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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 18th 2021



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


    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 aAaA 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.


    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.

    Related concepts


    • 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.

    v1, current