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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    added some indication of the actual construction, below the statement of the theorem.

    (This might deserve to be re-organized entirely, but I don’t have energy for this now.)

    diff, v23, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021
    • (edited May 7th 2021)

    added a bunch of textbook references:

    • Gerard Murphy, Section 3.4 of: C*-algebras and Operator Theory, Academic Press 1990 (doi:10.1016/C2009-0-22289-6)

    • Konrad Schmüdgen, Section 8.3 of: Unbounded operator algebras and representation theory, Operator theory, advances and applications, vol. 37. Birkhäuser, Basel (1990) (doi:10.1007/978-3-0348-7469-4)

    • Kehe Zhu, Section 14 of: An Introduction to Operator Algebras, CRC Press 1993 (ISBN:9780849378751)

    • Richard V. Kadison, John R. Ringrose, Theorem 4.5.2 in: Fundamentals of the theory of operator algebras – Volume I: Elementary Theory, Graduate Studies in Mathematics 15, AMS 1997 (ISBN:978-0-8218-0819-1)

    diff, v23, current

  1. Create link request and reference to yet-to-be-created nLab page

    Tom Mainiero

    diff, v24, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2022

    Namely you added a link to Functorial Aspects of the GNS Representation (with its own nForum thread now here).

    It’s not clear to me yet that this deserves a separate entry. It looks a lot like material for a subsection.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2024

    This entry used to refer to “Ghez, Lima and Roberts” without more details. I have added pointer to:

    • P. Ghez, R. Lima, John E. Roberts, Prop. 1.9 in: W*-categories, Pacific J. Math. 120 1 (1985) 79-109 [euclid:pjm/1102703884]

    diff, v28, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2024

    I have expanded the proof of the standard GNS construction (here), making explicit the use(s) of the Cauchy-Schwarz inequality.

    Also I added the assumption that the given state sends the star-involution to complex conjugation, which is needed to make the inner product on the resulting vector space be Hermitian.

    diff, v31, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2024

    added a brief paragraph mentioning the notion of the “folium” of states induced by a GNS construction.

    diff, v34, current