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It would be nice to finish the description of the theorem at GNS construction, if someone has the head for doing that. :-)
@davidoslive: Please feel free to edit! :-)
It is better to have BOTH versions (with the easier version first), as many readers will not comprehend the horizontally categorified version. The logical inclusion does not include the expositional inclusion.
I have given GNS construction an Idea-section and a bunch of references, amplifying also the generalization from $C^\ast$-algebras to general unital star-algebras.
Also I renamed the section “From the nPOV” to “For C-star categories”, since the statement there is a horizontal categorification, but in itself does not offer any category-theoretic perspective on the construction.
An actual nPOV is proposed in Parzygnat 16, but besides adding this reference to the entry, I haven’t added any details on this yet.
An actual nPOV is proposed in Parzygnat 16,
also Jacobs 10
(which Alexander Schenkel tells me serves to make all the universal AQFT constructions in Benini-Schenkel-Woike 17, surveyed in Schenkel 17b, generalize to star-algebras)
That makes me think whether it is interesting to consider the generalisation of correspondences of $C^\ast$-algebras (ie a kind of directional Hilbert bimodule) to more general $\ast$-algebras. There are versions of the Eilenberg-Watts theorem for representation categories of $C^\ast$-algebras, but it’s not immediately straightforward, and even some the notions bifurcate, depending on analytic considerations. See for instance this answer to a recent MO question of mine.
Thanks, Todd.
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