Topics in Operator Theory, Lecture 5 and on
by Orr Shalit
Last week (which was the fourth week, not really the fourth lecture) we finished the proof of Pick’s interpolation theorem, and then I gave a one hour crash course in C*-algebras. The main topics we covered were:
- Positive functionals and states on C*-algebras and the GNS construction.
- For a linear functional on a C*-algebra, .
- The Gelfand-Naimark theorem .
- A Hahn-Banach extension theorem: If is a unital C*-algebra and is a unital C*-subalgebra, then every state on extends to a state on .
From now on we will begin a systematic study of operator spaces, operator systems, and completely positive maps. I will be following my old notes, which for this part are based on Chapters 2 and 3 from Vern Paulsen’s book , and I will make no attempt at writing better notes.
As I start with some basic things this week, the students should brush up on tensor products of vector spaces and of Hilbert spaces.
UPDATE DECEMBER 4th:
I decided to record here in some more details the material that I covered following Paulsen’s book, since my presentation was not 1-1 according to the book. In what follows, will denote a unital operator space, an operator system, and and are C*-algebras. Elements in these spaces will be denoted as etc.
1. Material from Chapter 2,3,4 in Paulsen’s book
Proposition 1: For a positive map , .
Example A: The map from into is unital positive, and satisfies , so the previous proposition gives the best bound.
The reader is invited to use this example to check that several of the results below cannot be improved.
Proposition 2: For a completely positive (or just two-positive) map .
Proposition 3: (Kadison-Schwarz inequality) For a completely positive (or just two-positive) map and .
Proposition 4: If is a unital contraction, then the map given by
is well defined and positive.
Proposition 5: Consequently, if above is completely contractive, then is completely positive and completely contractive. In particular, a unital map is completely positive if and only if it is completely contractive.
Proposition 6: Let be a linear map. If is a commutative C*-algebra (that is ) then , so in particular it is completely contractive if and only if it is contractive. Likewise, if is an operator system (and still assumed to be a commutative C*-algebra), then is completely positive if and only if it is positive.
An analogue of the above proposition “from the other side” is as follows:
Proposition 7: If is a linear map from a commutative C*-algebra into and arbitrary C*-algebra , then is completely positive if and only if it is positive (and a similar statement about cb norms does not hold).
Theorem 8: (Stinespring’s theorem). Let be a completely positive map. Then there exists a Hilbert space , a *-representation , and a linear map , such that
for all .
Remarks: 1) Moreover, one can choose in such a way that . In this case, we say that is the minimal Stinespring representation of . As the terminology suggests, the minimal Stinespring representation is unique (up to unitary equivalence that respects ).
2) If is unital, then , that is, is an isometry. In that case, it is common to identify with and then the Stinespring representation is written as a dilation theorem:
2. Discussion – Applications of Stinespring’s theorem
1. As a first application of Stinespring’s theorem, we described the general form of CP maps on . First, we noted in class that every *-representation of has the form (up to unitary equivalence)
where is a finite family of isometries with mutually orthogonal ranges. (If is a normal *-representation of , then , where is now a finite or infinite family of isometries with mutually orthogonal ranges.) From Stinespring’s theorem we find that every CP map from (or on in the normal case) has a so-called Choi-Krauss decomposition
2. As a second application of Stinespring’s theorem as well as of the above results, we considered a “dilation machine”. As an example, consider the unital operator space , which is just the closure of all polynomials in the supremum norm in . Given a contraction , the map (defined for polynomials, and extended by continuity to all ) is clearly a contraction, thanks to von Neumann’s inequality. By Proposition 4, extends to a well defined, unital and positive extension of from into , which we also call . By Proposition 1, is bounded. Now, since is dense in , we obtain a positive unital continuous extension . By Proposition 7, is completely positive. (Now by Proposition 2 we find that is in fact completely contractive, but we don’t require it directly). By Stinespring’s theorem, dilates to a *-representation , and
Defining , we see that is unitary (since it is the image of a unitary under a representation), and we have found a unitary dilation:
, for all .
Remark: There was a certain amount of cheating involved in the above application, since we used von Neumann’s inequality to pull it off, whereas we proved von Neumann’s inequality using the unitary dilation of a contraction. Don’t feel cheated: first, von Neumann’s inequality can be proved by other methods, which do not go through a unitary dilation. Moreover, this is an outline for a general dilation machine.
Attempting to prove Ando’s dilation theorem using Stinespring’s theorem.
Suppose that we have a pair of commuting contractions , and that we want to prove that there exists a unitary dilation for and . Let’s try to pull of the above trick. Using Ando’s inequality, getting a unital contractive map given by . We can extend this to a map . At this stage we get stuck: the space is not dense in . So we get a unital positive map , but we don’t know that it is CP, and we don’t know that it extends to . Without the extension we don’t know that is UCP. Even if we did, we wouldn’t be able to apply Stinespring’s theorem (which would give unitaries and , the sought after dilation).
Thus we are led to seek a Hahn-Banach type extension theorem for CP maps.
3. Material from Chapters 6 and 7 from Paulsen’s book
Theorem 10: (Arveson’s extension theorem). Let be an operator system in a C*-algebra , and let be a CP map. Then there exists a CP map that extends .
Remark: The extension also has the same norm as .
The proof of Arveson’s extension theorem involved some tools that are worth recording. The proof consists of three steps.
Proposition 11: (Krein’s extension theorem): Let be as above, and assume that is a positive map. Then extends to a positive map .
Recall that when the target C*-algebra is commutative then a positive map is CP (Proposition 7), then Krein’s extension theorem is actually Arveson’s extension theorem for the special case where . We then used Krein’s theorem to obtain Arveson’s extension theorem for the case where . This is :
A key gadget for this is as follows. If be a linear map. Then we can define a linear functional by
Now, one can check that the map is bijective from to , with inverse , where
As one of the students pointed out to me in class, although the above clunky definitions were set up to have the proof (of a statement below) boil down to it might be better to view this in a coordinate free manner. This is done as follows.
For linear spaces , then
by associating with the map . More generally, see “Tensor-Hom Adjunction“.
If we let , and in the above, then (using that ), we get the isomorphism
Let us chase the identifications. A map is associated to the linear functional . Here, we used the identification of with the linear functionals on it. The standard way to do this is to identify every matrix with the linear functional , where is the normalized trace . Thus we see that under the above isomorphism, using this particular identification of with its dual, a map is identified with the linear functional
It is now easy to check (for example on elements of the form ) that this linear functional is precisely the one that we defined above in (*).
With these gadgets in place, the following Proposition is STEP II in the proof of Arveson’s extension theorem.
Proposition 12: With the above notation in the case , the following are equivalent:
- is CP.
- is -positive.
- is positive.
The proof actually goes through noting that (1) implies (2) implies (3) is immediate, and then proving that (3) implies
4. extends to a positive linear functional .
And then proving that the linear map corresponding to is CP. This is a CP extension of , so itself must be CP. In particular, this proves Arveson’s extension theorem in the case .
Finally, the last step of the proof of Arveson’s extension theorem uses the result in the finite dimensional case to obtain the general case. The idea is, that we can take a net of finite dimensional projections and define a net of CP maps by . Every extends to by STEP II.
Now the net is a bounded net of CP maps. It turns out that every closed ball in can be equipped with a topology that makes it compact. This is the BW topology (bounded weak topology), which is the topology where a bounded net converges to if and only if for all , . (For more information consult pp. 84-85 in Paulsen’s book).
Now the extension of is simply any cluster point of the net . That completes the idea of the proof of Arveson’s extension theorem.