Talk:Hilbert algebra

Hilbert algebras

User:D.Lazard proposes that an article be created on this topic. There has been no consensus for that at all. D.Lazard wrongly writes that a Hilbert algebra is a "sort of von Neumann algebra". Hilbert algebras are never von Neumann algebras. Taking the commutative case, the von Neumann algebra is $L^{\infty }(X,\mu )$ and the Hilbert algebra is $L^{2}(X,\mu )$. The semifinite case was covered by Godement, Dixmier and Segal (the commutation theorem). In the case of type III von Neumann algebras, the relation is even more complicated (cf the monographs of Stratila & Zsido). The modular operator and its essential spectrum play an important role, as evidenced by Connes' work. That is summarised in Tomita–Takesaki theory. In the approximately finite-dimensional case, elementary treatments are available.

Like the entry hyperfinite, this disambiguation page is completely appropriate. Mathsci (talk) 13:24, 4 January 2022 (UTC)

I agree that it may be true that it may be true that Hilbert algebras are not von Neumann algenras. However the two articles that define Hilbert algebras are so badly written that only experts of this particular subject can understand the relationship between these two clsses of algebras.

Presently the two links of this page refer to the same definition of Hilbert algebras provided in two different article. This is therefore not a disambiguation page, but a poor way for dealing with an inappropriate WP:content fork. The right way for dealing with it is to make this page a true article, or, at least a stub. At, least, for not confusing readers, it is essential to say, the two links refer to the same definition. D.Lazard (talk) 13:49, 4 January 2022 (UTC)

I don't know anything about the subject matter, but from what's written above I can conclude that 1) Hilbert algebras are a thing and that they ideally deserves an article, and 2) that a two-sentence stub without sources is probably not the best start. Maybe it's best try to write up something more substantial? Otherwise, treating this as a dab page is not optimal. Because of the principles (the term "Hilbert algebra" is not ambiguous), and for practical reasons too: a dab page discourages the creation of incoming links, among other things. – Uanfala (talk) 17:31, 4 January 2022 (UTC)

My maths is rusty but I think both sides have something correct and useful to say. I agree with D.Lazard that parts of Commutation theorem#Hilbert algebras and Tomita–Takesaki theory#Hilbert algebras describe the same concept, rather than telling us about two distinct concepts which have confusingly both been named "Hilbert algebra". It seems sensible to define and describe Hilbert algebras in general in one article, called Hilbert algebra, by splitting, merging and ideally refining that text. However, the rest of the text, which explains how Hilbert algebras can be applied specifically to commutation theorems and to Tomita–Takesaki theory, should be left where it is, and summarised briefly in the new article. Certes (talk) 17:42, 4 January 2022 (UTC)

• This disambiguation page was created by user:R.e.b., who is easily identifiable. He asked me to create the page commutation theorem as a favour as I don't usually edit in my own expertise (von Neumann algebras). The commutation theorem for type $II_{1}$ factors was proved in the 1930s by von Neumann, and his proof is easy: Dixmier, Godement and Dieudonné polished the proof in the 50s. For colleagues like the late Vaughan Jones and Sorin Popa, students are expected to know how to prove this in their sleep. For type III factors $M$, the theory is harder because the so-called modular group appears. It is a positive unbounded operator $\Delta$. The 1-parameter unitary group $\Delta ^{it}$ normalises $M$. There is also a modular conjugation operator $J$ which satisfies the same semifinite commutation theorem $JMJ=M'$. There is another important relation $J\Delta ^{it}J=\Delta ^{it}$. For approximately finite-dimensional von Neumann algebras, Roberto Longo gave an elementary proof by proving it for finite-dimensional algebras and then passing to the limit. When lecturing, it's usual to use ancient commutation theorem for von Neumann algebras with trace; and for AF von Neumann algebras, Longo's method works for states with cyclic-separating vectors. General proofs follow Marc Rieffel's bounded approach to TT theory; other simplifications can be found in Bratteli & Robinson and Uffe Haagerup's work. Haagerup developed a spatial theory where unbounded modules appear as non-commutative $L^{p}$ spaces. A detailed treatment can be found in Stratila & Zsido and Stratila, but is extremely technical. Takesaki's 3-volume book is a standard reference; but the main examples come from ergodic theory and Wolfgang Krieger's generalisation of von Neumann's group-measure space construction.

The theory of left and right Hilbert algebras (which correspond to type III von Neumann algebras) is often considered too technical to be lectured. It is often summarised without proofs. Possibly Takesaki's book or UCLA lecture notes provide an easier treatment. In the theory of von Neumann algebras, the case of Type I and II is quite distinct from that of Type III. (George Mackey's lectures give a nice account of that from the point of view of projections.) Similarly the relatively elementary theory of Hilbert algebras corresponding to type I and II von Neumann algebras contrasts with the theory of left and right Hilbert algebras arising in type III von Neumann algebras.

But there is a huge gulf between the relatively simple commutation theorem for type I or II von Neumann algebras (going back to von Neumann's proof in the 30s) and the type III case. That is why historically and now, there has been a separation between the two cases. Perhaps V. S. Sunder's "An invitation to von Neumann algebras" is a good place to look; Sunder only gives the treatment when $\Delta$ is bounded, referring to Vol. I of Bratteli & Robinson for the relatively short general proof. Gert Kjærgaard Pedersen is another place to look, but from a C* point of view.

When Tomita reported on the commutation theorem using left Hilbert algebras, there was some scepticism amongst non-Japanese mathematicians; as a result Takesaki wrote his 1970 lecture notes. In conclusion, the 1930s commutation theorem cannot really be improved; but the details of Tomita-Takesaki theory could be cleaned to include a combination of Sunder's presentations and Bratteli & Robinson's trickery. It is not hard after that to spell out the main properties of left and right Hilbert algebras. Mathsci (talk) 20:19, 4 January 2022 (UTC)

I agree with Certes and Uanfala that a true article on Hilbert algebras is needed. It is in this spirit that I transformed the dab page into a stub. I agree also that a two lines stub is not a good solution. Unfortunately, it seems that nobody is, for the moment, able to write a longer article (Apparently, Mathsci is technically able to write it, but he seems unable to write something that is understandable by anybody, but people who are specialists of von Neumann algebra). So, I suggest the following:

• Linking each section "Hilbert algebras" to the other (I did this, but I was reverted by Mathsci with only personal attacks in the edit summaries)
• Redirecting this article to Tomita–Takesaki theory#Hilbert algebras, with a tag {{R with possibilities}}. Redirecting to Commutation theorem##Hilbert algebras would be possible, but the section in Tomita–Takesaki theory and better written (although needing strong improvements).

Normally I would boldly implement this, but in view of Mathsci's disruptive editing and the fact that Mathsci considers that he owns the articles he has edited, I'll wait a consensus here. D.Lazard (talk) 21:05, 4 January 2022 (UTC)