### The isomorphism problem: update

Ken Davidson, Chris Ramsey and I recently uploaded a new version of our paper “Operator algebras for analytic varieties” to the arxiv. This is the second paper that was affected by a discovery of a mistake in the literature, which I told about in the previous post. Luckily, we were able to save all the results in that paper, but had to work a a little harder than what we thought was needed in our earlier version. The isomorphism problem for complete Pick algebras (which I like to call simply “the isomorphism problem”) has been one of my favorite problems during the last five years. I wrote four papers on this problem, with five co-authors. I want to give a short road-map to my work on this problem. Before I do so, here is  link to the talk that I will give in IWOTA 2014 about this stuff. I think (hope) this talk is a good introduction to the subject. The problem is about the classification of a large class of non-selfadjoint operator algebras – multiplier algebras of complete Pick spaces – which can also be realized as certain algebras of functions on analytic varieties. These algebras all have the form $M_V = Mult(H^2_d)\big|_V$

where $V$ is a subvariety of the unit ball and $Mult(H^2_d)$  denotes the multiplier algebra of Drury-Arveson space (see this survey), and therefore $M_V$ is the space of all restrictions of multipliers to $V$. The hope is to show that the geometry of the variety $V$ is a complete invariant for the algebras $M_V$, in various senses that will be made precise below.

#### A road map to the papers

Here are the works we did on this problem. In all cases the links I give lead to the best and most up-to-date version (which is on the arxiv).

1. “The isomorphism problem for some universal operator algebras“, with Ken Davidson and Chris Ramsey. Here we started by studying the universal operator algebras generated by a row contraction subject to homogeneous polynomial relations, or operator algebras coming from subproduct systems. This also includes the noncommutative case. We classified these algebras up to isometric isomorphism. As a special case we also considered algebras generated by a row contraction of commuting operators, subject to relations coming from a radical, homogeneous ideal. These turn out to be algebras of the type $M_V$, where $V$ is a homogeneous variety. Our main results regarding these algebras are summarised in the following theorems.

Theorem 1: Let $V$ and $W$ be homogeneous subvarieties in the unit ball $\mathbb{B}_d$ of $\mathbb{C}^d$, $d < \infty$. Then $M_V$ is isomorphic to $M_W$ if and only if there is an invertible linear map $A$ such that $A(V) = W$. This happens if and only if $V$ and $W$ are biholomorphic, by which we mean that there are maps $f,g : \mathbb{B}_d \rightarrow \mathbb{C}^d$ such that $f(V) = W, g(W) = V$ and the restrictions of $f$ and $g$ to $V$ and $W$ are mutual inverses.

It should be noted that this theorem was only fully completed with the work of Michael Hartz “Topological isomorphisms for some universal operator algebras” (Michael managed to eliminate some ugly technical conditions that we needed to impose). Apropos the title of Michael’s paper, let me to mention that since our algebras are semi-simple, any algebraic isomorphism is automatically bounded (with bounded inverse), hence the phrase “topological isomophisms”.

Theorem 2: Let $V$ and $W$ be homogeneous subvarieties in the unit ball $\mathbb{B}_d$ of $\mathbb{C}^d$, $d < \infty$. Then $M_V$ is isometrically isomorphic to $M_W$ if and only if $M_V$ is unitarily equivalent to $M_W$, if and only if there is a unitary $U$ such that $U(V) = W$

Since this paper is all around homogeneous algebraic varieties, we also had to learn and even prove results in (elementary, yet serious) algebraic geometry. (Since I don’t want to go on a tangent here, I’ll simply give the interested reader precise pointers: see Theorem 6.12 and Proposition 7.6 in the paper).

2. “Operator algebras for analytic varieties“, again with Ken Davidson and Chris Ramsey. In this paper we changed our point of view that we had in the first paper: in the first we started from the homogeneous polynomial relations and arrived at the algebras $M_V$ as a special case of of universal algebras for relations in a radical homogeneous ideal. In the second paper we start from a variety $V$ and consider the algebra $M_V$. This algebra can also be viewed as a universal algebra for certain relations – the relations in the ideal of multipliers vanishing on $V$. But that point of view does not help too much, and can be left in the background; a more useful point of view is that these algebras are all irreducible Pick algebras. It is significant that we now allow for general varieties determined by multiplier functions, and we also allow for $d=\infty$. Our most significant results in this paper are as follows:

Theorem 3: Let $V,W \subseteq \mathbb{B}_d$ be varieties (where we allow for $d = \infty$). $M_V$ and $M_W$ are completely isometrically isomorphic if and only if they are unitarily equivalent, and this happens if and only if there is a conformal automorphism of the ball $\alpha$ such that $\alpha(V) = W$. If $d< \infty$ then this is equivalent to the algebras being isometrically isomorphic.

This theorem is as good a generalisation of Theorem 2 as one could hope for.

Theorem 4:  Let $V,W \subseteq \mathbb{B}_d$ be varieties, where now we require $d < \infty$ plus some additional technical requirements. If $M_V$ and $M_W$ are isomorphic then $V$ and $W$ are biholomorphic.

This theorem is by far not as good a generalisation of Theorem 1 as we could hope for. Never mind the technical requirements or the requirement $d< \infty$ – we are missing the converse direction! We had some examples in the paper showing that a full converse cannot be hoped for (for example in the case of discrete varieties), but many questions remained unanswered. For example, if $\alpha : \mathbb{D} \rightarrow \mathbb{B}_\infty$ is given by $\alpha(z) = (b_1 z, b_2 z^2, b_3 z^3, \ldots)$,

then we were not able to completely characterise those sequences $(b_n)_{n=1}^\infty$ for which the algebra $M_V$ (where $V = \alpha(\mathbb{D})$) is isomorphic to $H^\infty(\mathbb{D})$. We were able to understand what sequences give rise to an operator algebra $M_V$ that is naturally isomorphic to $H^\infty$, meaning that the isomorphism is induced by a similarity of the Hilbert spaces and is given by $M_V \ni f \mapsto f \circ \alpha \in H^\infty$.

But we could not settle the problem whether there are other, “unnatural” isomorphisms.

This paper resulted in a major shift of my research focus. While working on this paper I understood that I cannot move forward in this field without learning several complex variables very seriously, and this topic has been taking up more and more of my time and thoughts since.

3. “On the isomorphism question for complete Pick algebras“, with Matt Kerr and John McCarthy. In this paper we study the converse of Theorem 4: that is, when does a biholomorphism induce an isomorphism between the multiplier algebras. We treat a very large class of one dimensional varieties (but not all), and show that if $V,W$ are “nice” (in a precise technical sense) images of finite Riemann surfaces, then a biholomorphism induces an isomorphism. This work generalizes and builds on a paper of Alpay, Putinar and Vinnikov where this result was obtained for embedded discs in a finite dimensional ball (before our work, this result was extended to planar domains in a paper by Arcozzi, Rochberg and Sawyer).

Special bonus: working on this paper forced me to (finally) begin to learn Riemann surfaces, which was delightful, and is not over.

4. “Multipliers of embedded discs“, with Ken Davidson and Michael Hartz. This is an unusual kind of paper, and contains all kinds of results, mostly about the case where $V$ is an embedded disc in $\mathbb{B}_d$; it is hard to say what is the “main” result. We treat both the finite and infinite dimensional cases.

One of the important achievements is that we were able to solve the problem mentioned above, that is, for an embedding $\alpha : \mathbb{D} \rightarrow \mathbb{B}_\infty$ is given by $\alpha(z) = (b_1 z , b_2 z^2, \ldots)$, we were able to describe precisely when $M_V$ is isomorphic to $H^\infty$ (where $V = \alpha(\mathbb{D})$). In particular, we showed that an isomorphism from $M_V$ onto $H^\infty$ must be “natural”, that is: it must be induced by a similarity between the Hilbert spaces and must be given by composition. We also exhibited an uncountable family of non-isomorphic algebras arising from an embedding of the kind $\alpha(z) = (b_1 z, b_2 z^2, \ldots)$.

What about embedded discs in finite dimensional balls? We were able to construct an example (actually, a family of examples) of a biholomorphic embedding of the disc into $\mathbb{B}_2$ (the two dimensional complex ball) such that the multiplier algebra is not isomorphic to $H^\infty$.

A final result I will mention here is that we found that if $M_V$ and $M_W$ are isomorphic as algebras via composition, then the induced map must be a bi-Lipschitz map between $W$ and $V$ (that can be used to show that the example discussed in the previous paragraph is indeed a counter-example, that is: we show that the biholomorphism embedding the disc into the ball is not bi-Lipschitz with respect to the pseudo-hyperbolic distance).

There are some other interesting results that I won’t explain here. Just to make you curious, we discovered that in infinite dimensions there exist compact one dimensional varieties – something that cannot happen in finite dimensions.

If the second paper mentioned above led me to understand that I want to understand better function theory in several complex variables, this paper has led me to understand that I should not stop at several, and that I must learn function theory on infinite dimensional Hilbert (and Banach, too) spaces. And really I have been going in a circle all along, since I started (in the first paper) from universal operator algebras generated by row contractions subject to polynomial relations, and these operator algebras should really be thought of as noncommutative analogues of algebras of bounded analytic functions on “noncommutative” varieties.