nLab
localizable Boolean algebra

Contents

Contents

Idea

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

Definition

A localizable Boolean algebra is a complete Boolean algebra AA such that 1A1\in A equals the supremum of all aAa\in A such that the Boolean algebra aAa A admits a faithful continuous valuation ν:A[0,1]\nu\colon A\to[0,1]. Here a valuation ν:A[0,]\nu\colon A\to[0,\infty] is faithful if ν(a)=0\nu(a)=0 implies a=0a=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 AA to the complexification of the completion of the free real algebra on AA, given by the left adjoint to the functor that takes idempotents.

References

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