Souvenirs from the Black Forest

by Orr Shalit

Last week I attended a workshop titled “Hilbert modules and complex geometry” in MFO (Oberwolfach). In this post I wish to tell about some interesting things that I have learned. There were many great talks to choose from. Below is a sample, in short form, with links.

1. A major advance on Arveson’s essential normality conjecture

One of the most exciting talks for me was the one given by Miroslav Englis, on a recent paper of his together with Joerg Eschmeier on the essential normality conjecture (I explained what this conjecture is about in this previous post). In their paper, Englis and Eschmeier treat what we like to call “the geometric Arveson-Douglas conjecture”, which says that quotient modules of Drury-Arveson space by a radical ideal I is p-essentially normal for every p> dim(I). Englis and Eschmeier prove the conjecture for the case where the variety of the ideal is smooth away from the boundary. Their methods use the theory of generalised Toeplitz operators developed by Boutet de Monvel and Guillemin three decades ago. The theory used is quite technical and perhaps scary for one who is not used to it, but after the talk by Miroslav Englis I felt that the approach is natural, in that the theory used is well suited to deal with self commutators of the shift and estimate them. The authors had to adapt this theory also to the non smooth case (since a homogeneous variety is never smooth).

This result is biggest step forward on this problem since Guo and Wang’s paper in 2008.

At the conference I also learned that Ron Douglas, Xiang Tang and Guoliang Yu also very recently published a paper making the same advance, but using a different approach. What both papers have in common, besides solving the problem, is that they are both based on figuring out how to adapt tools that have been available for thirty years to this problem. Both the existence of the tools, as well as the knowledge of how to carefully use them to this problem, are crucial.

2. Projective spectrum – a spectrum for non commuting tuples

Rongwei Yang gave a very interesting talk about the “projective spectrum”. The outset for this talk is that the spectrum of a Banach algebra element is a central and useful concept in functional analysis. Several notions of spectrum for commuting tuples of Banach algebra elements have been developed over the years (starting immediately after Gelfand’s work). Yang proposes a definition for joint spectrum for a tuple of not-necessarily commuting elements in a Banach algebra.

Definition: Let A = (A_0, A_1, \ldots, A_n) be an n+1-tuple of elements in a unital Banach algebra B. For every z = (z_0, z_1, \ldots, z_n) denote A(z) = z_0 A_0 + \ldots + z_n A_n. Let P(A) = \{z \in \mathbb{C}^{n+1} : A(z) \notin B^{-1}\} (here B^{-1} is the set of invertible elements in B). Let p(A) denote the projection of P(A) in projective space \mathbb{P}^n.

It is easy to see that when A_0 = 1, A_1 = a, then P(A_0,A_1) and p(A_0,A_1) are closely related to the usual spectrum \sigma_B(a). Also, Yang proves that p(A) is always a nonempty compact subset of p(A) (Note that P(A) will never be compact).

Yang and his collaborators studied the topology and the geometry of the projective spectrum. Their theory has interesting consequences to operators. Here is one result that struck me.

Theorem: Let B = M_k(\mathbb{C}). The tuple A = (A_0, \ldots, A_n) is commuting if and only if P(A) is a union of hyperplanes.

This has the following surprising result:

Corollary: A k \times k matrix T is normal if and only if the polynomial p(z_1, z_2) = \det(I + z_1 T + z_2 T^*) is a product of linear factors.

For more, see this first paper on the subject by Yang. (Yang says that the corollary is a later observation due to Kehe Zhu).

3. Noncommutative function theory

John McCarthy gave a beautiful talk on noncommutative function theory. There have been several approaches to developing noncommutative function theory, involving J. Taylor, D. Voiculescu, V. Vinnikov and D. Kalyuzni-Verbovetski, B. Solel and P. Muhly, G. Popescu and many others too. McCarthy presented his take at the subject, which he developed together with J. Agler (here are three papers on the arxiv related to the talk: one, two, three).

So what is “noncommutative function theory”? Let us recall first that given an operator T one can always define p(T) whenever p is a polynomial. One can go further and develop a holomorphic functional calculus (see these two previous posts): there is a way to define f(T) whenever f is a holomorphic function in a neighbourhood of \sigma(T).

Now suppose we have several operators T_1, \ldots, T_n. If the operators commute, there is an obvious way in which one can apply polynomials in several variables to them – just plug the operators into the polynomial (and there is also a way to extend this to functions that are holomorphic in a neighbourhood of the joint spectrum). If the operators do not commute, then we are in trouble, since the functional calculus is no longer a homomorphism. Indeed, suppose that we have two non commuting operators T_1 and T_2 and that we have the polynomials p_1(z_1,z_2) = z_1 and p_2(z_1, z_2) = z_2. Then we would like to define p_1(T_1, T_2) = T_1 and p_2(T_1, T_2) = T_2, nothing else makes sense. But if q denotes the polynomial q = p_1 p_2, then q(z_1, z_2) = z_1 z_2 = p_1 p_2(z_1, z_2) = p_2 p_1(z_1, z_2). However, p_1(T_1, T_2) p_2(T_1,T_2) \neq p_2(T_1,T_2) P_1(T_1,T_2), so we can’t define q(T_1, T_2) in a consistent way.

This problem is easily overcome by defining a functional calculus for noncommutative polynomials, that is polynomials in non commuting variables. A polynomial in non commuting variables is just a polynomial with variables where the order in which the variables are written matters. Whenever we have n operators T = (T_1, \ldots, T_n) (commuting or not) and a noncommutative polynomial p(x_1, \ldots, x_n) then there is an obvious way to evaluate p at T = (T_1, \ldots, T_n).

However this simple functional calculus is not enough. We would like to have a holomorphic functional calculus, that is we would like to be able to evaluate the holomorphic functions in n non commuting variables at the tuple T. But what is a holomorphic function in non commuting variables? (There is also the question “why would we like to do this?”, and I hope to show below that indeed it is a fruitful thing to do.)

As I wrote above, there are several approaches to the subject, and this is how Agler and McCarthy answer the above question.

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 \oplus y := \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 this is all d-tuples of n \times n matrices, running over all n.

Definition: An nc-set 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-set, 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 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-set \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. Without going into detail, let us assume that there is a “good” topology defined on M^{[d]}. This allows us to define holomorphic functions.

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

(Actually, Agler and McCarthy define a holomorphic function to be just a locally bounded nc-function, but they prove that continuity follows, so for simplification I throw continuity into the definition.)

A remarkable consequence of this definition – just the bare algebraic requirements together with continuity – is that a holomorphic function is differentiable. The simple proof is as remarkable as the result.

Proposition: If f is a noncommutative holomorphic function on \Omega, then for every a \in \Omega \cap M_n^d and h \in M_n^d, the limit

Df(a)[h] = \lim_{t\rightarrow 0} \frac{f(a+th) - f(a)}{t}

exists.

Proof: The idea of the proof: do some block 2 \times 2 matrix computations, and read the derivative from the (1,2) entry. Here is what this means. First write down the simple identity :

A B A^{-1} = \begin{pmatrix} a + th & h \\ 0 & a \end{pmatrix},

where A = \begin{pmatrix} 1 & -1/t \\ 0 & 1 \end{pmatrix} and B = \begin{pmatrix} a + th & 0 \\ 0 & a \end{pmatrix} (and t is a small complex number). Applying f to both sides, and using the properties of nc functions, we find

\begin{pmatrix} f(a+th) & (f(a+th) - f(a))/t \\ 0 & f(a) \end{pmatrix} = f\left(\begin{pmatrix} a +th & h \\ 0 & a \end{pmatrix} \right).

Now letting t \rightarrow 0 we find that Df(a)[h] exists, and can simply be read off the top-right entry in the right hand side of the above equation!

(Note that there is content here also when there is just one variable.)

This neat theorem is just the starting point. One of the subtleties is that different choices of topologies lead to rather different notions of holomorphic functions, and different theorems are available. If the topology is “free”, then holomorphic functions can be locally approximated by noncommutative polynomials, in analogy to analytic functions in one complex variable.

On the other hand, if the topology is “fat”, then they obtain the following implicit function theorem.

Implicit function theorem (d=2 version): Suppose that \Omega is a “fat” open set, and let f = f(x,y) be holomorphic on \Omega. Suppose that f(x_0,y_0) = 0 and that h \mapsto \frac{\partial f}{\partial y}(x_0,y_0)[h] is a full rank map. Then there exists an open neighbourhood U of x_0 and a holomorphic function g of one variable such that near (x_0,y_0) the set \{f = 0\} is given as \{(x,g(x)) : x \in U\}.

(Remark: Note that differentiation here is defined slightly differently from above. It is an exercise to sort out what this means.)

The implicit function theorem has the following remarkable consequence: a generic matrix solution to a (noncommutative) polynomial equation in two variables is, in fact, commuting. For example, consider

p(x,y) = x^3 + 2xy + 3 yx .

We calculate \frac{\partial p}{\partial y}(x,y)[h] = 2xh + 3hx. This is a full rank map unless \sigma(2x) intersects \sigma(-3x). This usually does not happen. Thus if we find a solution (x,y) \in M^2_n that satisfies the algebraic equation

x^3 + 2xy + 3yx = 0

where \sigma(2x) does not intersect \sigma(-3x), then the implicit function theorem tells us that y = g(x), and so y and x commute! (The implicit function theorem is from this paper. I hope this application is convincing that developing noncommutative function theory is worthwhile.)

4. More on multipliers of Drury-Arveosn space

Jingbo Xia presented a joint work with Quanlei Fang where (among other things) they answer in the negative the following question about the characterisation of multipliers on Drury-Arveson space H^2_d:

Question: Let h \in H^2_d, and suppose that \sup_{z\in\mathbb{B}_d}\|h \hat{k}_z\|_{H^2_d} < \infty. Does it follow that h \in Mult(H^2_d)?

Here \hat{k}_z denotes the normalised reproducing kernel at z. As I indicated above, the answer to this question is No. This does not come as a gigantic surprise, but it fills a hole in our knowledge, and the proof is hard.

I found this interesting because, as Xia points out, this answers an even simpler question that we did not know the answer to. Recall that since 1 \in H^2_d, every multiplier is in H^2_d (in particular an analytic function). Moreover, it is standard that every multiplier (on every Hilbert function space) is a bounded function. Thus

Mult(H^2_d) \subseteq H^\infty(\mathbb{B}_d) \cap H^2_d .

Fang and Xia’s result shows that this containment is strict.

5. Ken Davidson awarded “Distinguished Career Award”

Finally, some happy news for anybody working in operator algebras and operator theory. (This was not a talk in the workshop, of course, but something I learned while eavesdropping to a conversation.) The Canadian Math Society decided to award Kenneth R. Davidson the 2014 David Borwein Distinguished Career Award. Here is the press release. Congratulations Ken!