Several months ago I informed both MathSciNet as well as Zentralblatt that I would like to stop reviewing papers for these repositories. If you don’t know what I am talking about (your PhD thesis advisor should be fired!), then MathSciNet and Zentralblatt are databases that index published papers in mathematics, contains some bibliographic information (such as a reference list for every paper, as well as a list of papers that reference it), and, significantly, has a review for every indexed paper. The reviews are written by mathematicians who do so voluntarily (they get AMS points or something). If the editors find nobody willing to review, then the abstract appears instead of a review. This used to a very valuable tool, and is still quite valuable.

I quite because:

- I don’t have time for the voluntary work for free.
- This service is very useful for old papers that are hard to get a hand on, or that are written in a language that is not English. It used to be a very good way to stay up-to-date with works in the field. Today, the standard is that almost all papers are written in English and are available freely online. The actual added value of having this external review available is significantly lower than it used to be.
- I think that we, as a community, are not doing a good enough job of refereeing papers (I feel this as referee, author, and now also as an editor). I think that if we have some time that we are going to spend volunteering for reviewing papers, we shouldn’t split it up between refereeing and reviewing for databases. We should concentrate on refereeing, which is a crucial part of the mathematical eco-system, and not waste it on reviews, which are in a large part redundant.
- Reviewing papers has advantages also for the reviewer: it can discipline and focus the reviewer for staying up-to-date and working through current papers. However, in the current system, the papers are reviewed
**after they appeared in print (or online)**. This is ridiculously late. I do like to review papers some time, but the appropriate time to do this is after they appear as preprints on the arxiv. Then I can use my blog to post these reviews. Yes, this is not a standard platform, but nothing is perfect.

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This is post is reply to (part of) a post by Scott Aaronson. I got kind of heated up by his unfair portrayal of the blog “Stop Timothy Gowers!!!“, and started writing a reply which got to be ridiculously long, so I moved it here.

Dear Scott,

I think that, as others remarked in the comments, you unfairly portray sowa’s blog. It is much more than just a rant against Gowers, and contains some “positive” contributions (agreed, the “positive” ones are mostly historical/philosophical/other and not Gowers-style exposition, so what?). But even if it was true that that blog just had “negative” comments, I think it has a place. Here are some points to consider.

(Before the points, this is written in defense of sowa, and not in damnation of Gowers. I have never met either, I didn’t read their papers, I don’t agree with everything sowa said about Gowers, and I am willing to bet that Gowers is a very nice guy and a gentleman.)

**1) “Lack of exposition” I.** You wonder why doesn’t sowa for once take a break from discussing (say) the epochal greatness of Grothendiek, and “walk us through examples”. Well, he uses his blog to write about things he cares about. For serious mathematics he has others outlets. He wants to discuss the politics of mathematics, and he wants to oppose the what he sees as the current trends and power structure. There are politics in mathematics and there are power structures, fads, trends, celebrities, etc. These things affect the development of mathematics, where people go, where the money goes. These are totally legitimate issues to address.

**2) “Lack of exposition” II.** The kind of blogging that tries to teach some mathematics, expose it in a simplified way that non-experts can understand, is very difficult to do. I try to do it on my own blog, and honestly, I sometimes wonder whether the piece I wrote has any value at all. It happens (to me, and maybe also to you) that by the time you reach the beef, you run of breath, or out of time, or you realize that you cannot do this technical part any better than original paper or book that you linked to. And as a reader, when reading expositions on certain blogs or expository journals (or colloquium talks) I sometimes say to myself: the author really tried to walk me through this piece of mathematics/science or through their thought process, but unfortunately was unsuccessful in conveying any substantial information. So I can totally understand a blogger who feels that writing these friendly expository pieces is not useful, and spend no time on that.

**3) Symmetry.** You mention that there is asymmetry between them: Gowers writes about math, and sowa writes about Gowers. Well, you are right, there really isn’t symmetry: Gowers is at the center, and sowa is peripheral. Gowers has power and influence, and sowa thinks that Gowers has too much. So it is ridiculous to point out that sowa is just complaining and not talking math, and that Gowers isn’t wasting time complaining about politics. When it comes to the power structure in mathematics, Gowers doesn’t have much to complain about (although, being human, he does actually complain and rant on his blog, when the issues are not the ones where he happens to be up).

I want to emphasize a fallacy you have made: You point to the asymmetry as an answer to a question you raise: “How could a neutral observer possibly decide who was right?” (You mean, if the neutral observer didn’t care to weigh the actual statements made?) Interesting question, but your answer seems all wrong to me. The person complaining might have a strong point – that’s why he is so upset! – and the person not complaining might be comfortable enough.

**4) The three cultures in this discussion. **Scott, you are an American, watching from the side an exchange between an (apparently) Eastern European raised mathematician and an English mathematician. To you, it might seem like the first is shouting, and the second is being the most polite and maybe even gallant person ever. These differences in culture can distract from the actual points made. So the best thing to do would be to concentrate on the points themselves, and not on the volume.

**5) The point of the matter I.** As pointed out by eminent mathematicians, there is a certain periodic movement in the mainstream culture of mathematics, between the abstract and theoretical developments, on the one hand, and more concrete, problem-driven work, on the other. Very roughly speaking, Sowa on his blog advocated a certain style of mathematics, or a certain way of doing mathematics, which he felt was the best one. His point of view on what is good mathematics can be summarized in one word: “Grothendiek”. He very often used Gowers as an example of bad trends in mathematics, giving arguments against points-of-view publicized by Gowers. But in the beginning that blog did not look like a crusade against Gowers, and had the pleasant name: “Notes of an owl”. Sowa was just another force affecting the perpetual periodic motion in mathematical philosophy.

If I get the story, sowa really lost his top when it became known that Gowers would be presenting the work of Abel prize winner Pierre Deligne (and as his blog says, that’s when he changed the title of his blog to the current one). He stated his opinion that Gowers is unqualified to speak about Deligne’s work. Is it unacceptable to raise such a point? I think that it is (though I am in no way competent to answer the question of what Gowers is capable of). He also made a point that it was the third time in a row that Gowers was chosen to present the life work of an Abel prize winner. This is an even more valid point to make.

I know that everybody says that Gowers is a brilliant expositor. Well, I also saw a video of the lecture “The importance of mathematics” by Gowers and it was, indeed, a wonderful talk. I recall thinking that it is one of the best lectures I saw in my life (and for sure the best one that I saw on video). So I am convinced that he has the capability of expositoring exquisitely. But nobody is perfect, and no-one irreplaceable.

I stopped reading Gowers’s Blog some time ago, but there was a time that I tried to read a lot. I know what people are talking about when they speak of his posts as an intellectually honest journey, where he takes you by the hand and leads you through his thought process; I know what they are talking about, but I interpret this “leads you through his thought process” as lazy writing. Reading some of his old posts I got the notion that he hasn’t thought it all out before writing, and that his “delete” button is broken. Now, I don’t want to go and search for the old posts that I read and did not like (as Gowers once said: “I don’t have the information at the tip of my fingers”…), I am not out to prove that Gowers is a **bad** expositor, of course he isn’t; my point is just that different people might find different styles of expositoring appealing or useful. So the question, whether it is correct to have the same person present the prize three times in a row is, seems to me to be right on. And maybe, if you liked the style of the guy who did it the first time, then you wouldn’t have raised that question “is he the right person”, when he was chosen for the third time in a row. But once the question is raised, you cannot ignore it just because it is kind of rude to ask it.

**6) The point of the matter II****. **Another harsh criticism of sowa on Gowers (too harsh, I think, but basically right) is on the matter of publishing in mathematics. It is ironic that one of the good things that you (Scott) have to say about Gowers is that “He’s also been a leader in the fight to free academia from predatory publishers”. Google “predatory publishers”, I don’t think it means what you think it does. Indeed he played a creditable role as a leader in the boycott against Elsevier (about which I had doubts, I won’t go into that). But Gowers, in my opinion, abused his reputation and played a very dangerous role in actually **vindicating** predatory publishers, when he helped to set up Gold Open Access journals (see also this). In his defense, he seems to be very thoughtful and careful about these matters, is aware of the dangers, and has also later set up an arXiv overlay journal. Sowa has a lot to say on this matter, and here too, and I agree with some of the points he makes.

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Recall, that by Sz.-Nagy’s dilation theorem, given contraction acting on a Hilbert space , one can always construct a unitary acting on a Hilbert space , such that

(*) ,

(Here denotes the orthogonal projection of onto .) The operator is called a **unitary dilation** of . This simple theorem is the starting point of a ton of developments in operator theory on Hilbert spaces.

In the setting of operator on Banach spaces, we say that that an operator acting on a Banach space **has a dilation**, if there exists a Banach space , an invertible isometry , and two contractions and , such that

(**) ,

It is quite easy to see that if both and are Hilbert spaces, then this boils down to the definition (*). Moreover, invertible isometry seems like the right generalization of unitary, and examining (**) for , we see that must be isometric, and is the projection onto . In this setting it is understood that we are looking for invertible isometric dilations, and no adjective is used alongside the word “dilation”. (Other kinds of dilations can also be considered, i.e., one can search for a positive dilation, etc.) Note that for an operator to have a dilation it must be a contraction, and we shall always understand that operators for which we seek a dilation are contractions.

One very simple thing I learned from this paper is that the existence of a dilation for every contraction in the setting of **all** Banach spaces is a ridiculously trivial matter: one just constructs , (the bounded functions ), defines

,

(where the is in the th place), one lets be the left shift, and be the projection onto th summand. (A similar construction is given in the paper, using .) The key point of this paper is that this might not be very helpful unless shares with some regularity properies, such as being a Hilbert space, reflexivity, being an space on a finite measure space, etc. For example, if one wants to remain in the realm of Hilbert spaces, the above construction does not work, and one needs to proceed differently (the usual proof of the dilation theorem in Hilbert spaces (see Wikipedia) uses the existence of a square root; basic, but not trivial). In this post we will always understand that the we seek is to be chosen from within a well defined class of Banach spaces.

The authors don’t concentrate on the problem of finding a dilation for a single operator. They treat a more general problem, and this generality is actually a key to their proof. They make the following definition:

**Definition: **Let be a class of Banach spaces and let . A set of bounded operators on , say , is said to have a **simultaneous dilation **(in ), if there exists a and a set of invertible isometries , together with contractions and , such that

for all and all .

The main theorem is as follows (Theorem 2.9 in the paper):

**Theorem: ***Suppose that is a family of reflexive Banach spaces, that is closed under finite direct sums (for some fixed ) and closed under ultra-products. If is a family of bounded operators on has simultaneous dilation in , then so does the weak-operator closure of the convex hull of . *

For example, the family of all unitaries on a Hilbert space have simultaneous dilation (trivially). Since the weak operator convex hull of unitaries contains all contractions, we find that all contractions on a Hilbert space have simultaneous dilation (here we used the Theorem in the case where is the class of all Hilbert spaces, and ).

The existence of a simultaneous dilation for all contractions on a Hilbert space is only epsilon harder than Sz.-Nagy’s dilation theorem, and is brought just to illustrate. A more interesting example is that positive invertible isometries on are weak-operator dense in the set of all positive contractions, we get that the set of all positive contractions on has simultaneous dilation. The paper doesn’t exhaust all the dilation possibilities that it opens up (I guess that is why it is called a “toolkit”), and the authors suggest that the methods could be used in other situations; for example, maybe it can be used to find -endomorphic dilations to CP maps on C*-algebra.

Two very nice surprises were:

- I learned of an application of N-dilations (see this link overview of the notion in the context of a single or commuting operators on Hilbert spaces). In fact, N-dilations seem to be essential for the proof. The authors prove that a family has simultaneous dilation if and only if it has simultaneous N-dilation for every N (this is similar to a Theorem 1.2 from this paper (in a slightly different setting), but curiously there the easy direction was the direct implication. I wonder if the reverse implication there could also be proved with ultra products…).
- I found references to several earlier work regarding dilations (even N-dilations), unfortunately, a couple of them are in languages that I cannot read. In particular, I learned that the existence of dilations in the context of spaces allows to obtain pointwise ergodic theorems in spaces, as in this paper of Akcoglu and this paper of Akcoglu and Shucheston (I knew that Sz.-Nagy’s unitary dilation quickly reduces the mean ergodic theorem for contractions in to von Neumann’s mean ergodic theorem for unitaries, which is rather basic given the spectral theorem; however, the mean ergodic theorem for contractions in Hilbert spaces has a very elegant proof, it is not much different from von Neumann’s original proof, if I’m not mistaken. Pointwise ergodic theorems are harder, and is the easiest, so this is a far better application, even in the case, than what I was aware of).

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I decided to read this book primarily because I like to read the books I have, but also because I am teaching graduate functional analysis in the coming semester and I wanted to amuse myself by toying with the possibility of de-emhasizing Banach spaces and giving a more general treatment that includes topological vector spaces. I enjoyed thinking about whether it can and/or should be done (the answers are ** yes** and

Oh sister! I was pleasantly surprised with how much I enjoyed this book. They don’t write books like that any more. Published in 1964, the authors follow quite closely the tradition of Bourbaki. Not too closely, thankfully. For example they restrict attention from the outset to spaces over the real or complex numbers, and don’t torture the reader with topological division rings; moreover, the book is only 158 pages long. However, it is definitely written under the influence of Bourbaki. That is, they develop the whole theory from scratch in a self-contained, clean, efficient and completely rigorous way, working their way from the most general spaces to more special cases of spaces. Notions are given at the precise place where they become needed, and all the definitions are very economical. It is clear that every definition, lemma, theorem and proof were formulated after much thought had been given as to how they would be most useful later on. Examples (of “concrete” spaces to which the theory applies) are only given at the end of the chapters, in so called “supplements”. The book is rather dry, but it is a very subtly tasty kind of dry. The superb organization is manifested in the fact that the proofs are short, almost all of them are shorter than two (short) paragraphs, and only on rare occasion is a proof longer than a (small) page. There is hardly any trumpet blowing (such as “we now come to an important theorem”) and no storytelling, no opinions and no historical notes, not to mention references, outside the supplement. The authors never address the reader. It seems that there is not one superfluous word in the text. Oh, well, perhaps there is *one* superfluous word.

After the definition of a **precompact set** in a (locally convex) topological vector space, the authors decided to illustrate the concept and added the sentence *“Tapioca would make a suitable mental image”*. This happens on page 49, and is the first and last attempt made by the authors to suggest a mental image, or any other kind of literary device. It is a little strange that in this bare desert of topological vector spaces, one should happen upon a lonely tapioca, just one time…

* * * * *

So, why don’t people write books like that any more? Of course, because this manner of writing went out of style. It had to become unfashionable, first of all, simply because old things always do. But we should also remember that mathematical style of writing is not disconnected from the cultural and philosophical surroundings. So perhaps in the 1930s and up to 1950s people could write dogmatically and religiously about mathematics, but as time went by it was becoming harder to write like this about anything.

In addition to this, it is interesting that there were also some opposition to Bourbaki, from the time not much after the project took off, and until many many years later.

Not that I myself am a big fan. I personally believe that maximal generality is not conducive for learning, and I prefer, say, Discussion-SpecialCase-Definition-**Example**-Theorem-Proof to Definition-Theorem-Proof any day. I also don’t believe in teaching notions from the most general to the more specific. For example, in my opinion, set theory should not be taught-before-everything-else, etc. For another example, when I teach undergraduate functional analysis I start with Hilbert spaces and then do Banach spaces, which is inefficient from a purely logical point of view. But this is how humans learn: first we gurgle, then we utter words, then we speak; only much later do we learn about the notion of a *language*.

So, yes, I do find the books by Bourbaki hard to use (reading about all the pranks related to the Bourbaki gang, one cannot sometimes help but wonder wether it is all a gigantic prank). But I have a great admiration and respect for the ideals that group set and for some of its influences on mathematical culture. The book by Robertson and Robertson is an example of how to take the Bourbaki spirit and make something beautiful out of it. And because of my admiration and respect for this heritage, it is a little sad to know that Bourbaki was quite violently abused and denounced.

If you have ever read some harsh and mean criticism of the Bourbaki culture, if you have heard someone try to insult someone else by comparing them to Bourbaki, then please keep in mind this. Nobody really teaches three-year-olds set theory before numbers. In the beginning of every Bourbaki book (“To the reader”), it is explicitly stated that, even though in principle the text requires no previous mathematical knowledge on the part of the reader (besides the previous books in the series) “it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course”. Bourbaki didn’t “destroy French mathematics” or any other nonsense. The source of violent opposition is not theological or pedagogical, but psychological. In my experience, the most fervent opponents of the Bourbaki tradition who I heard of, are people of non-neglible egos (and their students), who were simply very insulted to find out that a self-appointed, French-speaking(!) elite group decided to take the lead, without asking permission or inviting them (or their teachers). That hurt, and a crusade, spanning decades, ensued.

* * * * *

Well, let us return to the pleasant Robertsons. Besides the lonely tapioca, I found one other curious thing about this book. On the first page the names of the authors are written:

**A.P. Robertson**

(Professor of Mathematics

University of Keele)

AND

**Wendy Robertson**

So, what’s the deal with A.P. and Wendy? Is A.P. a man? I guessed so. Are they brother and sister? Why is he a professor and she isn’t? Are they father and daughter? I wanted to find out. I found their obituaries: Wendy Robertson (she passed away last year) and Alexander Robertson.

So they were husband and wife, and it seems that they had a beautiful family and a happy life together, many years after writing this book together. I remained curious about one thing: whose idea was it to suggest tapioca? Did they immediately agree about this, or did they argue for weeks? Was it a lapse? Was it a conscious lapse?

* * * * *

In the course that I will teach in the coming semester, I am not going to use the language of topological vector spaces. I will concentrate on Banach spaces, then weak and weak-* topologies will enter. These are, of course, topological vector spaces, but there is no need to set up the whole framework to notice this, and there is no need to prove everything in the most general setting. For example, the students will be able to prove a Hahn-Banach extension theorem for, say, weak-* continuous functionals, by imitating the proof that I will give in class in a similar setting.

On Saturday I went to my nephew’s Bar-Mitzva, and they had tapioca for desert (not bad), and I thought about Wendy and Alex Robertson. Well, especially about Wendy. I think that it was her idea.

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Anyone who wishes to report mistakes can use the comment section in this page, or email me.

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There are two very interesting new submissions:

- “Hyperrigid subsets of graph C*-algebras and the property of rigidity at 0“, by our PhD. student Guy Salomon.
- “On fixed points of self maps of the free ball” by recently-become-ex postdoc Eli Shamovich.

There is also a cross listing (from Spectral Theory) to the paper “Spectral Continuity for Aperiodic Quantum Systems I. General Theory“, by Siegfried Beckus (a postdoc in our department) together with Jean Bellissard and Giuseppe De Nittis.

Finally, there is a new (and final) version of the paper “Compact Group Actions on Topological and Noncommutative Joins” by Benjamin Passer (another postdoc in our department) together with Alexandru Chirvasitu.

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I am not too happy about this review. It is not that it is a negative review – actually it has a rather kind air to it. However, I am somewhat disappointed in the information that the review contains, and I am not sure that it does the reader some service which the potential readers could not achieve by simply reading the table of contents and the preface to the book (it is easy to look inside the book in the Amazon page; of course, it is also easy to find a copy of the book online).

The reviewer correctly notices that one key feature of the book is the treatment of as a completion of , and that this is used for applications in analysis. However, I would love it if a reviewer would point out to the fact that, although the idea of thinking about as a completion space is not new, few (if any) have attempted to actually walk the extra mile and work with in this way (i.e., without requiring measure theory) all the way up to rigorous and significant applications in analysis. Moreover, it would be nice if my attempt was compared to other such attempts (if they exist), and I would like to hear opinions about whether my take is successful.

I am grateful that the reviewer reports on the extensive exercises (this is indeed, in my opinion, one of the pluses of new books in general and my book in particular), but there are a couple of other innovations that are certainly worth remarking on, and I hope that the next reviewer does not miss them. For example, is it a good idea to include a chapter on Hilbert function spaces in an introductory text to FA? (a colleague of mine told me that he would keep that out). Another example: I think that my chapter on applications of compact operators is quite special. This chapter has two halves: one on integral equations and one on functional equations. Now, the subject of integral equations is well trodden and takes a central place in some introductions to FA, and one might wonder whether anything new can be done here in terms of the organization and presentation of the material. So, I think it is worth remarking about whether or not my exposition has anything to add. The half on applications of compact operators to integral equations contains some beautiful and highly non-trivial material that has never appeared in a book before, not to mention that functional equations of any kind are rarely considered in introductions to FA; this may also be worth a comment.

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The “Multivariable operator theory workshop at the Technion, on occasion of Baruch Solel’s 65th birthday”, is over. Overall I think it was successful, and I enjoyed meeting old and new friend, and seeing the plan materialize. Everything ran very smoothly – mostly thanks to the Center for Mathematical Sciences and in particular Maya Shpigelman. It was a pleasure to have an occasion to thank Baruch, and I was proud to see my colleagues acknowledge Baruch’s contribution and wish him the best.

If you are curious about the talks, here is the book of abstracts. Most of the presentations can be found at the bottom of the workshop webpage. Here is a bigger version of the photo.

I will not blog about the workshop any further – I don’t feel like I participated as a mathematician. I miss being a regular participant! Luckily I don’t have to wait long: Next week, I am going to Athens to participate in the Sixth Summer School in Operator Theory in Athens.

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(The reference for this lecture is mostly Takesaki, Vol. I, Chapters 2 and 3).

Fix a Hilbert space (no need to worry about dimension; on the other hand, even finite dimensional spaces are interesting). In this lecture we will write to denote the space of all bounded functionals on a normed space . Even though our normed spaces will be linear subspaces of , we will not have an opportunity to be confused regarding whether the denotes the adjoint operator.

For every we define

- A linear functional given by .
- A linear operator given by .

It customary to also write , for reasons which I hope are obvious (or by “physicists”). Both and are sometimes denotes by .

Let denote the norm closed two sided ideal of compact operators in . Every vector functional can also be considered as an element of . We shall see below that in a natural way (where “in a natural way” is meant in a loose way). Let us begin by taking a closer look at compact operators.

**Lemma 1:** *For every , there exist two orthonormal sequences and a sequence of positive numbers that is either finite or convergent to zero, such that is given as the norm convergent sum *

.

*If , then has the form *

.

**Proof: **For self-adjoint operators, this is simply the spectral theorem for compact self-adjoint operators (see here for the formulation I am using). The result for positive operators follows at once.

For a general compact operator , let be the polar decomposition of . Then is compact, and . Then

,

so putting (and recalling that is a partial isometry with ), we are done.

Now let us consider a functional . Since for every , the rank one operator is compact, we can apply to . This gives rise to a sesqui-linear form

for .

(For example, . )

Now, , so is bounded. It follows from a familiar consequence of the Riesz representation theorem (see Exercise A below) that there is some such that

for all ,

or, in other words,

for all .

If you never encountered the “familiar consequence of Riesz theorem” that we used above, then prove it (now, please!).

**Exercise A:** Let be a Hilbert space, and let be a bounded sesqui-linear form, meaning that there is some such that for all and all ,

- ,
- ,
- .

Prove that there exists with , such that , for all .

Note that , so that the map is a bounded linear map. Moreover, it is not hard to see that this map is injective (as a map from to ).

**Lemma 2:** . Consequently, if , are orthonormal sequences, and , then ; moreover, is the unique element in mapped to .

**Proof:** We check

,

while, on the other hand,

.

It follows that . The remainder follows from the above remarks, that is a bounded, linear and injective map.

Now suppose that is finite dimensional. Then

.

Therefore (by linearity), for all . In other words, every linear functional on is given by , for some . Our discussion below will show that this is also true, in the appropriate sense, when has infinite dimension; however, not every will give rise to a bounded operator. The ones that do are said to be the * trace class operators*, and corresponds precisely to .

**Exercise B: **If , and , then define . Prive that .

**Lemma 3:** *For all , the operator is compact. Moreover, for every orthonormal sequence , *

* . *

*In fact, . *

**Proof:** For all , let us write for some complex number of absolute value one. Then for every finite set of indices ,

.

This shows that the series converges.

To show that is compact, we assume that it is a positive operator, and leave it to the reader to reduce the general case to this one.

Now, suppose that (note that in this case, the first part of the proof shows that can be supported on at most countably many basis vectors; we may therefore assume that is separable. This is not crucial.)

By the spectral theorem (Lecture 1), we may assume that is a multiplication operator for . To prove that is compact, it suffices to prove that the spectral measure of satisfies that (which is ) has finite rank for all (recall Exercise M in Lecture 1). But on the range of , so for all in the range; if the range was infinite dimensional this would contradict the summability of the series established above.

**Exercise C**: Complete the above proof, by showing that is compact in the general case (use Exercise A).

Now we can compute . For every , it is clear that , and in particular it restricts to a bounded functional on . Moreover, since , it certainly holds that for every sequence and every two orthonormal sequences , the series

converges to a bounded functional in and in . Now we shall see that all bounded functionals of have this form.

**Theorem 4: ***Let . Then there is a **sequence and two orthonormal sequences , such that *

*.*

**Proof:** From what we have gathered until now, we know that there is an operator such that

for all . By Lemma 3, . By Lemma 1, for a series of positive numbers that is either finitely supported or converges to zero, and two orthonormal sequences. For every finite , we can define . Then is a compact operator, and

.

But . We see that .

**Definition 5: **Let denote the space of all bounded operators for which there is some such that . Equivalently, is the space of all such that

for any (or every) orthonormal sequence . The space is called the space of * trace class operators*. For every we define the

**Remark 6: **By Theorem 4, every extends naturally in a unique way to a bounded functional on (the extension is unique only among “natural” extensions). For every , if satisfies , we write , and more generally, for any we write . Recall that in functional analysis, one writes the “pairing” between a space and its dual as

for .

In our setting, we identify with , and if , we write

.

**Exercise D:** Prove that .

**Exercise E: **Every is WOT continuous on , and hence extends to a unique such functional on .

Every (where and are orthonormal sequences) extends to be a bounded linear functional on , defined by . Therefore, every determines a bounded linear functional on by

.

The map is called * the canonical map*.

**Theorem 7: ***The canonical map is an isometric isomorphism of onto . *

**Proof: **For every pair of unit vectors , the functional . This can be used to show that the canonical map is an injective map, an in fact it is norm-nondecreasing:

which approximates .

On the other hand, if , then for any net of finite rank projections increasing to , so , therefore the canonical map is isometric.

Now if , then we define a bounded sesqui-linear form on :

.

One finds that there is some bounded operator , such that

for all .

Since the linear span of the functionals of the form is dense in , we see that is the image of under the canonical map.

We henceforth identify with , and with . It is also common to denote , and to refer to it as * the predual* of (an alert student should worry about the word “the”; until we show that the predual of a von Neumann algebra is unique, we can refer to it more precisely as

**Exercise F: (Tying all loose ends) **Prove that for every positive ,

,

where is an orthonormal basis for . Show that the right hand sum is independent of the particular orthonormal basis. (Recall that we defined to be the norm of the corresponding functional in ). For a not-necessarily positive , prove that , and that

Converges absolutely, and to the same value, for every choice of orthonormal basis . Prove that is an ideal in . Prove that for every and ,

.

Since , we can consider the weak-* topology on it.

**Definition 8:** The * -weak operator topology* on (or just

Thus, in the -weak operator topology, if and only if

for all .

Equivalently, the -weak topology is determined by the seminorms

where satisfy .

(Recall that a topology is said to be * generated by a family of seminorms* if convergence of a net is determined by convergence for all . Thus, the strong (operator) topology is the topology generated by the family of seminorms , .)

**Definition 9:** The * -strong operator topology *(or simply the -strong topology) is the topology generated by the seminorms

,

where .

**Definition 10:** The strong * topology is the topology generated by the seminorms , . The -strong * topology is defined similarly.

**Exercise G:** The *whatever*-strong topology is strictly stronger than the *whatever*-weak topology. The –*whatever* topology is strictly stronger than the *whatever* topology, but they coincide on the unit ball . The *whatever* * topology is strictly stronger than the *whatever* topology. All are strictly weaker than the norm topology.

****

**Theorem 11:** *Let be a von Neumann algebra. Then is closed in all of the above topologies. Consequently, is a dual Banach space. To be precise, if we let denote the subspace of consisting of all -weakly continuous functionals, then can be isometrically isomorphically identified with . *

**Proof:** Since is WOT closed, it is closed in the weakest, and hence in all, of the topologies. To see that is a dual space, we consider it as a weak-* closed subspace of . Let

for all .

Then, being weak-* closed,

for all .

(see Proposition 13 in this old lecture). But by standard results on dual spaces (see Theorem 7 in this old lecture),

by a natural map (where is the quotient map). Thus

,

and it remains to observe that this isomorphism respects the -weak functionals. For this, note that the restriction map , given by , induces an isomorphism of with its image – the weak-* functionals on .

We saw above that . In this section, we will see that the double dual of any C*-algebra is a von Neumann algebra. In contrast with everything we have done until this point, our C*-algebras will be just abstract C*-algebras: Banach *-algebras that satisfy the C*-identity: . We will use basic results of the theory (with due apologies) when we need them.

**Lemma 12: ***Let be a C*-algebra, let be a *-representation, and define . Let be the canonical embedding. Then there exists a unique linear map , which is surjective, continuous with respect to the and -weak topologies, which extends (in the sense that ). Moreover, maps the unit ball of onto the unit ball of . *

**Proof: **Consider as a map between the Banach spaces . Let be the adjoint map. Let be the restriction of to the space of -weakly continuous functionals. So . Now define . This satisfies for all , that

.

This shows that . The continuity is a “general nonsense” fact which always holds for adjoints: indeed, if in , then for all .

Finally, to show the map is surjective, it suffices to show that it takes the closed unit ball onto the closed unit ball . Since is continuous, it takes onto a -weakly compact set. But this compact set contains the open unit ball of (because that’s what *-homomorphisms do, being the composition of a quotient and an injective (hence isometric) *-homomorphism). But by Kaplansky (and Exercise F) the open unit ball of is -weakly dense in .

Let be a unital C*-algebra. Now we recall *the universal representation*

,

where (here, given a state , is the GNS representation of ). Since is an isometric isomorphism (by the Gelfand-Naimark theorem), we can identity with and consider as a C*-subalgebra of .

**Definition 13:** The algebra is called **the universal enveloping von Neumann algebra of . **

**Theorem 14:** *The map (given by Lemma 12) is isometric, hence . Every bounded functional on extends to a -weakly continuous continuous functional on . The universal enveloping von Neumann algebra has the following universal property: if is a *-representation, then there exists a unique -weakly continuous of onto such that . *

**Proof: **For simplicity of notation, put , , and we shall use the notation of Lemma 12 and its proof. By construction, every state on extends to a vector state on . By Exercise H below, every bounded functional is the linear combination of four states, and from this it is easy to show every functional extends to a vector functional, and therefore extends -weakly.

We will now show that is surjective, this will show that the map is injective, and since it maps an open unit ball onto an open unit ball it must be isometric and surjective. To see that is surjective, we just recall that it is defined to be . But is the restriction map given by (being the conjugate of an inclusion ), and since every extends to a map in (by the previous paragraph), this shows that is surjective.

Finally, given , let be as in Lemma 12. Then is a surjective and -weakly continuous linear map. Since it is a *-homomorphism on the -dense subspace , it is a representation. We get , and restricting to we get the final assertion.

**Exercise H: **Prove that every is the linear combination of four states. Conclude that every can be written as , for some representation and .

**Exercise I:** Prove that if is a *-representation, , and if have the same universal property as , then is -weakly continuously *-isomorphic to by a map that fixes .

]]>

Let and be two Hilbert spaces. Our goal is to construct a new Hilbert space, formed from and , called the * Hilbert space tensor product *and denoted .

**Definition 1:** Let be a vector space. A * semi-inner product* is a function such that for all and all :

- ,
- ,
- .

Of course, if occurs only for , then is said to be an ** inner product**.

**Definition 2:** Given a semi-inner product, we define the associated * semi-norm * by

.

**Exercise A:** A semi-inner product satisfies the Cauchy-Schwarz inequality:

.

Consequently, the semi-norm arising from a semi-inner product is really a semi-norm. It follows that is a subspace, and that for all and . Therefore, on we can define an inner product on it

.

Finally, the inner product space can be completed in a unique way to form a Hilbert space .

**Definition 3:** Given a semi-inner product on a vector space , the Hilbert space constructed above is called the * Hausdorff completion of *.

**Definition 4:** Given Hilbert spaces and , let denote the free vector space with basis ; that is, is just the space of all finite (formal) linear combinations . On define a semi-inner product

.

**Exercise B:** This is indeed a semi-inner product. (**Hint:** The only thing that requires proof is positive semi-definiteness. You can find a proof in all kinds of books, e.g. Takesaki. But I think the following might be an elegant approach: Given two finite dimensional Hilbert spaces and , and given and , one has the * rank one operator * given by matrix multiplication. Observe that defines a semi-inner product (which is actually an inner product) on the linear maps . Notice further, that is a semi-inner preserving map from to the linear maps from to .)

**Definition 4:** The * Hilbert space tensor product *of two Hilbert spaces and , denoted , is the Hausdorff completion of . The image of in is denoted . Vectors of the form are called

Note that

.

**Example: **The Hilbert space tensor product of and can be identified with , as in the hint of Exercise B.

**Exercise C: **For every , and , it holds that

(likewise with the roles of and reversed) and

.

**Exercise D:** If is an orthonormal basis for and is an orthonormal basis for , then is an orthonormal basis for .

**Exercise E: **If and are sets such that and , then

Let us fix notation for what follows. Let be Hilbert spaces, let be an orthonormal basis for and be an orthonormal basis for . By Exercise D, every element in can be written as the norm convergent sum . Rearranging, we see that every element in can be written as the norm convergent sum , where and the summands are all orthogonal. In fact This gives rise to an identification

,

where every is a copy of .

Keep the notation from above. Given and , we define **the tensor product of *** and *, denoted , by first defining it on simple tensors:

.

One then wishes to extend this definition from simple tensors, first to finite sums of simple tensors, and then to the whole space . It suffices to show that defines a bounded operator on finite linear combinations of simple tensors. In fact, it is enough to consider , because proving that is bounded is analogous, and then .

We shall make one more reduction: what we will actually work to show is that is an isometry, whenever (in fact will be unitary, which is very easy to see once one knows that it is well defined). This will suffice because every bounded operator is the sum of four unitaries (Lecture 1). Now, if is unitary, then

.

This shows that preserves inner products, hence is an isometry on the space of finite sums of simple tensors. Whence extends to an isometry (actually a unitary) on .

**Remark: **The explanation we gave here is different than the one I gave in class. If you attended the lecture, can you see why I gave a different explanation?

Now that we know that , it is a simple matter to obtain

.

The tensor product of operators enjoys some other nice properties:

- (and likewise it is linear in the right factor),
- ,
- .

**Definition 5: **Let and be von Neumann algebras. The * von Neumann algebra tensor product* (or simply: the

.

i.e., is the von Neumann algebra generated by all tensor products , where and .

Now for all , let be the operator

.

Letting be the isomorphism mentioned above (given by ) can be considered also as the restriction , so can be considered as an operator in .

For every and every define . This gives a (usually infinite) “operator block matrix” . If , then . There is a bijection

.

Operator block matrices follow the usual algebraic rules, and act on elements in by matrix-versus-column multiplication.

**Proposition 6: **. *Moreover, is in the above set if and only if there is some such that . *

**Proof:** A direct calculation shows ““. Conversely, if then multiplying by from both sides, and using the fact are isometries with orthogonal ranges, one sees that for some . An operator with such a diagonal block operator matrix is easily seen to operator as .

Now define a *-representation by .

**Exercise F:** Is WOT/SOT continuous?

Now let be the rank one operator on , given by . A direct calculation shows that .

**Proposition 7: ***Let be a von Neumann algebra. Then is a von Neumann algebra. To be precise: *

.

**Proof:** Since , we have by the previous proposition . Therefore if then for some . Since , commutes with for all . Therefore so . This shows that , and since is tautological, the proof is complete.

As a consequence, we obtain

.

**Corollary:** .

**Theorem 8:** *Let be a von Neumann algebra. Then *

.

*Thus,*

- .
- .

**Proof: **See the following exercise.

**Exercise G:** Complete the details.

**Project C: **What about ? It is very reasonable and elegant to conjecture that . This is true, but (maybe surprisingly) highly non-trivial. For Project C, show that

- , and
- .

**Exercise H: **Let be a von Neumann algebra, and suppose that there exists a family such that

- ,
- ,
- .

(Such a family is called a *system of matrix *** units in **). Let for some . Then . Prove that

.

Using the above exercise, you can now prove:

**Exercise I: **If is a type factor on a separable Hilbert space, then there is a type factor and a separable Hilbert space such that

.

We now meet a special, particular factor, called the * hyperfinite factor*. It is constructed as follows.

For every , let , and let be the unique unital trace on . The algebra can be identified as a unital subalgebra of , via the unital, injective and trace preserving *-homomorphism given by

.

Let be the normed *-algebra formed as the increasing union of the s (all the algebraic operations are performed in one of the s, that is, if , we find some so that , and then we define in ; same for , , ). On , we define a functional by , if . Since the inclusions are trace preserving (), the functional is well defined.

The hyperfinite factor is defined using the ingredients . The construction itself – called * the GNS construction* – is recurrent in the theory of operator algebras, so let us give it special attention.

In this subsection, we let denote a *-algebra satisfying some of the nice properties of the algebra we just considered above. Thus, does not have to be an increasing union of matrix algebras, it can also be a unital C*-algebra, or the increasing union (or direct limit) of unital C*-algebras. (I am not sure what is the optimal category of *-algebras for which the construction works.)

We also let denote, not necessarily the tracial state treated above, but any * state* on , by which we mean a

**Example:** If is a *-subalgebra and is a unit vector, then is a state. Such a state is called a * vector state*. is faithful if and only if is separating.

The GNS representation will show that essentially all states are vector states (of course, not every state is really actually a vector state).

**Theorem 9 (GNS representation): ***Given a pair of a nice unital normed *-algebra as above, there exists a Hilbert space , a *-representation , and a unit vector such that *

(*) *(“ is cyclic“)*

*and*

*(**) for all .*

*The triple is called the GNS representation of , and is the unique such triple satisfying (*) and (**). *

**Proof:** We begin by defining an inner product on :

.

It is plain to see that is a sesqui-linear form, and it is positive because is positive. We define to be the Hausdorff completion of , and we write for the image of in . Put .

Define for every , the linear map be given by

.

After showing that is bounded on , one can extend it to a linear operator on . Boundedness follows from

,

which follows from (because .

It is then routine to check that is a *-representation, and we omit this. We cannot omit the gratifying step of verifying that it satisfies (**):

.

The uniqueness is left as an exercise.

**Exercise J:** Prove the uniqueness of the GNS representation (make sure you first explain what uniqueness means).

**Example: **Let , where is a probability measure. Let be the state

.

Then is the completion of with respect to the inner product

,

that is, . Also, , and the GNS representation is the representation by multiplication operators, given by .

The above example leads the following notation: given a von Neumann algebra and a state , one writes for , and .

**Example: **Consider , for a countable group , with the state . The the GNS representation is the identity representation.

Now we leave the general GNS construction and return to the particular choice of (where ) with its state . Letting be the GNS representation of we now define to be the von Neumann algebra generated by :

.

is called * the hyperfinite factor*. Since is faithful on , there is no quotient required in the Hausdorff completion, just completion; moreover is faithful:

,

if . By Theorem 1 in Lecture 3, every is isometric, so is isometric. We can therefore push to :

.

Being a vector state, extends from to its WOT/SOT closure . This state is, in fact, a trace: if , we invoke Kaplansky’s density theorem to find bounded nets so

.

is faithful : if then . But then for all , using that is a trace,

.

Since is dense in , .

Now, we will show that is a factor. It will follow that it is a factor, since it has a faithful (normal) trace.

Let . Define for all . The functional is also a trace, because is central. Therefore is also a trace, and by uniqueness of the trace (in finite factors, in particular uniqueness of the trace in ), it must hold that for some constant . Since the inclusions are trace preserving, it must be that for some . Since is WOT dense in , . But , so

.

We conclude that either or . Since is faithful, we have that either or .

This shows that , so is a factor.

**Definition 10:** A von Neumann algebra is said to be * hyperfinite* (or AFD – approximately finite dimensional) if it contains a SOT dense increasing union of finite dimensional C*-algebras, that is , where are all finite dimensional C*-algebras.

The hyperfinite factor is hyperfinite, by construction. By Exercise B in Lecture 4, is also hyperfinite (and also a factor). The reason that is called **THE** hyperfinite factor, is because; **it turns out that every hyperfinite factor is *-isomorphic to **. This is not trivial – it was proved in Murray and von Neumann’s fourth joint paper on the subject. We don’t have time to cover the proof. If you want a heavy project, this is a good choice.

**Project D:** Uniqueness of the hyperfinite factor (this project might be too big, and may spill over into the summer break. But if you are interested this can be a nice experience, we can discuss it, and see how much of it you can do).

Here is another way to look at the hyperfinite factor.

and

.

The imbedding is given by

.

Therefore, one thinks of as the infinite tensor product , and .

Using different finite dimensional algebras and different **states** (not necessarily traces), one gets different kinds of von Neumann algebras with states. Replacing the trace with the state

,

for , Powers obtained infinitely many type factors.

]]>