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 author 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 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 .

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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.

]]>

Most of the time we will stick to the assumption that all Hilbert spaces appearing are separable. This will only be needed at one or two spots (can you spot them?).

In addition to “Exercises”, I will start suggesting “Projects”. These projects might require investing a significant amount of time (a student is not expected to choose more than one project).

**Definition 1:** Let be a von Neumann algebra. Two projections in are said to be * equivalent *(or

Note: a crucial part of the definition is that . One can think of the subspaces corresponding to and as subspace the “look the same in the eyes of .”

**Exercise A: **Murray-von Neumann equivalence is an equivalence relation.

**Exercise B: **Describe when two projections are equivalent in (i) , and (ii) .

Recall that in Lecture 3 (Definition 2), we defined the range projection of an operator to be equal to the orthogonal projection onto ; equivalently is the smallest projection such that . One sometimes denotes , and calls it the ** left support **of . Similarly, the

**Proposition 2: ***If is a von Neumann algebra and , then . *

**Proof: **Use the polar decomposition of to find the partial isometry that provides the equivalence.

**Definition 3:** Let be a von Neumann algebra, and let be projections in . We write , and say that is (Murray-von Neumann) * sub-equivalent to * , if there exists a partial isometry such that and (in other words, if is equivalent to a sub-projection of ). If is sub-equivalent but not equivalent to , then we write .

Murray-von Neumann sub-equivalence is a partial ordering on the set of projections in a von Neumann algebra. The reflexivity and transitivity of this relation is straightforward. We will soon prove that it is anti-symmetric.

**Lemma 3:** *Let and be two families of orthogonal projections. If (respectively, ) for all , then (respectively, ) for all . *

**Proof: **If and [respectively, ] then converges strongly to a partial isometry and , while [respectively, ] (note that as and for , so , and likewise ).

The next proposition is an analogue of the Cantor-Bernstein-Schroder theorem, and shows that sub-equivalence is indeed a partial ordering on .

**Proposition 4:** *If and , then . *

**Proof: **Suppose that

and

and

and .

We define two decreasing sequences of projections and by induction. Write and , and define

and for all .

We have that , by assumption, and , by induction (likewise for ).

Now, since for all , we have

,

where . Likewise,

.

Now, for every , we have by definition that . But then is a partial isometry setting up an equivalence between and . Thus, .

Likewise, and . Now we cleverly put this together, obtaining

.

**Proposition 5:** *For two projections and in a von Neumann algebra , TFAE: *

*(i.e., , where and are the central covers of and ).**.**For all nonzero and , and are not equivalent.*

**Proof:** If 1 holds, then for all , so 2 holds. On the other hand, if , then we set . Then is weakly closed ideal, so by Theorem 6 in Lecture 3 for some . But , so , and therefore . But then , and it follows that . Therefore so , and it follows that . So 2 implies 1.

Next, if and where and , then , thus 2 implies 3.

Finally, suppose that 2 fails. If is in , then , so and . By Proposition 2, , so 3 fails as well.

**Definition 6:** Two projections in a von Neumann algebra satisfying the conditions of the previous proposition are said to be **centrally orthogonal. **

The relation of Murray-von Neumann sub-equivalence is a partial ordering, but it is not full: if and are projections in a von Neumann algebra , it may happen that neither nor hold. The following comparison theorem shows how one may always bring projections to a position where they are comparable.

**Theorem 7 (the comparison theorem):** *If and are projections in a von Neumann algebra , then there exists a projection such that and . *

**Proof: **Let be a maximal pair of projections that satisfy , and . Well, if or then we are done. Otherwise, consider the projections and ; these do not have any subprojections and such that , for otherwise the pair would not be maximal. By the previous proposition, and are orthogonal. We find that , so

,

where we used the fact for all central . Likewise, , so

.

**Corollary:** *In a factor, every two projections are comparable. *

**Definition 8: **Let be a projection in a von Neumann algebra . is said to be:

if is abelian.**abelian**if implies that .**finite**if it is not finite.**infinite**if for every central projection , is either infinite or zero.**properly infinite**if for every projection , is either infinite or zero.**purely infinite**

If the identity in a von Neumann algebra is finite/infinite/properly infinite/purely infinite, then is said to be finite/infinite/properly infinite/purely infinite.

**Examples: **

- In an abelian von Neumann algebra, all projections are abelian.
- Perhaps surprisingly, is finite. In fact, every abelian projection is finite (why?).
- It is easy to see which projections in are finite, which are abelian, which are infinite.
- is properly infinite, but not purely infinite. Can you find an example of a projection in a von Neumann algebra that is infinite but not properly infinite? (I bet you can).
- Can you find an example of a projection in a von Neumann algebra that is purely infinite? (I bet you can’t).

**Corollary (to Proposition 5): ***A nonzero projection in a factor is abelian if and only if it is minimal. *

**Proof: **Let be a nonzero abelian projection. Then it must be minimal, because if , then and are not centrally orthogonal, so by Proposition 5 they dominate a pair of equivalent projections, and this would show that is not abelian.

Conversely, if is minimal, then , so is abelian.

**Exercise C: **Let be a family of centrally orthogonal projections (i.e., for ). If every is abelian (finite), then is abelian (finite).

**Definition 9:** A von Neumann algebra is said to be (of):

if for every nonzero central projection , there exists a nonzero abelian in .**Type I**if has no nonzero abelian projections, but for every nonzero central projection , there exists a nonzero finite in .**Type II**if has no nonzero finite projections (i.e., if is purely infinite).**Type III**

For the sake of addressing an issue that better not be addressed, let us say that the algebra acting on the Hilbert space is a von Neumann algebra of any type.

**Theorem 10:** *Let be a von Neumann algebra. Then there exists a unique decomposition of into a direct sum*

* of a type , a type and type von Neumann algebra. *

**Proof:** If there are no abelian projections in , let . Otherwise, let be a maximal family of centrally orthogonal abelian projections. Then, by Exercise C is abelian. Let be the central cover of . Then is a von Neumann algebra, and we claim that it is of type I. Indeed, if , that is, if is a nonzero central projection in , then is a nonzero abelian projection dominated by (if it was zero, then would contradict that fact that is the central cover of ).

By design, is a von Neumann algebra with no abelian projections. Let be a maximal family of centrally orthogonal finite projections in , and let , which is finite thanks to Exercise C. Now let be the central cover of in . Then is a von Neumann algebra, and as in the previous paragraph, one shows that it is of type II.

Finally, letting , we find that is central, and is a type III von Neumann algebra.

We leave it to the reader to check that the decomposition is unique.

Thus, a von Neumann algebra in general does not have to be of a particular type. But for factors, things are nicer.

**Corollary:** *A factor is either of type I, type II, or type III. *

There is a theory, going back to von Neumann, that describes how every von Neumann algebra can be decomposed uniquely into a direct integral of factors. We shall not go into that direction. Since a considerable amount of interesting work on classification theory is concentrated on factors, and there are many interesting examples, we shall mostly speak about factors.

**Example 11: **As our first example, we note that, trivially, commutative von Neumann algebras are always type I. A slightly deeper fact is this: if ( separable) is an abelian type I algebra, then is also of type I. To see this, let be a central projection. We need to show that dominates an abelian projection in . For this end, let be a nonzero vector, and let . If , then is nonzero and . Then is a cyclic and commutative von Neumann algebra on , and we may also assume that it is singly generated. Therefore, is unitarily equivalent to , so is abelian. Therefore, is abelian, and this shows that is type I.

**Example 12: **As an example at the opposite extreme, let us consider , where is a Hilbert space. We have already seen that this is a factor, and it is a type I factor as a special case of the previous example, since . Alternatively, to see that it is type I, one needs to show that it contains a nonzero abelian projection. But clearly, if is a minimal projection (which must have the form ), then is abelian.

In fact, the previous example contains all type I factors (up to isomorphism, not up to unitary equivalence), but we will have to wait a little bit for this. Before the following lemma, the reader might want to review Proposition 9 in Lecture 3.

**Lemma**:* Let be a von Neumann algebra, and let such that . Then is a *-isomorphism from onto . *

**Proof: **We know that is a WOT continuous and surjective *-homomorphism, and so the kernel of this isomorphism is for a central projection . Therefore, , so is a central element dominating . Since we must have , and is injective.

**Theorem 13:** *Let be a von Neumann algebra. The following conditions are equivalent: *

*is type I.**is type I.**There exists a faithful and WOT continuous representation such that is abelian.*

**Remark: **Before the proof, let us remark that in 3 above, will be a von Neumann algebra, because the unit ball of will be WOT compact.

**Remark: **One last remark before the proof: it also true that is type II (respectively, type III) if and only if its commutant is type II (respectively, type III). You may take this as a challenging:

**Exercise D:** Take care of the remark above (for reference to start with, see Section 9.1 in Kadison-Ringrose, Vol II).

**Proof of Theorem 13:** Suppose that is type I. As in the proof of Theorem 10, let be a maximal sum of abelian projections. We then have . The lemma now implies that . But is the commutant of an abelian von Neumann algebra, so by Example 11, is type I, and therefore is type I.

If is type I, then the previous paragraph (with the roles of and reversed) shows that there exists some so that the map is WOT continuous *-isomorphism onto the commutant of an abelian von Neumann algebra.

Finally, if is abelian, then is type I, so is type I.

**Corollary :** *If is a type I factor, then there is a Hilbert space such that is isomorphic to . *

**Proof: **If is a type I factor, then is clearly a factor too, and it is type I by the theorem. Let be an abelian projection. By the corollary to Proposition 5, is minimal, so as a von Neumann subalgebra of . Thus . Since ( being a factor), the lemma shows that .

**Definition 14: **A type I factor is said to be of * type * if it is isomorphic to where . One write if .

The classification problem of type I factors up to isomorphism is therefore settled: there is exactly one type I factor of type for every cardinal , and there aren’t any other examples up to isomorphism (except non-popular examples living on non-separable Hilbert spaces ).

We also see that the equivalence classes of projections in a type I factor, as a partially ordered space, is isomorphic either to for some or to .

If one wants to classify type I factors up to unitary equivalence, there is another issue that comes in, which are not technically prepared to handle at the moment. Roughly, type I factors look like , where and are Hilbert spaces. The meaning of the tensor notation will be made precise in the upcoming lectures.

Finally, we mention that one can describe all type I algebras on separable Hilbert spaces. Roughly, these are just direct sums of “matrix algebras with coefficients in commutative von Neumann algebras”. We leave it to the interested student to work or dig this out.

**Project 1:** Determine the structure theory of type I von Neumann algebras. You might be able to go a significant part of the way on your own. Once stuck, help can be found in the following references: Conway (A Course in Operator Theory), Kadison-Ringrose (Fundamentals of the Theory of Operator Algebras, Vol. II), or Takesaki (Theory of Operator Algebras, Vol. I).

**Definition 15: **A type II factor is said to be a factor if it is finite (that is, if is a finite projection); otherwise it is said to be a factor.

In this section, we will show that the group von Neumann factors are type . Let us recall some notation. If is a countable group, we let be the standard orthonormal basis of , and let the left and right regular representations be given by

and

.

The (left and right) group von Neumann algebras are defined to be and . In the previous lecture, we saw that and vice versa, and that is a factor if and only if was an ICC group (the conjugacy class of every element, except the identity, is infinite).

Recall that on we defined** the trace**

.

“The trace” is a WOT continuous, faithful tracial state, a notion we recall in the following definition:

**Definition 16: **Let be a C*-algebra. A * trace *on is a positive ( for ) and tracial ( for all ) linear functional. A trace is called a

Sometimes we will just say * trace* instead of “tracial state.”

**Theorem 17:** *Let be a countable ICC group. Then is a type factor. *

**Proof: **We already know that is a factor. Now, is finite, which just means that is a finite projection. Indeed, if and , then and , so . This argument shows that is finite.

Being finite, cannot be type , , or . The only remaining possibilities are for , or . Since is infinite dimensional, only the case remains.

To argue a little more “constructively”, we have to show simply that has no abelian projections (since we already know that it is a finite factor). But if it had an abelian projection, the arguments used in the type I case would show that is isomorphic to , which we have seen cannot happen.

Note that the above proof actually shows that any infinite dimensional factor with a tracial state is a type factor. Let us record this.

**Corollary:** *If is an infinite dimensional factor, and if has a faithful tracial state, then is a type factor. *

Nice, we see that there exist factors. Are there many of them? Yes. We will be able to show that there are *some*, not just there is one. Dusa McDuff proved that there are uncountably many non isomorphic ones, in fact uncountably that arise as group von Neumann algebras.

**Project 2: **Read and present McDuff’s paper “A countable infinity of factors” (there is also a second paper “Uncountably many factors”, if you are ambitious).

What about factors? It turns out that a von Neumann algebra is a (separably acting) type factor, if and only if there exists a type factor such that

for an infinite dimensional separable Hilbert space. We plan to discuss tensor products in the next lecture.

In the previous section we saw that ICC groups give rise to type factors. The fact that these factors are type followed from the existence of a faithful (and WOT continuous) tracial state. It can in fact be shown that the existence of such a tracial state characterizes type factors.

**Theorem 18: ***An infinite dimensional factor is of type if and only if it has a faithful tracial state (“trace”, for short). In this case, the trace is unique, and is in fact WOT continuous. *

Before sketching the idea of the proof of the Theorem, we collect some more definitions and propositions.

**Definition 19: **A von Neumann algebra is said to be * diffuse* if it contains no minimal projections.

Thus a factor is diffuse if and only if it is type II or type III.

**Proposition 20 (the halving lemma): ***Let be a diffuse factor. For every , there exist in , such that . *

**Proof: **Since is not minimal, there is some . By Proposition 5, there are mutually equivalent nonzero projections and , and these satisfy .

Now we consider a maximal family of pairs such that and such that all are mutually orthogonal. Set and . Then , and by maximality (and the first part of the proof) .

**Exercise E:** Use the halving lemma to show that if is a factor with a tracial state , then is the unique tracial state, and it is faithful. Conversely, prove that if is a von Neumann algebra with tracial state , and if is the unique tracial state, then must be a factor (hence a factor).

**Exercise F: **Show that if is an infinite projection, then the halving lemma can be improved: *there exist in , such that * (note the difference: and are also equivalent to ).* *

**Idea of the proof of Theorem 18: **Since all the examples come equipped with such a trace, we will not prove this theorem (at least for now). But let us go over the idea of the proof. The corollary to Theorem 17 says that the existence of a trace implies type .

Conversely, let be a type factor. The factor is finite, and the equivalence classes of projections form a totally ordered set. Since there are no minimal projections, we might think of it as being something like – which is indeed what it turns out to be. Using the halving lemma, we construct inductively a sequence of orthogonal projections such that and (equivalently, ). [Indeed, we start by finding such that , then we throw away and find such that , so and so forth. ]

One then proceeds to show that every the sequence can be used to give a “binary expansion” for every projection, i.e., every is equivalent to sum partial sum (this requires work). One then defines , and uses the binary expansion to define

.

The function , currently defined on , is call * the dimension function*. If this can be extended to a WOT continuous state, there is only one way in which it could, since is generated by its projections. One then works and works to show that this indeed extends to a WOT continuous, faithful tracial state.

Uniqueness you have already shown in Exercise E (by slightly less sophisticated technology) basically follows by the same ideas: the value of a (normalized) trace on must be (because and induction), and this determines that value of on any projection , hence on .

We finish this section by showing that the equivalence classes of projections in a factor is isomorphic (as a partially ordered set) to .

**Theorem 21:** *Let be a factor, and let be the tracial state on . Then , and for any pair of projections, , (resp. ) if and only if )resp. ). *

**Proof: **The first assertion really follows from the proof of the above theorem. Next, if then clearly . If , then , so because of positiveness and faithfulness. This basically finishes the proof.

The (non-normalized) trace on a matrix algebra, when evaluated on a projection, gives the dimension of the range of the projection. The trace on type II factor therefore serves as a kind of generalized “dimension function”. von Neumann was fascinated by the fact the dimension of projections in a type II factor can vary continuously.

**Definition 22: **Let be a von Neumann algebra. A * tracial weight* on is a map such that

- for all and .
- for all .

Some immediate consequences: , for all and , and implies .

**Definition 22 (continued): **A tracial weight is said to be ** normal **if (equivalently, ) for every increasing net . It is said to be

Sometimes, we will abbreviate * semi-finite normal trace* instead of the longer “semi-finite normal tracial weight”.

**Example: **Let be a von Neumann algebra with a tracial state . Then is a tracial weight. (A semi-finite tracial weight for which is said to be a * finite* weight.)

**Example: **Let (with Lebesgue measure), and define by

.

Then is indeed a tracial weight (obvious). It is semi-finite because the Lebesgue measure is regular, but it is not finite. It is normal because of the monotone convergence theorem.

**Example: **Let , and let be an orthonormal basis for . Define by

.

When , this is just the usual trace. When , this is just the sum over the diagonal elements in the matrix representation of in the basis . This is, too, a normal, faithful and semi-finite tracial weight, which is not finite (you can prove this with your bare hands; it will also follow from the proof of Proposition 24 below).

**Proposition 23: ***Let be a nonzero normal semi-finite trace on a factor . Then *

*is faithful.**For every , is infinite if and only if .*

**Remark:** Before the proof, note that this proposition also shows that a normal trace on a factor is faithful.

**Proof: **For (1), we will show that if is normal, semi-finite, and **not** faithful, then it is zero. It suffices to show that , for then positivity implies that for every , giving .

If is not faithful, then there is some such that . Then there is also some nonzero such that . Now let be a maximal family of projections equivalent to . Then , because is maximal. Since is a factor, we have by the corollary to the comparability theorem (Theorem 7) that . Thus

,

and

.

But now additivity and normality of implies that

.

That concludes the proof of (1).

For (2), first note that an infinite projection can be written as , where and are orthogonal projections equivalent to (the case of type I is immediate, and the case of types II and III is taken care of by Exercise F). But then

.

Since part (1) rules out , we must have .

Finally, let be a finite projection. By semi-finiteness, there is a nonzero such that and . Let be a maximal family of subprojections of , such that for all . Since is finite, this family has finitely many elements, say . As above, by maximality, so . Therefore we find

.

**Proposition 24 (existence of tracial weights): ***Every type factor and every type factor have a faithful, normal, semi-finite trace. This trace is unique up to a scalar factor. *

**Proof:** Since every factor type I factor has the form , the example given above (the usual trace ) shows that it carries such a tracial weight when (and if , then the usual trace is a finite tracial state satisfying all conditions). Thus, we need only consider the case of a type factor (the reader will notice though, that the proof could work for type just as well). Moreover, the previous proposition shows that faithfulness is immediate, so we only have to prove that there exists a nonzero normal semi-finite weight.

Since is type II, it has a nonzero finite projection .

**Claim:** *Let be a nonzero finite projection in factor. **Then there exists a family of orthogonal and projections, such that for all , and such that . *

Assuming the claim for the moment, we prove the existence of a semi-finite normal tracial weight as follows. The von Neumann algebra is finite, so it has a WOT continuous trace defined on it. Let be partial isometries so that , and . We define

by

.

First of all, this is well defined, because , and the summands are all non-negative. So we have a map , and by properties of infinite summation of non-negative numbers, we have the first item of Definition 22.

Let us drop the habit of skipping details that we have picked up, and show that is normal, semi-finite, tracial weight.

We begin by showing for every increasing net . Write . By positivity,

for all , so .

For the reverse, suppose that , and that . Our goal is to show that for all “sufficiently large” .

On the one hand, there exists a finite set of indices such that

.

(In case that acts on a separable Hilbert space, the family is an infinite sequence, and we could say that there exists an integer such that .)

On the other hand, we have

for all , because is WOT continuous. So there is some , such that

for all . For such , we find

.

This shows that , and normality is established.

Next, let us show that is tracial. Since we have already dealt with normality, the following formal calculations are legal:

.

Equations (*) follow from (SOT) and (**) follows from being a trace on .

It remains to show that is semi-finite. For every finite subset of indices , let . Now, for every , is a finite projection. If , then the net converges SOT to . Therefore, there is some such that . Now – a finite von Neumann algebra. Therefore, there is a projection , such that and . This shows that is semi-finite.

Finally, the uniqueness of follows from uniqueness of the trace on a finite type II factor. It seems like the good time to revert to the habit of skipping details

**Definition 25:** A von Neumann algebra is said to be * semi-finite *if it is type I or type II.

Thus, a factor is semi-finite if and only if it is not type III. We have seen that semi-finite algebras have normal, faithful, semi-finite traces. The converse will be established below. In the meanwhile, I did not forget that we owe ourselves the following:

**Proof of claim:** Let be a maximal family of orthogonal projections such that . Then by maximality, so . If , then we put , and we are done.

Assume that . Let be a partial isometry such that and . Note that , so this implies that the family is infinite (see Exercise G below). For the proof, we will assume that this family is an infinite sequence (by the end of the proof, it should be clear what to do if the cardinality of the family is greater than ; if acts on a separable Hilbert space, then of course the cardinality cannot be strictly greater than ).

Now, being equivalent to , every breaks up as , where and . Now we define a new family of orthogonal projections, by

and

.

Then and .

**Exercise G:** Prove that if and is finite, then is finite. Prove that if and is finite, then is finite. Prove that if are finite and orthogonal projections, then is finite.

After Murray and von Neumann initiated the program of classification into types, they determined all type I algebras and gave examples of type II factors, but at first it was not known whether there exist type III algebras. Then von Neumann provided an example, and later Powers found uncountably many examples, and the classification problem for type III von Neumann algebras is still today a very active field of research. We will see examples of type III factors later on in this course. For now, we record the following result that is one of the technical keys for showing that a factor is type III.

**Proposition 26: ***A factor is of type III if and only if there does not exist a semi-finite normal trace on . *

**Proof:** We already know, by the previous proposition, that if is not type III, then there exists a normal semi-finite trace on it. On the other hand, if is type III, then all projections in are infinite. Proposition 23(2) now tells us that if there was a semi-finite normal trace on , then necessarily for all , but such a weight cannot be semi-finite. This completes the proof.

]]>

The purpose of this book is to serve as the accompanying text for a first course in functional analysis, taken typically by second- and third-year undergraduate students majoring in mathematics. As I prepared for my first time teaching such a course, I found nothing among the countless excellent textbooks in functional analysis available that perfectly suited my needs. I ended up writing my own lecture notes, which evolved into this book (an earlier version appeared on my blog).

The main goals of the course this book is designed to serve are to introduce the student to key notions in functional analysis (complete normed spaces, bounded operators, compact operators), alongside significant applications, with a special emphasis on the Hilbert space setting. The emphasis on Hilbert spaces allows for a rapid development of several topics: Fourier series and the Fourier transform, as well as the spectral theorem for compact normal operators on a Hilbert space.

I did not try to give a comprehensive treatment of the subject, the opposite is true. I did my best to arrange the material in a coherent and effective way, leaving large portions of the theory for a later course. The students who finish this course will be ready (and hopefully, eager) for further study in functional analysis and operator theory, and will have at their disposal a set of tools and a state of mind that may come in handy in any mathematical endeavor they embark on.

The text is written for a reader who is either an undergraduate student, or the instructor in a particular kind of undergraduate course on functional analysis. The background required from the undergraduate student taking this course is minimal: basic linear algebra, calculus up to Riemann integration, and some acquaintance with topological and metric spaces (in fact, the basics of metric spaces will suffice; and all the required material in topology/metric spaces is collected in the appendix).

Some “mathematical maturity” is also assumed. This means that the readers are expected to be able to fill in some details here and there, not freak out when bumping into a slight abuse of notation, and so forth. (For example, a “mathematically mature” reader needs no explanation as to what mathematical maturity is :-).

This book is tailor-made to accompany the course *Introduction to Functional Analysis* given at the Technion — Israel Institute of Technology. The official syllabus of the course is roughly: basic notions of Hilbert spaces and Banach spaces, bounded operators, Fourier series and the Fourier transform, the Stone-Weierstrass theorem, the spectral theorem for compact normal operators on a Hilbert space, and some applications. A key objective, not less important than the particular theorems taught, is to convey some underlying principles of modern analysis.

The design was influenced mainly by the official syllabus, but I also took into account the relative place of the course within the curriculum. The background that I could assume (mentioned above) did not include courses on Lebesgue integration or complex analysis. Another thing to keep in mind was that besides this course, there was no other course in the mathematics undergraduate curriculum giving a rigorous treatment of Fourier series or the Fourier transform. I therefore had to give these topics a respectable place in class. Finally, I also wanted to keep in mind that students who will continue on to graduate studies in analysis will take the department’s graduate course on functional analysis, in which the Hahn-Banach theorems and the consequences of Baire’s theorem are treated thoroughly.

This allowed me to omit these classical topics with a clean conscience, and use my limited time for a deeper study in the context of Hilbert spaces (weak convergence, inverse mapping theorem, spectral theorem for compact normal operators), including some significant applications (PDEs, Hilbert functions spaces, Pick interpolation, the mean ergodic theorem, integral equations, functional equations, Fourier series and the Fourier transform).

An experienced and alert reader might have recognized the inherent pitfall in the plan: how can one give a serious treatment of spaces, and in particular the theory of Fourier series and the Fourier transform, without using the Lebesgue integral? This is a problem which many instructors of introductory functional analysis face, and there are several solutions which can be adopted.

In some departments, the problem is eliminated altogether, either by making a course on Lebesgue integration a prerequisite to a course on functional analysis, or by keeping the introductory course on functional analysis free of spaces, with the main examples of Banach spaces being sequence spaces or spaces of continuous functions. I personally do not like either of these easy solutions. A more pragmatic solution is to use the Lebesgue integral as much as is needed, and to compensate for the students’ background by either giving a crash course on Lebesgue integration or by waving one’s hands where the going gets tough.

I chose a different approach: hit the problem head on using the tools available in basic functional analysis. I define the space to be the completion of the space of piecewise continuous functions on equipped with the norm , which is defined in terms of the familiar Riemann integral. We can then use the Hilbert space framework to derive analytic results, such as convergence of Fourier series of elements in , and in particular we can get results on Fourier series for honest functions, such as convergence for piecewise continuous functions, or uniform convergence for periodic and functions.

Working in this fashion may seem clumsy when one is already used to working with the Lebesgue integral, but, for many applications to analysis it suffices. Moreover, it shows some of the advantages of taking a functional analytic point of view.

I did not invent the approach of defining spaces as completions of certain space of nice functions, but I think that this book is unique in the extent to which the author really adheres to this approach: once the spaces are defined this way, we never look back, and *everything* is done with no measure theory.

To illustrate, in Section 8.2 we prove the mean ergodic theorem. A measure preserving composition operator on is defined first on the dense subspace of continuous functions, and then extended by continuity to the completion. The mean ergodic theorem is proved by Hilbert space methods, as a nice application of some basic operator theory. The statement (see Theorem 8.2.5) in itself is significant and interesting even for piecewise continuous functions — one does not need to know the traditional definition of in order to appreciate it.

Needless to say, this approach was taken because of pedagogical constraints, and I encourage all my students to take a course on measure theory if they are serious about mathematics, *especially* if they are interested in functional analysis. The disadvantages of the approach we take to spaces are highlighted whenever we stare them in the face; for example, in Section 5.3, where we obtain the existence of weak solutions to PDEs in the plane, but fall short of showing that weak solutions are (in some cases) solutions in the classical sense.

The choice of topics and their order was also influenced by my personal teaching philosophy. For example, Hilbert spaces and operators on them are studied before Banach spaces and operators on them. The reasons for this are **(a) **I wanted to get to significant applications to analysis quickly, and **(b)** I do not think that there is a point in introducing greater generality before one can prove significant results in that generality. This is surely not the most efficient way to present the material, but there are plenty of other books giving elegant and efficient presentations, and I had no intention — nor any hope — of outdoing them.

A realistic plan for teaching this course in the format given at the Technion (13 weeks, three hours of lectures and one hour of exercises every week) is to use the material in this book, in the order it appears, from Chapter 1 up to Chapter 12, skipping Chapters 6 and 11. In such a course, there is often time to include a section or two from Chapters 6 or 11, as additional illustrative applications of the theory. Going through the chapters in the order they appear, skipping chapters or sections that are marked by an asterisk, gives more or less the version of the course that I taught.

In an undergraduate program where there is a serious course on harmonic analysis, one may prefer to skip most of the parts on Fourier analysis (except convergence of Fourier series), and use the rest of the book as a basis for the course, either giving more time for the applications, or by teaching the material in Chapter 13 on the Hahn-Banach theorems. I view the chapter on the Hahn-Banach theorems as the first chapter in further studies in functional analysis. In the course that I taught, this topic was given as supplementary reading to highly motivated and capable students.

There are exercises spread throughout the text, which the students are expected to work out. These exercises play an integral part in the development of the material. Additional exercises appear at the end of every chapter. I recommend for the student, as well as the teacher, to read the additional exercises, because some of them contain interesting material that is good to know (e.g., Gibbs phenomenon, von Neumann’s inequality, Hilbert-Schmidt operators). The teaching assistant will also find among the exercises some material better suited for tutorials (e.g., the solution of the heat equation, or the diagonalization of the Fourier transform).

There is no solutions manual, but I invite any instructor who uses this book to teach a course, to contact me if there is an exercise that they cannot solve. With time I may gradually compile a collection of solutions to the most difficult problems.

Some of the questions are original, most of them are not. Having been a student and a teacher in functional and harmonic analysis for several years, I have already seen many similar problems appearing in many places, and some problems are so natural to ask that it does not seem appropriate to try to trace who deserves credit for “inventing” them. I only give reference to questions that I deliberately “borrowed” in the process of preparing this book. The same goes for the body of the material: most of it is standard, and I see no need to cite every mathematician involved; however, if a certain reference influenced my exposition, credit is given.

The appendix contains all the material from metric and topological spaces that is used in this book. Every once in while a serious student — typically majoring in physics or electrical engineering — comes and asks if he or she can take this course without having taken a course on metric spaces. The answer is: yes, if you work through the appendix, there should be no problem.

There are countless good introductory texts on functional analysis and operator theory, and the bibliography contains a healthy sample. As a student and later as a teacher of functional analysis, I especially enjoyed and was influenced by the books by Gohberg and Goldberg, Devito, Kadison and Ringrose, Douglas, Riesz and Sz.-Nagy, Rudin, Arveson, Reed and Simon, and Lax. These are all recommended, but only the first two are appropriate for a beginner. As a service to the reader, let me mention three more recent elementary introductions to functional analysis, by MacCluer, Hasse, and Eidelman-Milman-Tsolomitis. Each one of these looks like an excellent choice for a textbook to accompany a first course.

I want to acknowledge that while working on the book I also made extensive use of the Web (mostly Wikipedia, but also MathOverflow/StackExchange) as a handy reference, to make sure I got things right, e.g., verify that I am using commonly accepted terminology, find optimal phrasing of a problem, etc.

This book could not have been written without the support, encouragement and good advice of my beloved wife, Nohar. Together with Nohar, I feel exceptionally lucky and thankful for our dear children: Anna, Tama, Gev, Em, Shem, Asher and Sarah.

I owe thanks to many people for reading first drafts of these notes and giving me feedback. Among them are Alon Gonen, Shlomi Gover, Ameer Kassis, Amichai Lampert, Eliahu Levy, Daniel Markiewicz, Simeon Reich, Eli Shamovich, Yotam Shapira, and Baruch Solel. I am sorry that I do not remember the names of all the students who pointed a mistake here or there, but I do wish to thank them all. Shlomi Gover and Guy Salomon also contributed a number of exercises. A special thank you goes to Michael Cwikel, Benjamin Passer, Daniel Reem and Guy Salomon, who have read large portions of the notes, found mistakes, and gave me numerous and detailed suggestions on how to improve the presentation.

I bet that after all the corrections by friends and students, there are still some errors here and there. Dear reader: if you find a mistake, please let me know about it! I will maintain a page on my personal website in which I will collect corrections.

I am grateful to Sarfraz Khan from CRC Press for contacting me and inviting me to write a book. I wish to thank Sarfraz, together with Michele Dimont the project editor, for being so helpful and kind throughout. I also owe many thanks to Samar Haddad the proofreader, whose meticulous work greatly improved the text.

My love for the subject and my point of view on it were strongly shaped by my teachers, and in particular by Boris Paneah (my Master’s thesis advisor) and Baruch Solel (my Ph.D. thesis advisor). If this book is any good, then these men deserve much credit.

My parents, Malka and Meir Shalit, have raised me to be a man of books. This one, my first, is dedicated to them.

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As for exercises:

**Exercise A: **Prove that has the ICC property.

**Exercise B: **Prove that there is an increasing sequence of von Neumann subalgebras of , such that is *-isomorphic to and such that .

**Exercise C: **Prove that the free group () has the ICC property.

**Exercise D: **Prove that . What can you say about ? (May require more advanced material: What can you say about , where is a countable discrete abelian group?).

**Exercise E:** We will later see that is not isomorphic to . It might be a nice exercise to think about it now (it might also be not a nice exercise, take your chances).

**Exercise F: **Let be a left convolver, and let be the corresponding convolution operator. Find the adjoint .

**Exercise G: **Prove that is a commutative group, if and only if (or ) is commutative, and that this happens if and only if .

**Exercise H:** Prove that (where is the usual trace) is the unique linear functional on that satisfies and for all .

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