nLab localizable Boolean algebra

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.

Related concepts

• commutative von Neumann algebra

• hyperstonean space

• enhanced measurable space

• duality between geometry and algebra

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, arXiv:2005.05284.