Introduction to von Neumann algebras, Lecture 3 (some more generalities, projection constructions, commutative von Neumann algebras)
by Orr Shalit
In this lecture we will describe some projection construction in von Neumann algebras, and we will classify commutative von Neumann algebras.
So far (the first two lectures and in this one), the references I used for preparing these notes are Conway (A Course in Operator Theory) Davidson (C*-algebras by Example), Kadison-Ringrose (Fundamentals of the Theory of Operator Algebras, Vol .I), and the notes on Sorin Popa’s homepage. But since I sometimes insist on putting the pieces together in a different order, the reader should be on the look out for mistakes.
1. *-isomorphisms are isometric
We begin by proving an interesting rigidity property of *-homomorphisms from C*-algebras: they are automatically contractive, and if they are injective then they are isometric.
Theorem 1: Let be a (concrete) C*-algebra possessing a unit, and let be a *-homomorphism (meaning that is a linear map that also satisfies and ). Then for all . Moreover, if is injective, then for all .
In the above theorem, a C*-algebra possessing a unit is simply an algebra that has a multiplicative identity element, not necessarily . If we want to say that the unit of is actually equal to , then we will say that is a unital C*-subalgebra of .
For the proof, we require the following lemma, which is sometimes referred to as the spectral permanence theorem. If is a unital Banach algebra, and , then the spectrum of relative to is the set
has no inverse in .
Thus, the spectrum of an operator as we defined it in the first lecture, is the spectrum relative to , i.e., . It is conceivable that if is an element of a unital C*-algebra , then is bigger than (in other words, it is possible, that an element in has an inverse in which is not contained in ). One of the remarkable properties of C*-algebras is that this does not happen.
Lemma: If is an operator in a unital C*-subalgebra , then is an invertible operator if and only if has an inverse . Consequently, for every unital C*-subalgebras ,
Proof of the lemma: Consider first the case of a selfadjoint operator . Then by the spectral theorem, we may as well assume that is a multiplication operator . A moment of thought reveals that if is a bounded invertible operator, then the inverse must be equal to . Since by the bounded inverse theorem is bounded, we find that there is some such that almost everywhere. This implies that Now, the function is continuous on , and by the continuous functional calculus we have that is contained in the C*-algebra generated by and .
If is a general invertible element in , then is also invertible, contained in , and is selfadjoint. Thus, by the previous paragraph, , so is in .
Finally, the assertion regarding spectrum follows immediately from the assertion about invertible operators.
Exercise A: Give an example of an operator and a unital norm closed operator subalgebra containing such that
Proof of Theorem 1: We may assume that is a unital C*-subalgebra of and that (because otherwise, and are projections, and everything orthogonal to the ranges of these projections is irrelevant). For every , if is an invertible operator, then it is invertible in , and it follows that is invertible in (with inverse ). Thus, . Then, for every ,
but – as – the right hand side is less than . We find that for every .
In particular, we find that every *-homomorphism is continuous.
Now, suppose that is injective, and assume for contradiction that is not isometric. Then there exists some such that and (the fact that we can assume that there is a positive element on which norm preservation fails follows from the C* identity). Let be such that on , and . Since for every polynomial, and since is continuous, we have that . By the continuous functional calculus, , so , but this is a contradiction to injectivity, because .
Remark: In these notes we are working with concrete C*-algebras. We have reached a point that nicely exemplifies the deficiency in this concrete approach. If we were using abstract C*-algebras, we would easily be able to show that the image of a C*-algebra under a *-homomorphism is a C*-algebra (the key issue is that it is closed). This is done by noting that, since a *-homomorphism is continuous, its kernel is a closed ideal. Therefore, we would be able to take the quotient , and we get an injective *-homomorphism which must be isometric, hence its image is closed. But the image of is equal to the image of , so the image of is a C*-algebra. Thus, we see that the abstract approach has significant advantages, even at the early parts of the theory.
2. Some constructions involving projections
Definition 2: For every , the projection onto is called the range projection of (also called the left support of ). The range projection of is denoted . Our first goal is to show that von Neumann algebras are closed under the operation of “passing to the range projection”. In solving the exercise below (as well as some following exercises), Theorem 11 from Lecture 2 will be useful.
Exercise B: Prove that if is a von Neumann algebra and , then . (Hint: First, assume that , and without loss of generality assume , and consider the monotone sequence (note that having the functional calculus always on your mind, this is a natural sequence to consider). To see the result for general operators, find the relationship between and .)
If is a family of projections on , we define to be the projection on the closed subspace spanned by the subspaces , and we define to be the projection onto the intersection . The projections and are called the sum and intersection of the family. Sometimes, one writes for and for .
Proposition 3: If all the projections in the family are contained in a von Neumann algebra , then the sum and intersection are also in .
Proof: It is enough to prove the claim for the sum (why?). Recall that a projection belongs to if and only if the range of is invariant for (see the lemma used in the proof of the double commutant theorem, in Lecture 2). Reversing the role of and , we see that if every is in , then every space is invariant for . It follows that is invariant for , thus .
The following two exercises give an alternative way of proving the above proposition.
Exercise C: If are projections, then and ( if this is difficult, consult Kadison-Ringrose, Vol. I, Section 2.5).
Exercise D: Prove that if is a von Neumann algebra, and if , then and , by making use of the previous exercise, and applying the theorem on monotone nets of operators.
3. The center of a von Neumann algebra, factors
Definition 4: The center of a von Neumann algebra is defined to be . A projection is said to be a central projection if it is contained in the center of .
The center of a von Neumann algebra is an commutative von Neumann algebra. Note that the although the commutant of a von Neumann algebra depends on the particular representation of the von Neumann algebra and not on the *-algebraic structure, the center depends only on the *-algebraic structure.
Definition 5: A von Neumann algebra is said to be a factor if .
Clearly, a von Neumann algebra is a factor if and only it has no central projections.
- The algebra is a factor (we have essentially seen already that so ). At the other extreme, every commutative von Neumann algebra is its own center. The only commutative von Neumann algebra that is a factor is .
- We will see in a later lecture that if is a countable group and if for every , the conjugacy class is infinite, then the group von Neumann algebra is a factor. (A group for which for every is called an ICC group; examples of countable ICC groups are the free groups , and the group consisting of all permutations of the natural numbers that fix all but finitely many elements.)
In a way that can be made precise (but will probably not be made precise in this course) factors form the “building blocks” of von Neumann algebras. We will now see that is a factor if and only if it has no weakly closed ideals, so that the factor are in a sense the “simple” von Neumann algebras.
If , then it is easy to see that is a *-subalgebra of , and in fact, it is WOT closed. To see that it is WOT closed, note that every element in can be identified with the operator . The map can therefore be considered as a (contractive) *-homomorphism of onto , which maps onto (onto, because this map is “the identity” on ). Since this map is WOT continuous and is WOT compact, we obtain that is WOT compact. Invoking Corollary 15 of the previous lecture, we see that can be considered to be a von Neumann algebra in . It follows that is a WOT closed, *-closed, two sided ideal in . Note that in this case we have the decomposition , i.e.,
It turns out that all WOT closed (two sided) ideals have this form.
Theorem 6: Let be a von Neumann algebra, and let be a WOT closed two sided ideal. Then , and, moreover, there exists a central projection such that .
Exercise E: Prove Theorem 4. (Hint: To prove that is selfadjoint, use the the polar decomposition. To prove the existence of the form , note that the projection , if it exists, must be equal to the sup of all projections in , thus one can define as the supremum and prove that such a supremum does what we want. To show that there are sufficiently many projections in use the Borel functional calculus. The notion of range projection (see the next section) will also be useful for proving that .)
Corollary 7: A von Neumann algebra is a factor if and only if has no non-trivial WOT closed ideals.
Proof: Indeed, is a factor precisely when has no non-trivial projections, and this corresponds to the situation where there are no non-trivial WOT closed ideals.
Definition 8: Let be a von Neumann algebra. For every operator , the central cover of (also called the central support or central carrier) is the projection
Exercise F: Suppose that is a projection in a von Neumann algebra . Prove that is the orthogonal projection onto the subspace
4. More projection constructs
Suppose that is a von Neumann algebra, an let be a projection. We can define a set by
If is either in or in , then is a *-algebra, called the reduced or the induced von Neumann algebra, respectively. In the case , then can be identified with the *-subalgebra . In the case that , then the compression map is a *-homomorphism (make sure you understand why), and can be identified with the map or .
Proposition 9: Let be a von Neumann algebra, and let . Then and are both von Neumann algebras on , and are mutual commutants: .
Proof: To prove that that is a von Neumann algebra, one runs an arguent similar to the case where , which we treated above. We will show that is a von Neumann algebra (this will also follow from , but the proof is interesting in itself).
To see that is a von Neumann on , we consider the map . is a WOT continuous *-homomorphism. Its kernel is therefore a WOT closed ideal, hence, by Theorem 4, for some . On the map must be injective. Therefore, it is isometric (Theorem 1). So , so the latter – as the WOT continuous image of a WOT compact set – is WOT compact, and therefore – by Corollary 15 in the previous lecture – is a von Neumann algebra.
Let . Then for all ,
and this shows that .
For the reverse inclusion (which we will prove as , using the fact that we already established that is a von Neumann algebra), it suffices to show that every unitary is the compression of some (recall Exercise D in the first lecture, which shows that a C*-algebra is spanned by its unitary elements). For such a , we define an operator by
for and , and extending linearly. We see that
for , and .
So, is an isometry on . We extend to be a partial isometry on by defining on . (The partial isometry satisfies that is the orthogonal projection onto , that is, it is (computed in )). For every we have (1) for all and
and (2) for every we have , so
Therefore, , so . But , and the proof is complete.
Exercise G: Prove that also in the case that , it also holds that is a von Neumann algebra and that . Moreover, show that if or , then .
5. Every countably generated commutative von Neumann algebra is singly generated
Theorem 10: Let be a separable Hilbert space and a commutative von Neumann algebra. Then there is a selfadjoint operator such that .
Proof: By Corollary 15 in Lecture 2, the unit ball of is WOT compact, and by Exercise E in that lecture, it is metrizable. Thus, is separable (as every compact metric space is). Since every selfadjoint operator can be approximated by its spectral projections associated with intervals with rational endpoints, we see that there is a sequence that generates as a von Neumann algebra.
We now claim that the operator generates as a von Neumann algebra. To establish this, it is enough to show that for all . And really, it suffices to concentrate on showing that , because if that’s true then will be in and one can proceed inductively.
We have the direct sum (recall that is a commutative algebra). Let as above. Since ,
It follows that . Likewise, , so , therefore .
We see that . Therefore, is continuous on , and , as required.
6. Separating vectors and cyclic vectors
Definition 11: Let be a *-algebra. A vector is said to be cyclic if . It is said to be separating for if for all , implies that .
Lemma 12: If is cyclic for a *-algebra , then is separating for . The converse is also true when is non-degenerate.
Proof: Exercise (easy).
Proposition 13: Every commutative von Neumann algebra on a separable Hilbert space has a separating vector.
Proof: We will prove that for every von Neumann algebra has a cyclic vector; the result then follows from the lemma above.
As in Exercise I of Lecture 1, we may write as a direct sum of “cyclic subspaces”, where for some . Since every is invariant for , the projection onto belongs to (the inclusion following from commutativity). It follows that the vector is cyclic for . Indeed, for every , , so , and this completes the proof.
7. Classification of commutative von Neumann algebra
We now describe what commutative von Neumann algebras “look like”, and this will lead to a classification of all commutative von Neumann algebras.
Recall that the support of a Borel measure on , denoted , is the closure of the set of all points , that satisfy
for every neighborhood , .
A measure is said to be compactly supported if its support is compact.
Theorem 14: Let be a commutative von Neumann algebra on a separable Hilbert space. Then there exists a regular, compactly supported, Borel probablity measure on , such that is *-isomorphic to .
Proof: Let be a separating vector for , given by Proposition 13. Let , and . Then , so is a *-homomorphism from onto . In fact, it is a *-isomorphism, because is a separating vector. Will show that is unitarily equivalent to for some .
Let be an operator – the existence of which is guaranteed by Theorem 10 – for which . As the map is a WOT continuous *-isomorphism, we have that is generated as a von Neumann algebra by . The vector is cyclic for , so by a previous result (see Exercise I in Lecture 2) we have that is unitarily to , where is as we require. That completes the proof.
Now it is our goal to understand what kind of C*-algebras arise as for a regular, compactly supported, Borel probablity measure on the real line.
Given such a measure, let . Then the cardinality of is at most . Define by
Generally, a measure is said to be discrete if , and continuous if for every point in the space. The measures and are then seen to be discrete and continuous, respectively. We have the decomposition . The measures and are called the discrete and continuous parts of , and the decomposition above is unique up to sets of measure zero. Both measures are also regular Borel measures if is.
Exercise H: Prove that is *-isomorphic to .
Exercise I: Prove that if is a discrete probability measure, then there exists a countable set such that is *-isomorphic to .
It remains to understand for finite continuous measures. It turns out that the only von Neumann algebra that arises this way, is (where the measure is the Lebesgue measure).
Theorem 15: Let be a compactly supported and continuous regular Borel probability measure on the real line. Then is *-isomorphic to .
Proof: Assume that the support of is contained in , and assume also that . Thus, we can think of as a measure on . It will be convenient also to set , and to think of as a Borel probability measure on . It is straightforward to show that , and we switch back and forth between the two viewpoints as convenient.
We will construct a map such that is a *-isomorphism of onto .
The map is defined by
(In order for this to have meaning it is convenient to think of as a measure on . In fact we can also think of as a map defined ). Then is a non-decreasing function, so it is Borel measureable. In fact, the map is strictly increasing on , with the exception of a countable number of pairs for which (there at most such pairs, because for all i, the interval is disjoint from but contained in , thus the total length of these intervals is finite). Moreover, is onto, because is continuous. We can therefore invert , missing at most countably many points in . The map is also increasing, so it Borel measurable.
Now, is measure preserving. By this, we mean that if , then is equal to the Lebesgue measure of . Since both measures are regular, and since maps intervals to intervals, it is enough to consider the case of . Let such that . Then
but the latter is the Lebesgue measure of , as required. The converse is also true, so preserves the measure in both directions.
Now one has to show that the map is a well defined *-isomorphism of onto . The final details are left to the reader.
Conclusion: Every commutative von Neumann algebra on a separable Hilbert space has one of the following forms (up to *-isomorphism):
- , for a countable set ,
- , for a countable set .