Introduction to von Neumann algebras, Lecture 5 (comparison of projections and classification into types of von Neumann algebras)
by Orr Shalit
In the previous lecture we discussed the group von Neumann algebras, and we saw that they can never be isomorphic to . There is something fundamentally different about these algebras, and this was manifested by the existence of a trace. von Neumann algebras with traces are special, and the existence or non-existence of a trace can be used to classify von Neumann algebras, into rather broad “types”. In this lecture we will study the theory of Murray and von Neumann on the comparison of projections and the use of this theory to classify von Neumann algebras into “types”. We will also see how traces (or generalized traces) fit in. (For preparing these notes, I used Takesaki (Vol I) and Kadison-Ringrose (Vol. II).)
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).
1. Murray- von Neumann equivalence
Definition 1: Let be a von Neumann algebra. Two projections in are said to be equivalent (or Murray-von neumann equivalent) if there exists a partial isometry such that and . If and are equivalent, then we write .
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 right support of is the projection onto , it is equal to , and is the smallest projection such that . In Exercise B of Lecture 3, you proved that if is an element of a von Neumann algebra, then (and therefore also ).
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
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.
2. Types of projections and types of vN algebras
Definition 8: Let be a projection in a von Neumann algebra . is said to be:
- abelian if is abelian.
- finite if implies that .
- infinite if it is not finite.
- properly infinite if for every central projection , is either infinite or zero.
- purely infinite if for every projection , is either infinite or zero.
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.
- 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):
- Type I if for every nonzero central projection , there exists a nonzero abelian in .
- Type II if has no nonzero abelian projections, but for every nonzero central projection , there exists a nonzero finite in .
- Type III if has no nonzero finite projections (i.e., if is purely infinite).
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.
3. Type I algebras and 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).
4. Existence of type II factors
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
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 tracial state if . A trace is said to be faithful if for implies .
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.
5. A little more on factors
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.
6. Semifinite tracial weights on type and factors
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 faithful if, as usual, if . It is semi-finite if every nonzero , there is some and such that and .
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
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
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
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.
7. III factors
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.