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:

  1. Positive functionals and states on C*-algebras and the GNS construction.
  2. For a linear functional f on a C*-algebra, f\geq 0 \Leftrightarrow f(1) = \|f\|.
  3. The Gelfand-Naimark theorem .
  4. A Hahn-Banach extension theorem: If A is a unital C*-algebra and B is a unital C*-subalgebra, then every state on B extends to a state on A.

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, M will denote a unital operator space, S an operator system, and A and B are C*-algebras. Elements in these spaces will be denoted as a,b etc.

1. Material from Chapter 2,3,4 in Paulsen’s book

Proposition 1: For a positive map \phi : M \to B(H), \|\phi\| \leq 2 \|\phi(1)\|.

Example A: The map \phi(a + bz + c \overline{z}) = \begin{pmatrix} a & 2b \\ 2c & a \end{pmatrix} from span\{1,z,\overline{z}\} \subseteq C(\mathbb{T}) into M_2 is unital positive, and satisfies \|\phi\| = 2 = 2\|\phi(1)\|, 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 \|\phi\| = \|\phi(1)\|.

Proposition 3: (Kadison-Schwarz inequality) For a completely positive (or just two-positive) map \phi(a)^* \phi(a) \leq \|\phi(1)\|\phi(a^*a) and \|\phi(a^*b)\|^2 \leq \|\phi(a^*a)\|\|\phi(b^*b)\|.

Proposition 4: If \phi : M \to B(H) is a unital contraction, then the map \tilde{\phi} : M+M^* \to B(H) given by

\tilde{\phi}(a+b^*) = \phi(a) + \phi(b)^*.

is well defined and positive.

Proposition 5: Consequently, if \phi above is completely contractive, then \tilde{\phi} is completely positive and completely contractive. In particular, a unital map \phi is completely positive if and only if it is completely contractive.

Proposition 6: Let \phi : M \to B be a linear map. If B is a commutative C*-algebra (that is B = C(X)) then \|\phi\|_{cb} = \|\phi\|, so in particular it is completely contractive if and only if it is contractive. Likewise, if M is an operator system (and B still assumed to be a commutative C*-algebra), then \phi 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 \phi is a linear map from a commutative C*-algebra A = C(X) into and arbitrary C*-algebra B, then \phi 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 \phi : A \to B(H) be a completely positive map. Then there exists a Hilbert space K, a *-representation \pi: A \to B(K), and a linear map V \in B(H,K), such that

\phi(a) = V^* \pi(a) V

for all a \in A.

Remarks: 1) Moreover, one can choose (\pi,K,V) in such a way that [\pi(A) VH] = K. In this case, we say that (\pi,K,V) is the minimal Stinespring representation of \phi. As the terminology suggests, the minimal Stinespring representation is unique (up to unitary equivalence that respects V).

2) If \phi is unital, then V^*V = I, that is, V is an isometry. In that case, it is common to identify H with VH and then the Stinespring representation is written as a dilation theorem:

\phi(a) = P_H \pi(a) \big|_H.

2. Discussion – Applications of Stinespring’s theorem

1. As a first application of Stinespring’s theorem, we noted in class that every *-representation of M_n has the form (up to unitary equivalence)

\pi(A) = A \oplus A \oplus \cdots A = \sum_i V_i A V_i^*,

where \{V_i\} is a finite family of isometries with mutually orthogonal ranges. (If \pi is a normal *-representation of B(H), then \pi(A) = \sum_i V_i A V_i^*, where \{V_i\} is now a finite or infinite family of isometries with mutually orthogonal ranges.) From Stinespring’s theorem we find that every CP map from M_n (or on B(H) in the normal case) has a so-called Choi-Krauss decomposition

\phi(A) = V^* \pi(A) V = \sum W_i^* A W_i,

where W_i = V_i^* V.

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 A(\mathbb{D}), which is just the closure of all polynomials in the supremum norm in C(\mathbb{T}). Given a contraction T \in B(H), the map p \mapsto p(T) (defined for polynomials, and extended by continuity to all p \in A(\mathbb{D})) is clearly a contraction, thanks to von Neumann’s inequality. By Proposition 4, \phi extends to a well defined, unital and positive extension of \phi from A(\mathbb{D}) + \overline{A(\mathbb{D})} into B(H), which we also call \phi . By Proposition 1, \phi is bounded. Now, since A(\mathbb{D}) + \overline{A(\mathbb{D})} is dense in C(\mathbb{T}), we obtain a positive unital continuous extension \phi : C(\mathbb{T}) \to B(H). By Proposition 7, \phi is completely positive. (Now by Proposition 2 we find that \phi is in fact completely contractive, but we don’t require it directly). By Stinespring’s theorem, \phi dilates to a *-representation \pi : C(\mathbb{T}) \to B(K), and

p(T) = \phi(p) = P_H \pi(p) \big|_H.

Defining U = \pi(z), we see that U is unitary (since it is the image of a unitary under a representation), and we have found a unitary dilation:

p(T) = P_H p(U) \big|_H,    for all   p \in \mathbb{C}[z].

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 used 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 T_1,T_2 \in B(H), and that we want to prove that there exists a unitary dilation for T_1 and T_2. Let’s try to pull of the above trick. Using Ando’s inequality, getting a unital contractive map \phi : A(\mathbb{D}^2) \to B(H) given by p \mapsto p(A,B). We can extend this to a map \phi : A(\mathbb{D}^2) + \overline{A(\mathbb{D}^2)} \to B(H). At this stage we get stuck: the space S:= A(\mathbb{D}^2) + \overline{A(\mathbb{D}^2)} is not dense in C(\mathbb{T}^2). So we get a unital positive map \phi : S \to B(H), but we don’t know that it is CP, and we don’t know that it extends to C(\mathbb{T}^2). Without the extension we don’t know that \phi is UCP. Even if we did, we wouldn’t be able to apply Stinespring’s theorem (which would give unitaries \pi(z_1) and \pi(z_2), 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 S \subseteq A be an operator system in a C*-algebra A, and let \phi : S \to B(H) be a CP map. Then there exists a CP map \tilde{\phi} : A \to B(H) that extends \phi.

Remark: The extension \tilde{\phi} also has the same norm as \phi.

The proof of Arveson’s extension theorem involved some tools that are worth recording. The proof consists of three steps.

STEP I. 

Proposition 11: (Krein’s extension theorem): Let S \subseteq A be as above, and assume that \phi : S \to \mathbb{C} is a positive map. Then \phi extends to a positive map \tilde{\phi} : A \to \mathbb{C}.

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 \dim H = 1. We then used Krein’s theorem to obtain Arveson’s extension theorem for the case where \dim H = n < \infty. This is :

STEP II. 

A key gadget for this is as follows. If \phi : S \to M_n be a linear map. Then we can define a linear functional s_\phi : M_n(S) \to \mathbb{C} by

(*)   s_\phi([a_{ij}]) = \frac{1}{n} \sum_{i,j}\phi(a_{ij})_{ij}.

Now, one can check that the map \phi \mapsto s_\phi is bijective from L(S,M_n) to $L(M_n(S),\mathbb{C})$, with inverse s \mapsto \phi_s, where

\phi_s(a) = n \sum_{i,j} s(a \otimes E_{ij}) E_{ij} .

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 V,W,Z, then

L(V,L(W,Z)) \cong L(V \otimes W, Z),

by associating \phi : V \to L(W,Z) with the map v \otimes w \mapsto \phi(v)(w). More generally, see “Tensor-Hom Adjunction“.

If we let V = S, W = M_n and Z = \mathbb{C} in the above, then (using that M_n \cong L(M_n,\mathbb{C})), we get the isomorphism

L(S,M_n) \cong L(S \otimes M_n, \mathbb{C}) = L(M_n(S), \mathbb{C}).

Let us chase the identifications. A map \phi \in L(S,M_n) is associated to the linear functional a \otimes T \mapsto \phi(a)(T). Here, we used the identification of M_n with the linear functionals on it. The standard way to do this is to identify every matrix X \in M_n with the linear functional Y \mapsto tr(X^t Y), where tr is the normalized trace \frac{1}{n} Trace. Thus we see that under the above isomorphism, using this particular identification of M_n with its dual, a map \phi \in L(S,M_n) is identified with the linear functional

a \otimes T \mapsto \frac{1}{n}Trace(\phi(a)^t T).

It is now easy to check (for example on elements of the form a \otimes E_{ij}) that this linear functional is precisely the one s_\phi 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 \dim H = n < \infty, the following are equivalent:

  1. \phi is CP.
  2. \phi is n-positive.
  3. s_\phi is positive.

The proof actually goes through noting that (1) implies (2) implies (3) is immediate, and then proving that (3) implies

4. s_\phi extends to a positive linear functional \tilde{s} : A \to B(H).

And then proving that the linear map \tilde{\phi} corresponding to \tilde{s} is CP. This \tilde{\phi} is a CP extension of \phi, so \phi itself must be CP. In particular, this proves Arveson’s extension theorem in the case \dim H < \infty.

STEP III. 

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 \{P_n\} and define a net of CP maps \phi_n : S \to B(P_n H) by \phi_n(a) = P_n \phi(a) \big|_{H_n}. Every \phi_n extends to \tilde{\phi}_n :A \to B(P_nH) by STEP II.

Now the net \{\tilde{\phi}_n\} is a bounded net of CP maps. It turns out that every closed ball in CP(A,B(H)) 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 \psi_n converges to \phi if and only if \langle \psi_n(a) h, g \rangle \to \langle \psi(a) h,g \rangle for all a \in A, h,g \in H. (For more information consult pp. 84-85 in Paulsen’s book).

Now the extension \tilde{\phi} : A \to B(H) of \phi is simply any cluster point of the net \{\tilde{\phi}_n\}. That completes the idea of the proof of Arveson’s extension theorem.