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I started rewriting von Neumann algebra from the nPOV. So far I rewrote the definition and added some remarks about Sakai's theorem and preduals, but you can already see a proposed list of sections to be written.
I also edited the remarks section to stress the nPOV.
Good, we need that.
I put terminological remarks higher up to head off potential confusion early.
This is beginning to look really good, Dmitri – thanks!
Re the Gelfand duality for commutative von Neumann algebras, I’m still groping to understand. It seems the “measurable locales” and locale morphisms are the “right” or in any case cleaner thing to consider (as opposed to the point-set stuff, where one is obliged to consider equivalence classes of measurable functions between what you are calling measurable spaces). But what is a measurable locale?
Passing to the associated frames, they are evidently some sort of complete Boolean algebra. Indeed, it was said elsewhere that the projection functor, taking a commutative von Neumann algebra to the (complete) Boolean algebra of self-adjoint idempotents, is full and faithful. So we have fully faithful functors; the second is just an inclusion (NB: not the functor taking a complete Boolean algebra to the topology of its Stone space):
$CvNAlg \stackrel{proj}{\to} CBool \hookrightarrow Frame.$This permits us to think of commutative von Neumann algebras as special type of frames which we can call a “measurable frames”. (That frame, considered as an object in the category $Locale$ opposite to $Frame$, is what we would call the associated measurable locale, if I am understanding this set-up correctly.)
But how does one characterize measurable frames or measurable locales, besides saying they are the ones coming from commutative von Neumann algebras? Is there some sort of lattice-theoretic description, for example?
I do not agree with the changes. I mean, the standard definitiion is now REMOVED and should now be inferred from a very complicated treatment of Sakai theorem, ultraweak topology, preduals and whatever. I have spent few months of my life in operator algebra study and find it incomprehensible and nonstandard. I agree that it is nice to have a categorical definition with “preduals” whatever it means (note: there is even no $n$Lab entry about it at the moment!), but this does not say we should make it convoluted for a working mathematician and REMOVE the standard definition. I should also say that the standard definition of von Neumann algebras does not refer to $C^\ast$-algebras at all; a posteriori one notices that every von Neumann is a $C^\ast$-algebra, the point of view which is also rarely useful. I will now restate the standard definition. Both definitions should stay at the beginning.
This kind of n-only-writing is exactly the reason why many people do not want to contribute to $n$Lab because they consider it esoteric ncategory theory not for them.
I have restated now also the standard definition. So now there are two definitions one after another.
@zskoda: The "standard" (standard for whom?) definition was meant to simply reappear below as a property. In fact, it already appeared as such in the Terminology section, except that "satisfying certain properties" should be replaced with "closed in ultraweak topology".
Furthermore, your claims about this definition being "nonstandard" are totally false. One of the standard textbooks in operator algebras is Sakai's textbook, which uses precisely this definition. By the way, Sakai's textbook does not use categories. In the future, please take a look at the literature before making such claims.
There is nothing "complicated" or "incomprehensible" about preduals. (Why did you put preduals in scare quotes? Is this a new notion for you?) As you can see from Takesaki's books, preduals are absolutely crucial for understanding von Neumann algebras.
I also totally fail to get your remarks about "n-only-writing". What's the point of nLab if not to emphasize the nPOV? What's the point of having a separate article on von Neumann algebras at nLab if it's identical to Wikipedia's article?
It is the standard for most mathematicians. I heard the standard definition possibly thousand times. Being hidden in some form as a side property somewhere at the bottom of a long article does not make it noticeable for average reader.
In the future, please take a look at the literature before making such claims.
I have had in my life several hundred references on operator algebras in my hand. One more or less does not make essential difference in impression what is a standard.
There is nothing “complicated” or “incomprehensible” about preduals.
I am not sure how you make it intrinsic to the category of C-star algebras: all duality/rigid/automous etc. in category theory make are always difficult to axiomatize. Second nLab does not have the entry for the term used (at least the basic definition should use defined terms). Of course, it is very standard to use this same characterization/definition descriptively without using the term and without mistifying with the category context. Finally, it is important that the theory of von Neumann algebra can be (and usually is) developed quite independently from the theory of C-star algebras.
Finally the formulation with subcategory was missing “full” subcategory (which I added but it might be wrong ? hence the formulation was confusing). If one does not talk essentially about which morphisms are included than it is no point of talking subcategory, but rather the subset!!
I still do not understand WHAT IS THE POINT OF ERASING one definition to replace it with completely different ? The point of $n$Lab is to have as much parallel points of view as possible, and being gentle to newcomers and not exclusive.
What’s the point of nLab if not to emphasize the nPOV?
There are many exclusive and elitisitic societies and mafias, defining one more is not the point, for sure. It would not be a very worthy activity to dedicate huge effort (I wrote many hundreds of articles in nLab, for example, working on daily bases for over 2 years an sacrificing lots of other work) just to make something $n$POV. The point is that $n$POV is a cohesive center of our small community defined largely by interest of a group of collaborators of Urs, centered around (but not limited to) mathematical physics, categories, Grothendieck-style geometry (including stacks and cohomology) and foundational questions.
What’s the point of having a separate article on von Neumann algebras at nLab if it’s identical to Wikipedia’s article?
It should not be identical but should be strictly better. Hence it should have things of the level of wikipedia plus much more. If it is lacking the wikipedia standard it may be in lower tier, for many users. Wikipedia existence and limitations are not to be used as a licence to erase (or hide) the work/formulations of your colleagues in $n$Lab unless it is about stylistic improvements or inappropriate material, reorganization into new sectioning. IMHO $n$Lab is more reliable than the (larger scope) wikipedia, and if anything falls fairly well within our scope I prefer not to open wikipedia at all for the same item.
I am not sure how you make it intrinsic to the category of C-star algebras: all duality/rigid/automous etc. in category theory make are always difficult to axiomatize.
I'm afraid I don't understand this statement. Would you mind elaborating? In particular, what do you mean when you say intrinsic? What do you mean by "difficult to axiomatize"?
Second nLab does not have the entry for the term used (at least the basic definition should use defined terms).
Which term? My definition was complete and did not use any undefined terms.
without mistifying with the category context
Additional categorical context is now considered mystifying?
Finally the formulation with subcategory was missing "full" subcategory
Missing?! The category of von Neumann algebras is not a full subcategory of C*-algebras. My definition made this crystal clear: it explicitly said that only those morphisms of C*-algebras that admit a predual are morphisms of von Neumann algebras. Thus the use of the term "subcategory" as opposed to "full subcategory" was not only justified but necessary.
Finally, it is important that the theory of von Neumann algebra can be (and usually is) developed quite independently from the theory of C-star algebras.
Important for whom? What benefits one might expect from such an independence? Are you aware of the fact that most references on von Neumann algebras deduce many basic properties of von Neumann algebras (e.g., properties of positive elements) from the theory of C*-algebras? This is the approach taken in Takesaki's book, for example. So your claim of "usually is" is false.
I still do not understand WHAT IS THE POINT OF ERASING one definition to replace it with completely different ?
Once again, nobody erased the classical definition, it appears in the Terminology section, not at the bottom of the article as you claim. I fail to see how this can be considered "erasing" or "hiding". Frankly, I do not see a reason for such an emotional reaction (e.g., SHOUTING at me) nor do I see a reason to immediately revert my changes and introduce mathematical errors (see above), instead of discussing them here first.
I am not sure how you make it intrinsic to the category of C-star algebras: all duality/rigid/automous etc. in category theory make are always difficult to axiomatize. Second nLab does not have the entry for the term used (at least the basic definition should use defined terms).
I don’t understand what in the world you are talking about. Rigid? autonomous? where is this coming from? The only category theory in Dmitri’s definition is a trivial remark in the first sentence that should confuse no self-respecting mathematician today; but it’s easy enough to remove it if it really is a problem. That doesn’t justify equipping every $W^*$-algebra with a representation and calling that ‘standard’. As for the term ‘predual’, it is defined immediately after it is used; there is no need to link to a nonexistent page and then complain that the page doesn’t exist. The missing definition is right there!
I added a detailed discussion of von Neumann's original definition together with a comparison with the usual definition (see the second section): http://ncatlab.org/nlab/revision/von+Neumann+algebra/34
However, it seems to be gone in the current revision.
Dmitri, something seems to have gone wrong with Instiki’s article-locking mechanism to prevent edit conflicts. The current version was just now produced by me by editing Zoran’s version, but the page history says that your version was made before mine. If you were editing before me, then I should have been warned and I would not have edited. Instead, I get a warning only now that you are editing! (I tried to edit again, because my version has typos that I want to fix.)
@Toby: Can you retract the changes or somehow combine the two versions? I don't want to do anything for the fear of further editing conflicts.
I didn't get any warnings from Instiki, by the way.
Are you still editing now? If not I will unlock it and combine the two versions.
@Toby: No, I am not editing right now.
By the way, in your version you don't say what a morphism of von Neumann algebras is in the traditional definition.
By the way, in your version you don’t say what a morphism of von Neumann algebras is in the traditional definition.
Well, it remained in the Terminology section. I think that ‘traditionally’ one is not so concerned with morphisms, and I was only trying to get Zoran’s definition (which didn’t mention morphisms) sufficiently high up.
@Toby: I am not sure that W*-morphism is a term I have ever seen in the literature; morphism of W*-algebras sounds better.
Besides, W*-morphism can be confused with a morphism in a W*-category, or a functor between W*-categories, which are totally different notions (von Neumann algebras do not form a W*-category).
OK, I’m done editing. I moved history and terminology up even earlier. I left out some of your essay, Dmitri, but I’d like to move some of that to concrete structure. (I also had ‘morphism’ some places where I meant to have ‘homomorphism’, but I fixed that.)
@Toby: I also noticed that you typeset W* and C* in words like W*-algebra and C*-algebra as formulas.
Is there a particular rationale behind this decision? Historically speaking, W* and C* are abbreviations for weak and closed, and the star was added to indicate closedness under involution, i.e., the operator *.
But W* is clearly not the result of * applied to W, it's simply a particular abbreviation. I don't think there is a way to interpret W* and C* as formulas.
One issue with typesetting W* and C* as formulas is that they are not bolded properly. Another issue is that they cannot be hyperlinked properly (only the word algebra is linked). A minor issue is that W and C are italicized, which in my opinion is wrong, since they are abbreviations, not mathematical variables.
I also don't see any particular benefits associated with typesetting W* and C* as formulas.
@Toby: Moving the "essay" to concrete structure might be a good idea, I was also wondering if it really belonged there.
I see arguments for the typesetting either way.
Although $C^*$ and $W^*$ aren’t the result of applying $*$ to anything, I still think of them as (irreducible) mathematical symbols. For example, if $A$ and $B$ are $W^*$-algebras, then $C^*(A,B)$ is the set of $C^*$-homomorphisms from $A$ to $B$ and $W^*(A,B)$ is the space of $W^*$-homomorphisms.
The linking is a problem that happens all over the Lab. Ideally, we would fix it in the software, but that’s proven difficult. Many of the Lab authors (but not me) will take things out of math formatting specially to be put in a link. (Example: $\omega$-category vs ω-category.)
@Toby: Sure, names of categories (like C*, W*, Set, Man) can appear in formulas, in particular, in notations like Set(X,Y) and Man(U,V), but I thought the nLab convention is to type them in roman font. In TeX's math mode this would be \hbox{C*} and \hbox{W*}.
Similarly, we get the category of representations of W∗-algebras on Hilbert spaces using instead the spatial morphisms of concrete von Neumann algebras.
We only get faithful representations. There are plenty of nonfaithful representations; these don't come from concrete von Neumann algebras.
the von Neumann algebras form a (particularly nice) class of concrete C∗-algebras.
I would say the opposite is true; von Neumann algebras are horrible from the viewpoint of C*-algebras, even though if we consider them on their own, they do have a lot of nice properties (Borel functional calculus etc.).
I still don't think the term "W*-homomorphism" appears anywhere in the literature. Perhaps "homomorphism of W*-algebras" will be better?
I don’t understand what in the world you are talking about. Rigid? autonomous? where is this coming from?
What is the point of category here ? If there is an internal characterization of the subcategory then it makes sense. The various duality notions are usually stated in terms of special kinds of monoidal categories, traces and so on, like autonomous, rigid categories and so on. I do not know if there is a possibility that introducing preduals can be put as additional structure here, if it is it is not trivial, regarding the other mentioned examples. So, I do not see categorical point of view here, just its false impression.
nobody erased the classical definition, it appears in the Terminology section, not at the bottom of the article as you claim
I was discussing version 31 of the page, which was actual at the time, which does not have terminology section at all. It has Elementary examples section where the standard definition is taken as example, without stating 1) that all examples fall in this category, 2) that this as a definition implies that it is norm closed (what was written before) 3) it quotes in terms of ultraweak rather than weak topology (it requires a bit again “predual” point of view to invoke the latter). Finally this section is sufficiently low in the long page to notice it.
I am happy with present version, but not version 31.
What is the point of category here?
Which category? The category of von Neumann algebras?
If there is an internal characterization of the subcategory then it makes sense.
What makes sense? What do you mean by "internal characterization"?
The various duality notions are usually stated in terms of special kinds of monoidal categories, traces and so on, like autonomous, rigid categories and so on.
True, but how is this statement relevant here? Sure, the Banach space dual of a Banach space X can be expressed as the internal hom from X to C. Is this what you mean here?
I do not know if there is a possibility that introducing preduals can be put as additional structure here, if it is it is not trivial, regarding the other mentioned examples.
As explained in the article, the predual is canonically isomorphic to the dual of von Neumann algebra in the ultraweak topology. So clearly the predual is not an additional structure.
So, I do not see categorical point of view here, just its false impression.
You don't see a categorical point of view in defining the category of von Neumann algebras as opposed to the (proper set-theoretical) class of von Neumann algebras?
Sure, the Banach space dual of a Banach space X can be expressed as the internal hom from X to C. Is this what you mean here? As explained in the article, the predual is canonically isomorphic to the dual of von Neumann algebra in the ultraweak topology. So clearly the predual is not an additional structure.
It is not an additional stricture on a Banach space, but if category has duals then the category has the additional structure of duality. For example, rigid categories are roughly a kind of monoidal categories with dual.
You don’t see a categorical point of view in defining the category of von Neumann algebras as opposed to the (proper set-theoretical) class of von Neumann algebras?
Well, defining an object as an object of a subcategory requires understanding the above category (so need to understand C-star to define von Neumann) plus understanding the morphisms (what is beneficial if the morphisms are the same). Hence if the above category is something simpler, more familiar, and if the morphisms are the same, then it is beneficial to use it in definition. Otherwise it is a fancy property. Or if there is an internal characterization. Internal means defining the subcategory internally, only in terms of ambient category as an abstract category (possibly with structure, like monoidal). E.g. the full subcategory of compact objects. Don’t you see that you make it hard for simple users ?
What I mean by categorical point of view making sense is when category theory helps to see something one does not see without it; when category is not just a bookkeeping device, but gives an insight. For most trivial example, when one enlarges some class and sees that one in fact has a clearly cut subcategory of presheaves on the original category. Or seeing that Shelah’s $T^{eq}$ for a complete first order theory $T$ is at the level of the syntactic category just a familiar operation of pretopos completion (well, for coherent theories). If one has to list all the same properties, without even economizing, then by the definition there is no point.
I do like that eventually the category of von Neumann algebras is defined somewhere in the entry. But it is better to avoid its definition (if the only one offered) depend on another category of comparable importance and familiarity plus a characterization involving new notions.
In any case the things are good in the present version. Thanks for expanding all the aspects.
nor do I see a reason to immediately revert my changes and introduce mathematical errors (see above)
I have past experience that when I complain to Urs about a similar issue he gets slightly irritated why I did not simply improve myself and notify of, rather than waiting for full discussion to evolve. Discussions tend to take more energy than collectively working further in the entry and commenting shortly on changes and reasoning there in $n$Forum. People who work here as much as Urs can easily see the source of his principle. Often when we make a move which is controversial, people who see it such offer immediate solutions which can be just approved or further improved, and this was here occasionally even called Urs’s principle. I did not revert your changes, which introduced new definition, what would mean erase new definition, but added the old simple definition, what is a synthesis of both, and expected that this would be acceptable. Of course, the iteration of checking by you and finding possible errors is expected part of the process.
Don't you see that you make it hard for simple users ?
I very strongly disagree with such a statement. The definition of von Neumann algebras with preduals is well-rooted in classical measure theory: Riesz representation theorem states that the dual space of the space of finite measures is precisely the space of bounded measurable functions.
Because of Gelfand duality for measurable spaces we know that the algebra of bounded measurable functions characterizes the measurable space up to a unique isomorphism. Hence one might wonder whether simply dropping the commutativity condition from the notion of commutative C*-algebra with a predual gives a nice notion of a noncommutative measurable space. It turns out that it does.
On the other hand, the definition of von Neumann algebras as subalgebras closed in the (ultra)weak topology is not rooted in anything.
Thus I would claim that for somebody who is familiar with classical measure theory it is much easier to absorb the definition with preduals, which is rooted in classical concepts of measure theory, than the definition with subalgebras, which appears out of the blue.
At this point, I think that Zoran is only complaining about phrasing the definition as
The category of von Neumann algebras is the subcategory of the category of C*-algebras consisting of C*-algebras that admit a predual and morphisms of C*-algebras that admit a predual.
(rev 31) instead of
A von Neumann algebra is a C*-algebra that admits a predual, and a morphism of von Neumann algebras is a morphism of C*-algebras that admits a predual. In this way, the category of von Neumann algebras becomes a subcategory of the category of C*-algebras.
(rev 35, modulo other changes).
By the way, Dmitri, I would be interested if you would briefly state at Riesz representation theorem the version of the theorem that holds for all localisable measurable spaces and how the form that I know is a version of it (if it is).
@Toby: Riesz proved many representation theorems. The one that you are referring to states that the dual of functions (i.e., some dense subset of L_0) is the space of measures (i.e., L_1). The theorem I am referring to states that the dual of the space of measures (i.e., L_1) is the space of functions (i.e., L_0). And another representation theorem of Riesz states that the dual of L_{1/2}=L^2 is L_{1/2}=L^2.
In fact Riesz proved that L_p is dual to L_{1−p} for all 0≤p≤1.
OK, the one that I know is as you said, specifically that the dual of the space of continuous functions vanishing at infinity on some locally compact Hausdorff topological space is the space of regular Borel measures on that space (and also a variation for the positive cone of the dual of the space of continuous functions with compact support). Wikipedia also lists that the dual of a Hilbert space is its complex conjugate; since L_{½} is a Hilbert space that comes with an involution, this gives its self-duality (and conversely after proving some stuff about basises).
Even for the theorem that the dual of L_{1} is L_{0}, I usually see that as referring to integrable functions on a measure space rather than to measures on a measurable space. Of course, the Radon–Nikodym theorem converts between these. So it is all fitting together!
So the theorem, as I gather it, is that a generalised (equipped with a σ-ideal of null sets) measurable space X has the property that the short linear map (morphism of Banach spaces) from the space L_{0}(X) of bounded measurable functions to the dual L_{1}^{*} of the space L_{1} of bounded measures, given by integration f ↦ (µ ↦ ∫fµ), is an isometric isomorphism iff X is localisable.
@Toby: Your statement is correct.
Hi all,
I am only coming to this now after a long vacation. I have not followed the discussion nor will I probably have time to do so soon.
But looking briefly at the entry I thought that the single sentence in the Idea-section deserved to be slightly expanded on. As a suggestion, I have turned it into the following:
A von Neumann algebra or $W^*$-algebra is an important and special kind of operator algebra, relevant in particular to measure theory and quantum mechanics/quantum field theory in its algebraic formulation as AQFT. Specifically, (non-commutative) von Neumann algebras can be understood as the formal duals of (non-commutative) measurable spaces, see the section Relation to measurable spaces below.
I am not claiming that this is perfect, but maybe you can see which points I would like to see the Idea-section mention right away. Feel free to improve on the formulation as you see the need.
By the way, currently the entries von Neumann algebra and measurable locale don’t point to each other at all. But I guess they should!?
Certainly, although they’re linked through the more standard notion of measurable space too.
I made some edits at von Neumann algebra. Mostly cosmetic I think, e.g., replacing a subsection title “Abstract” by “Abstract von Neumann algebras”, but in order to avoid possible confusions for a beginner. Quite a bit remains to be done obviously.
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