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nLab > Latest Changes: Hilbert W*-module

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 14th 2021

    Created with this content:

    Definition

    A Hilbert W*-module is a Hilbert C*-module M over a von Neumann algebra A such that M admits a predual as a Banach space.

    Properties

    For any von Neumann algebra A, the category of Hilbert W*-modules over A is equivalent to the category of W*-representations of A.

    The equivalence is implemented by the following functors.

    Given a Hilbert W*-module M, we send it to the completion of M⊗AL2(A), where L2(A) is the Haagerup standard form of A.

    Given a W*-representation R, we send it to the internal hom HomA(L2(A),R), which is a Hilbert W*-module over A.

    Related concepts

    v1, current

    • CommentRowNumber2.
    • CommentAuthorTom Mainiero
    • CommentTimeMay 9th 2022

    Added a few references, including the first reference I know of to W*-modules (Paschke).

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorTom Mainiero
    • CommentTimeMay 9th 2022

    Added extra technical condition needed in definition, along with appropriate reference.

    diff, v4, current

1 to 3 of 3

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