### Souvenirs from Amsterdam

(I am writing a post on hot trends in mathematics in the midst of war, completely ignoring it. This seems like the wrong thing to do, but my urge to write has overcome me. To any reader of this blog: I wish you a peaceful night, wherever you are).

Last week I returned from the yearly “International Workshop on Operator Theory and Applications”, IWOTA 2014 for short (see the previous post for the topic of my own talk, or this link for the slides).

This conference was very broad (and IWOTA always is). One nice thing about broad conferences is that you are able sometimes to identify a growing trend. In this talk I got particularly excited by a series of talks on “noncommutative function theory” or “free analysis”. There was a special session dedicated to this topic, but I was mostly inspired by a semi-plenary talk by Jim Agler, and also by two interesting talks by Joe Ball and Spela Spenko. I also attended nice talks related to this subject by Victor Vinnikov, Dmitry Kalyuhzni-Verbovetskyi, Baruch Solel, Igor Klep and Bill Helton. This topic has attracted the attention of many operator theorists, for its applications as well as for its inherent beauty, and seems to be accelerating in the last several years; I will only try to give a taste of some neat things that are going on, by telling you about Agler’s talk. What I will not be able to do is to convey Agler’s intense and unique mathematical charisma.

Here is the program of the conference, so you can check out other things that were going on there.

#### 1. Quick reminder on nc function theory

In a previous post (see Section 3 there) I gave the basic definitions of noncommutative (nc) function theory. For the reader’s convenience I will repeat it here (with a few small improvements):

For every $n$, denote by $M_n^d$ the set of all $d$-tuples of $n \times n$ matrices. For $x = (x_1, \ldots, x_d) \in M_n^d$ and $y = (y_1, \ldots, y_d) \in M_k^d$, we let $x \oplus y$ denote the $d$-tuple in $M_{n+k}^d$ with elements

$x_i \oplus y_i := \begin{pmatrix} x_i & 0 \\ 0 & y_i \end{pmatrix}$ , $i =1, \ldots, d$.

We denote my $M^{[d]} = \cup_{n\geq 1}M^d_n$, so $M^{[d]}$ is the set of all $d$-tuples of $n \times n$ matrices, running over all $n$.

Definition: An nc-domain is a set $\Omega \subseteq M^{[d]}$ such that the following conditions hold:

1. For all $n$, the intersection $\Omega \cap M^d_n$ is open,
2. If $x,y \in \Omega$ then $x \oplus y \in \Omega$,
3. If $x \in \Omega \cap M_n^d$ and $u \in M_n$ is unitary then $u^* x u \in \Omega$.

The set $M^{[d]}$ is an nc-domain, for example. So is the union of all tuples of matrices in the open unit ball, or union of all strict row contractions.

On these sets we may define noncommutative functions. A “noncommutative polynomial”, or a “free polynomial”, or a “polynomial in non-commuting variables” (all terms meaning the same thing) is defined in the obvious way. We denote by $\mathbb{P}_d$ the algebra of polynomials in $d$ non-commuting variables. A noncommutative function is supposed to be a generalization of a noncommutative polynomials.

A function $f : M^{[d]} \rightarrow M^{[1]}$ is called a graded function if it maps $M^d_n$ into $M_n$. (As an example, note that noncommutative polynomials are graded functions.)

Definition: An nc-function on and nc-domain $\Omega \subseteq M^{[d]}$ is a graded function such that

1. $f(x\oplus y) = f(x) \oplus f(y)$ ,
2. If $x \in \Omega \cap M_n^d$ and $s$ is an invertible $n \times n$ matrix such that $s^{-1} x s \in \Omega$, then $f(s^{-1} x s) = s^{-1} f(x) s$.

Again, it is easy to check that noncommutative polynomials are nc-functions. Indeed, it seems that these requirements are the bare minimum that we might ask from functions which are to be in some sense limits of noncommutative polynomials.

Until now everything is completely algebraic. In order to obtain analytic results, one needs to introduce a topology. There are several reasonable topologies one may define on $M^{[d]}$. The Free Topology is the topology on $M^{[d]}$ generated by the sets of the form

(1)         $G_\delta = \{x \in M^{[d]} : \|\delta(x) \| < 1\},$

where $\delta$ is a matrix whose entries are in $\mathbb{P}_d$.

This allows us to define holomorphic functions.

Definition: A free holomorphic function on an nc set $\Omega$ is a continuous nc function.

For a free open set $U \subset M^{[d]}$, we denote by $H^\infty(U)$ the bounded free holomorphic functions on $U$. Here by bounded we mean – as you may guess – that there exists $C$ such that $\|f(x)\| \leq C$ for all $x \in U$. The norm in $H^\infty(U)$ is defined as $\|f\|_{H^\infty(U)} = \sup \{\|f(x)\| : x \in U\}$.

#### 2. Jim Agler’s talk: The Nevanlinna-Pick and Cartan extension theorems in non-commuting variables

Jim Agler gave a very nice talk – probably my favourite talk in the conference – about his work on nc function theory, which he has done together with John McCarthy. I already reported on John McCarthy’s talk in Oberwolfach on the same general theme, but Agler’s talk emphasised a rather different aspect. The talk (and therefore this post too) is based on this paper.

Agler began with an interesting result about a Nevanlinna-Pick type interpolation problem. Let $U = G_\delta \subset M^{[d]}$ be a basic (free) open set (see (1) above), and let there be given $N$ points $z_1,\ldots, z_N \in U$. Suppose that one is also given $N$ targets, that is elements $w_1, \ldots, w_N \in M^{[1]}$. Then the interpolation problem is: does there exist a function in $f \in H^\infty(U)$ with norm less than or equal to $1$ such that

(2)          $f(z_i) = w_i$ ,  $i = 1, 2, \ldots, N$.

This problem and its generalizations are very well studied in the setting of classical function theory, or classical analytic-matrix-valued function theory (I explained a little bit in this previous post). In the free setting there are a couple of surprises.

First surprise: In the free setting, only $N=1$ matters. This is surprising, but very simple: in the free setting, $f$ satisfies (2) if and only if it satisfies $f(Z) = W$, where $Z = z_1 \oplus z_2 \oplus \ldots \oplus z_N$ and $W = w_1 \oplus w_2 \oplus \ldots \oplus w_N$. (Recall that both the domain and the function are assumed to respect direct sums).

In the classical (scalar valued analytic functions on the unit disc) case, the interpolation problem with one point “find $f$ analytic with norm less than or equal to one such that $f(Z) = W$” has a solution if and only if $\|W\| \leq 1$. In the free setting this is not so. Moreover, in the classical case, if one drops the norm constraint, then one may always find a bounded analytic $f$ satisfying (2), in fact one may choose a polynomial.

Second surprise: In the free setting, the interpolation problem may fail to have a solution, even if one drops the norm constraint (and even if there is only one point). As in the first surprise, the explanation is really simple. Since free nc functions are supposed to respect direct sums, if $Z$ is diagonal and $W$ is not, then there is no free nc function mapping $Z$ to $W$. (I don’t think this problem can be made to go away if one begins the discussion from free polynomials with matrix coefficients, but I did not think about this enough).

To formulate the theorem that Agler and McCarthy obtain, we need some more notation.

Denote

$I_Z = \{p \in \mathbb{P}_d : p(Z) = 0\}$

and

$V_Z = \{x \in M^{[d]} : \forall p \in I_Z . p(x) = 0\}$.

The set $V_Z$ should be thought of as the smallest noncommutative variety that contains $Z$. Note that if two polynomials agree at the point $Z$, then they must agree on all of $V_Z$. We are now ready to formulate the free interpolation theorem (Theorem 1.3 in their paper).

Theorem: Let $U = G_\delta$, $Z \in M^{[d]} \cap U$ and $W \in M^{[1]}$. There exists a function $f$ in the closed unit ball of $H^\infty(U)$ such that $f(Z) = W$ if and only if the following two conditions hold:

1. There exists $p \in \mathbb{P}_d$ such that $p(Z) = W$.
2. $\sup \{\|p(x)\| : x \in V_Z \cap U\} \leq 1$.

Thus a necessary condition for solvability is that the interpolation problem is solvable with polynomials – that’s something we took for granted in the classical problem. To see that the second condition is also necessary, one first proves that polynomials that agree with $f$ on $Z$ must agree with $f$ on all of $V_Z$.

That the two conditions together are sufficient is harder to prove and I won’t go into that. However, it is very interesting to note that their proof uses the lurking isometry argument (see Section 3 of this previous post for a simple application of this argument). One wonders if there is a theorem in mathematics that a sufficiently clever lurking isometrer cannot prove with that tool!

(BTW IMPORTANT NOTE: I am never sure what I I like better: the lurking isometry argument, or the name it got: “lurking isometry”. In any case, I wish to record here my discovery (which I made during the conference) that the lurking isometry argument’s name is due to Joe Ball. Joe says that the argument itself has been around forever, and does not take credit for the argument itself.)

So that was the “Nevanlinna-Pick interpolation” result of Agler-McCarthy. Now we move to their “Cartan extension” theorem. By “Cartan extension theorem” Agler-McCarthy mean the following classical theorem in several complex variables, which is a consequence of the very deep Cartan’s Theorem B:

Theorem: Let $V$ be an analytic variety in a domain of holomorphy $U \subseteq \mathbb{C}^d$. If $f$ is holomorphic on $V$, then there exists a function $F$ holomorphic on $U$ such that $F\big|_V = f$

To appreciate how remarkable this theorem is, recall that a) an analytic variety is a very thin set inside a domain that contains it, and b) the definition of holomorphicity is local, that is, $f$ is holomorphic on $V$ if for every point $x \in V$ there is an open neighborhood $U_x \subset U$ such and a holomorphic function $F_x$ on $U_x$ such that $F_x \big|_{U_x \cap V} = f\big|_{U_x \cap V}$. There is no easy to see reason why there should exist a globally defined $F$ that restricts to $f$, and indeed the proof of this extension theorem is extremely difficult and involved.

There are theorems of a similar nature, dealing with the problem of when a bounded holomorphic function $f$ on a variety $V$ can be extended to a bounded holomorphic function $F$ on a domain $U \supset V$. In this type of theorem, usually one gets also some kind of bound on sup norms $\|F\|_U \leq C \|f\|_V$. It is known, however, that almost never one gets $\|F\|_U = \|f\|_V$ (see, for example, this other paper of Agler-McCarthy).

Remarkably, in the free setting one is able to get norm preserving extensions as a relatively simple (compared to the classical kind of extension theorems) corollary to the Nevanlinna-Pick interpolation result. The result (Theorem 1.5 in their paper) is as follows:

Theorem: Let $U = G_\delta$ be a free open set, and let $V$ be a free algebraic subvariety of $U$ (thus $V$ is the joint zero set of a set of free polynomials). If $f$ is a function on $V$ that is free holomorphic in a neighborhood of every point of $V$, then there exists a free holomorphic $F \in H^\infty(U)$ that extends $f$ such that

$\|F\|_{H^\infty(U)} = \sup\{\|f(x)\| : x \in V\}$

These two theorems really whet ones appetite for more nc function theory. Especially interesting are questions about nc varieties (can a locally defined nc variety always be given by a global description) and nc domains (is there an nc analog of domain of holomorphy?). There are lots of people working on nc function theory, but it seems like there is a ton of work to be done.