localizable Boolean algebra

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

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.

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.

- 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.

Last revised on October 18, 2021 at 11:16:57. See the history of this page for a list of all contributions to it.