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.

]]>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]

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.

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

Tom Mainiero

]]>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)

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.)

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