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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeOct 18th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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. ## Related concepts * [[commutative von Neumann algebra]] * [[hyperstonean space]] * [[enhanced measurable space]] ## 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](https://arxiv.org/abs/2005.05284). <a href="https://ncatlab.org/nlab/revision/localizable+Boolean+algebra/1">v1</a>, <a href="https://ncatlab.org/nlab/show/localizable+Boolean+algebra">current</a>

   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.

   Related concepts

   • commutative von Neumann algebra

   • hyperstonean space

   • enhanced measurable space

   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.

   v1, current