalent to the category of commutative von Neumann algebras.

The equivalence sends a commutative von Neumann algebra to its localizable Boolean algebra of projections. It sends a localizable Boolean algebra $A$ to the complexification of the completion of the free real algebra on $A$, given by the left adjoint to the functor that takes idempotents.

[[!localizable Boolean algebras]]