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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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.) <a href="https://ncatlab.org/nlab/revision/diff/Gelfand-Naimark-Segal+construction/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/Gelfand-Naimark-Segal+construction/23">v23</a>, <a href="https://ncatlab.org/nlab/show/Gelfand-Naimark-Segal+construction">current</a>

   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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2021
   • (edited May 7th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a bunch of textbook references: * [[Gerard Murphy]], Section 3.4 of: _$C^\ast$-algebras and Operator Theory_, Academic Press 1990 ([doi:10.1016/C2009-0-22289-6](https://doi.org/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. Birkh&#228;user, Basel (1990) ([doi:10.1007/978-3-0348-7469-4](https://link.springer.com/book/10.1007/978-3-0348-7469-4)) * [[Kehe Zhu]], Section 14 of: *An Introduction to Operator Algebras*, CRC Press 1993 ([ISBN:9780849378751](https://www.routledge.com/An-Introduction-to-Operator-Algebras/Zhu/p/book/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](https://bookstore.ams.org/gsm-15)) <a href="https://ncatlab.org/nlab/revision/diff/Gelfand-Naimark-Segal+construction/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/Gelfand-Naimark-Segal+construction/23">v23</a>, <a href="https://ncatlab.org/nlab/show/Gelfand-Naimark-Segal+construction">current</a>

   added a bunch of textbook references:

   • Gerard Murphy, Section 3.4 of: $C^\ast$-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

3. • CommentRowNumber3.
   • CommentAuthornLab edit announcer
   • CommentTimeMay 8th 2022
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexCreate link request and reference to yet-to-be-created nLab page Tom Mainiero <a href="https://ncatlab.org/nlab/revision/diff/Gelfand-Naimark-Segal+construction/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/Gelfand-Naimark-Segal+construction/24">v24</a>, <a href="https://ncatlab.org/nlab/show/Gelfand-Naimark-Segal+construction">current</a>

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

   Tom Mainiero

   diff, v24, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 8th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexNamely you added a link to *[[Functorial Aspects of the GNS Representation]]* (with its own nForum thread now [here](https://nforum.ncatlab.org/discussion/14086/functorial-aspects-of-the-gns-representation/?Focus=98336#Comment_98336)). It's not clear to me yet that this deserves a separate entry. It looks a lot like material for a subsection.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 13th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis 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: *$W^\ast$-categories*, Pacific J. Math. **120** 1 (1985) 79-109 &lbrack;[euclid:pjm/1102703884](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-120/issue-1/Wast-categories/pjm/1102703884.full)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Gelfand-Naimark-Segal+construction/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/Gelfand-Naimark-Segal+construction/28">v28</a>, <a href="https://ncatlab.org/nlab/show/Gelfand-Naimark-Segal+construction">current</a>

   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: $W^\ast$-categories, Pacific J. Math. 120 1 (1985) 79-109 [euclid:pjm/1102703884]

   diff, v28, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJan 14th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have expanded the proof of the standard GNS construction ([here](https://ncatlab.org/nlab/show/Gelfand-Naimark-Segal+construction#ProofOfGNS)), 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. <a href="https://ncatlab.org/nlab/revision/diff/Gelfand-Naimark-Segal+construction/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/Gelfand-Naimark-Segal+construction/31">v31</a>, <a href="https://ncatlab.org/nlab/show/Gelfand-Naimark-Segal+construction">current</a>

   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