Dilations, inclusions of matrix convex sets, and completely positive maps

by Orr Shalit

In part to help myself to prepare for my talk in the upcoming IWOTA, and in part to help myself prepare for getting back to doing research on this subject now that the semester is over, I am going to write a little exposition on my joint paper with Davidson, Dor-On and Solel, Dilations, inclusions of matrix convex sets, and completely positive maps. Here are the slides of my talk.

The research on this paper began as part of a project on the interpolation problem for unital completely positive maps*, but while thinking on the problem we were led to other problems as well. Our work was heavily influenced by works of Helton, Klep, McCullough and Schweighofer (some which I wrote about the the second section of this previous post), but goes beyond. I will try to present our work by a narrative that is somewhat different from the way the story is told in our paper. In my upcoming talk I will concentrate on one aspect that I think is most suitable for a broad audience. One of my coauthors, Adam Dor-On, will give a complimentary talk dealing with some more “operator-algebraic” aspects of our work in the Multivariable Operator Theory special session.

[*The interpolation problem for unital completely positive maps is the problem of finding conditions for the existence of a unital completely positive (UCP) map that sends a given set of operators A_1, \ldots, A_d to another given set B_1, \ldots, B_d. See Section 3 below.]

1. Dilations

Given a tuple of operators A = (A_1, \ldots, A_d) \in B(H)^d, a tuple B = (B_1, \ldots, B_d) \in B(K)^d is said to be a dilation of A if H is a subspace of K and if for every i=1,\ldots, d,

A_i = P_H B_i \big|_H.

This is a classical, highly developed, and popular notion (see this survey by Ambrozie and Muller, or this exposition which I wrote with Eliahu Levy a few years ago, which has a somewhat non-classical twist).

Classically, the operators given at the outset were assumed commuting, and one searched for a dilation consisting of commuting normal operators. Helton, Klep, McCullough and Schweighofer’s dilation result is completely different, because they dilate noncommuting operators to commuting operators.

Theorem (Helton, Klep, McCullough, Schweighofer). Let H be a fixed (real) Hilbert space of dimension n. There exists a Hilbert space K, an isometry V: H \to K, a constant \vartheta_n and a commuting family \mathcal{C} of selfadjoint contractions on K, such that for every selfadjoint A \in B(H), there is some N \in \mathcal{C} such that 

A = \vartheta_n V^* N V

In other words, they say that all selfadjoint n \times n contractions can be simultaneously dilated (up to some scaling) to a commuting family of selfadjoint contractions. I remember my reaction when seeing the abstract of their preprint on the arxiv mailing list. My first reaction was denial: this can’t be true. Next: well, maybe it’s true, but then it is probably trivial. And then: in any case, it is probably not interesting – just another variation on a theme. Who cares?

Wrong, wrong, and wrong. It’s true (obviously), non-trivial (in particular because they “find” the optimal constant), and turns out to be very important (see the introduction of their paper to see connections to semidefinite programming). And it is very interesting from an operator theory/operator algebras perspective.

We were highly inspired by the HKMS paper, and in particular we were led to ask the following questions:

  1. Does a version of this work for operators on an infinite dimensional Hilbert space? Can one find a dilation constant that is independent of the ranks of the dilated operators?
  2. If one starts with operators on a finite dimensional Hilbert space, can they be dilated to commuting normal operators acting on a finite dimensional Hilbert space?
  3. Given additional information, can one obtain more precise spectral control on the dilation, something more specific than that they are all contractions?

(There is also a natural question of whether this works for complex instead of real spaces, but this is not hard and less interesting). For 1 and 2 one must restrict the number of operators being dilated, because of the sharpness of the HKMS result.

In our paper we show, regarding question 2 above, that starting from finite dimensional spaces one can dilate to operators on finite dimensional spaces – this is a general fact that follows from work of my former student David Cohen’s master thesis. So I will concentrate here only on answers to questions 1 and 3.

Theorem (Davidson, Dor-On, Shalit and Solel). Let K be a convex subset of \mathbb{R}^d which has some nice symmetry properties. For every d-tuple of selfadjoint operators A=(A_1, \ldots, A_d) such that the joint numerical range W_1(A) is contained in K, there is a d-tuple of commuting self-adjoint operators N = (N_1, \ldots, N_d) such that the joint spectrum \sigma(N) is contained in K and such that d N is a dilation of A

In other words, after rescaling by d, the tuple A can be dilated to a commuting normal tuple with joint spectrum contained in the joint numerical range of A.

Some explanations are in order:

  1. What is joint numerical range?
  2. What is joint spectrum?
  3. What do we mean by nice symmetry properties?

The joint numerical range of a tuple A = (A_1, \ldots, A_d) is just the set W_1(A) consisting of all points of the form (\phi(A_1), \ldots, \phi(A_d)) \in \mathbb{R}^d, where \phi is a state. (WARNING: The definition I am using here might be different from another one that comes to mind, that is the set of all such points where \phi ranges over all vector states, and this gives rise to a different notion, in general).

The joint spectrum of commuting normal operators is a basic notion, especially when the operators N_1, \ldots, N_d act on a finite dimensional space, so I won’t elaborate further.

By nice symmetry properties, I mean, roughly, that K is invariant under the projection onto each member of a frame. Anything that is nicely symmetric around the origin, like a convex regular polytope, or a ball, centred around the origin, satisfies this. In the case where K is a cube [-1,1]^d , the assumption W_1(A) \subseteq K corresponds precisely to the assumption that A is a tuple of contractions (that is, \|A_i\|\leq 1 for all i), as in the theorem of HKMS.

Note that our constant d, unlike HKMS’s constant \vartheta_n, is independent of the ranks of the operators, and this theorem holds for operators on an infinite dimensional Hilbert space. On the other hand, unlike HKMS, our constant depends on the number d of operators – we have to fix the number of operators to be d. Thus our theorem is not a generalization of their theorem – it is a different kind of theorem (our theorem is likely to be of less practical significance).

2. Inclusions of matrix convex sets

My goal now is to explain an application of our dilation result. A matrix convex set  is as set of the form S = \cup_n S_n where for all n, the set S_n is a subset of M_n^d (the set of all d-tuples of n \times n matrices) or a subset of (M_n)^d_{sa} (the set of all d-tuples of selfadjoint n \times n matrices), such that

  1. S is closed under direct sums;that is, if X, Y \in S, then X \oplus Y \in S, where X \oplus Y = (X_1 \oplus Y_1, \ldots, X_d \oplus Y_d).
  2. S is closed under the application of unital completely positive (UCP) maps; that is, if X \in S_n, then (\phi(X_1), \ldots, \phi(X_d)) \in S for every UCP map \phi: M_n \to M_k.

If S and T = \cup_n T_n are two matrix convex sets, we say that S \subseteq T if S_n \subseteq T_n for all n. Here is a natural question:

Suppose that we know S_1. What can we say about S? In particular, if we know that S_1 \subseteq T_1, what can we say about the relationship between S and T

(Helton, Klep and McCullough explain in this paper why this kind of question is interesting and even give references to why it is of practical significance.)

To address the “in particular” part of the question above, it is helpful to define, given a convex set K \subseteq \mathbb{R}^d, the matrix convex sets W^{max}(K) and W^{min}(K), which are the maximal and minimal matrix convex sets “living over” K. These maximal and minimal sets do exist, and we observed that W^{min}(K) is equal to the set of all tuples X that have a normal dilation N such that \sigma(N) \subseteq K. We thus obtain the following corollary.

Corollary (DDSS): For a nice convex set K as in the theorem above, W^{max}(K) \subseteq d W^{min}(K). In particular, if S and T are matrix convex sets, then S_1 \subseteq T_1 = K implies that S \subseteq dT

We also show by example that d is the best constant that works for all such K. For example, d is the best constant that works when K is the (Euclidean) unit ball. Embarrassingly, we do not know whether d is the best constant when K is equal to the unit cube [-1,1]^d.

In fact, we obtain a shaper inclusion theorem than the corollary above. I will only illustrate with one example – for the full theorem (involving frames and polar duals) see the last section in our paper.

Let D_d denote the d-dimensional “unit diamond”,

D_d = \{ x \in \mathbb{R}^d : \sum |x_i| \leq 1\}.

Theorem (DDSS): W^{max}(D_d) \subseteq W^{min}([-1,1]^d).

This is a significant strengthening of the corollary above, because [-1,1]^d \subset d D_d, and has a much smaller diameter. So W^{min}([-1,1^d]) is significantly smaller than d W^{min}(D_d).

3. Interpolation of UCP maps

Finally, I wish to mention one reason why one might be interested in the inclusion problems of matrix convex sets; this reason being the connection to interpolation of UCP maps. Among the many topics I won’t have time to discuss in my talk, this one is the one I wish to talk about most, because the interpolation problem was really the thing we had on our mind when starting this research. (Adam Dor-On will talk about this in his talk).

Given a tuple A = (A_1, \ldots, A_d) of operators on H, we define W_n(A) to be the set

W_n(A) = \{ (\phi(A_1), \ldots, \phi(A_d)) : \phi : B(H) \to M_n is a UCP map \}.

Then the set W(A) = \cup_n W_n(A) is a matrix convex set in \cup_n M_n^d.

Theorem (DDSS): Let A = (A_1, \ldots, A_d) and B = (B_1, \ldots, B_d) be two d-tuples of operators on Hilbert spaces H and K, respectively. Then there exists a UCP map \phi: B(H) \to B(K) such that \phi(A_i) = B_i for all i=1, \ldots, d if and only W(B) \subseteq W(A)

This theorem almost looks obvious, and indeed it is an easy extension of a Theorem of Arveson treating the case d=1. But from this theorem we recover and sharpen some results from the literature, for example, we show that for tuples of commuting normal operators A and B, there is a UCP map sending A to B if and only the joint spectrum of B is contained in the convex hull of the joint spectrum of A. In turn, this reduced to a theorem of Li and Poon (see this paper), that obtained this for commuting normal matrices.

We also obtain an approximate version of the above theorem: there is a UCP map \phi : B(H) \to B(K) such that \|\phi(A_i) - B_i\| \leq \epsilon if and only if, roughly, W(B) is contained in W(A) up to an error of at most \epsilon.

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