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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 18th 2021

Created:

## 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 $aA$ 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.

## References

• Dmitri Pavlov, Gelfand-type duality for commutative von Neumann algebras. Journal of Pure and Applied Algebra 226:4 (2022), 106884. doi:10.1016/j.jpaa.2021.106884](https://doi.org/10.1016/j.jpaa.2021.106884), arXiv:2005.05284.