On the isomorphism question for complete Pick multiplier algebras

by Orr Shalit

In this post I want to tell you about our new preprint, “On the isomorphism question for complete Pick multiplier algebras“,  which Matt Kerr, John McCarthy and myself just uploaded to the arXiv. Very broadly speaking, the motif of this paper is the connection between algebra and geometry; to be a little bit more precise, it is the connection between complex geometry and nonself-adjoint operator algebras.

Here is an excerpt from the introduction to our papaer:

The general theme of this note is the following question.

(Q) Given two specific function algebras \mathcal{M}_V and \mathcal{M}_W associated with two analytic varieties V and W, if there is a biholomorphic equivalence h: V \to W, does composition with h induce an isomorphism from \mathcal{M}_W to \mathcal{M}_V?

This type of question — and its converse, whether isomorphism of the algebras implies biholomorphic equivalence of the varieties — can be asked in many contexts, e.g. in algebraic geometry it is well known that two affine algebraic varieties are isomorphic if and only the algebras of polynomial functions on the varieties are isomorphic. Coming back to complex analysis, a theorem of L. Bers  says that two domains V and W in the plane are conformally equivalent if and only if the corresponding algebras of holomorphic functions O(V) and O(W) are isomorphic. Note that in the setting of Bers’s theorem, (Q) is trivial and it is the converse that is the interesting part. In our setting, (Q) turns out to be much harder than its converse.

A quick path to explain what is the setting in which we work in the paper is as follows. Fix an integer d. Let H^2_d be the Hilbert space of analytic functions f(z) = \sum_\alpha a_\alpha z^\alpha on the unit ball \mathbb{B}_d \subset \mathbb{C}^d for which

\sum_\alpha \frac{\alpha!}{|\alpha|!}|a_\alpha|^2< \infty .

(We are using here the standard multi-index notation). Now let \mathcal{M}_d be the algebra of functions

\mathcal{M}_d = \{f: \mathbb{B}_d \rightarrow \mathbb{C} : \forall h \in H^2_d . fh \in H^2_d\}.

This algebra is the multiplier algebra of the Hilbert function space H^2_d.  Given a variety V in the ball, one may construct the algebra

\mathcal{M}_V = \{f\big|_V : f \in \mathcal{M}_d\}.

You may ask: Of all algebras in the world, why these algebras? That is a long story to be told another day — now I wish just to describe the result. But a short answer for now is that all algebras which may arise as multiplier algebras of an irreducible complete Pick kernel arise as one of our \mathcal{M}_Vs.

In previous works (references may be found in the paper), it was shown that the answer to (Q) is yes in the case that V is a homogeneous variety, and in the case that V is (sufficiently regular) proper image of planar domain of finite connectivity. In our paper we show that the answer to (Q) is yes in the case where V is a (sufficiently regular) proper image of a finite Riemann surface, and also in the case where V and W are both disjoint unions of varieties, say V = V_1 \cup \dots \cup V_k, W = W_1 \cup \dots \cup W_k, where each W_i is the image of V_i under an automorphism of the ball (which may be a different for every i). There are other results of a more technical nature which I don’t want  to discuss here.

Understanding the isomorphism problem for these algebras (multiplier algebras of complete Pick kernels) has been an interest of mine for the past few years, and it is still one of my research objectives. It is a big challenge to obtain some positive answers for wider classes of varieties. I conjecture that the answer to (Q) is positive if V and W are algebraic varieties. I have no idea how to prove this.