### The disc trick (and some other cute moves)

This post is about a chain of little tricks that I discovered with collaborators and used in several papers. It is just a collection of simple moves that lets you deduce the existence of a zero preserving map of a certain class between two gauge invariant spaces, given the existence of a map from that class (things will be very clear soon, I hope). These tricks were later used by some other people, who applied it in different settings.

I am writing this post as notes for my upcoming Pizza & Beer seminar talk. The section at the end of the notes contains references and links to papers where this was used.

#### 1. Background

Since the disc trick is just a trick, we can explain it with an example. Let $\mathbb{B}_d \subset \mathbb{C}^d$ be the open unit ball in complex $d$-space. We wish to prove the following theorem.

Theorem A: Let $V, W \subseteq \mathbb{B}_d$ be two homogeneous analytic varieties. Suppose that there exists a biholomorphism $f : V \to W$. Then there exists a zero preserving biholomorphism $V \to W$.

(The statement of the theorem should remain on the board the whole talk.)

The intuitive idea is that zero is contained in every homogeneous variety, and that it is a distinguished point in varieties. The theorem really should seem obvious. If you think it’s trivial, then please don’t waste your time giving me your proof. Instead, try to prove it for the case where biholomorphism is replaced by homeomorphism.

If we could prove the above theorem, then we could obtain the following corollary (with simple proof):

Corollary: Let $V, W \subseteq \mathbb{B}_d$ be two homogeneous analytic varieties. Suppose that there exists a biholomorphism $f : V \to W$. Then there exists a linear map $A : \mathbb{C}^d \to \mathbb{C}^d$ such that $A(V) = W$.

This has following striking corollary (with complicated proof):

Corollary: Let $V, W \subseteq \mathbb{B}_d$ be two irreducible homogeneous analytic varieties. Suppose that there exists a biholomorphism $f : V \to W$. Then there exists a unitary $U$ such that $U(V) = W$.

These kinds of questions arose in my study of the isomorphism problem for universal operator algebras, and the varieties played the role of maximal ideal spaces of the algebras. That’s not so important to understand right now.

We will prove Theorem A in the next section. The rest of this section is devoted to a crash course in several complex variables. All of the following definitions are given in the form most convenient for my presentation, and are equivalent to the “usual” definitions (the equivalence might depend on very deep theorems in complex analysis; though I don’t remember whether I need Cartan’s Theorem A, Theorem B, or both).

(In the talk itself, I will begin by discussing Cartan’s Uniqueness Theorem, that a self map of a domain that fixes a point and has derivative equal to the identity at that point must be the identity map. Then I will show that a biholomorphism between circled domains that sends the origin to the origin must be a linear map, concluding that the ball and polydisc are not biholomorphic. Since these are well known results I will not type them up here. See Rudin’s book: Function Theory in the Unit Ball of $C^n$)

We shall write $\mathbb{B}_d$ for the open unit ball in $\mathbb{C}^d$, and we shall denote the unit disc as $D = \mathbb{B}_1 \subset \mathbb{C}$. We will use the word “disc” to refer to any intersection of $\mathbb{B}_d$ with a one dimensional subspace. Likewise, the word “ball” will refer to the intersection of a unit ball with a linear subspace.

Definition 1: A function $f : \mathbb{B}_d \to \mathbb{C}$ is said to be analytic (or holomorphic) if it is given by a power series that converges absolutely in $\mathbb{B}_d$:

$f(z) = \sum_{n \in \mathbb{N}^d} a_n z^n$,

where we use the multi-index notation: if $z = (z_1, \ldots, z_d) \in \mathbb{C}^d$ and $n = (n_1, \ldots, n_d) \in \mathbb{N}^d$, then we write $z^n = z_1^{n_1} \cdots z_d^{n_d}$.

Definition 2: A map $f = (f_1, \ldots, f_d) : \mathbb{B}_d \to \mathbb{C}^d$ is holomorphic (or analytic) if all its coordinates $f_1, \ldots, f_d$ are holomorphic.

(Sorry, I really do use the terms analytic/holomorphic interchangeably. Bear with me.)

Definition 3: An analytic variety in $\mathbb{B}_d$ (or an analytic subvariety of $\mathbb{B}_d$) is just a subset $V \subseteq \mathbb{B}_d$ that is given as the joint zero set of a family of analytic functions on $\mathbb{B}_d$.

Usually we’ll just say “variety” or “subvariety”, omitting the word “analytic”. All of our varieties will be subsets of the ball.

Definition 4: A variety $V \subseteq \mathbb{B}_d$ is said to be homogeneous if for every $\lambda \in \overline{D}$ and every $z \in V$, we have that $\lambda z \in V$, too.

It’s a fact that a homogeneous analytic subvariety in $\mathbb{B}_d$ is actually an algebraic subvariety, meaning that it is the zero set of a family of homogeneous polynomials.

Note that for every homogeneous variety $V$, the circle $T = \partial D$ acts on $V$ by gauge automorphisms

$\gamma_\lambda (z) = \lambda z$

for $\lambda \in T$ and $z \in V$.

Definition 5: A map $f : V \to W$ between two subvarieties in the ball is said to be holomorphic if it is the restriction of a holomorphic map on the ball. It is a biholomorphism if it has a holomorphic inverse $g = f^{-1}:W \to V$.

WARNING: A biholomorphism between subvarieties of the ball need not be the restriction of an automorphism.

#### 2. Proof of Theorem A

We will prove the theorem (and understand why it is not trivial) by considering a sequence of cases.

1. Points. The simplest kind of homogeneous variety is the singleton $\{0\}$. If $V = \{0\}$ and $f : V \to W$ is a biholomorphism, then $W = \{0\}$ and there is really nothing to show.

2. Subspaces. The second simplest kind of homogeneous variety is a subspace, or – if we replace a subspace by its intersection with the ball – a ball (or the point $0$). If $V, W$ are two balls inside $\mathbb{B}_d$, and $f : V \to W$ is a biholomorphism, then $V$ and $W$ have to be balls of the same dimension, and essentially we have an automorphism (by which I mean self-biholomorphism) of the unit ball $B:= \mathbb{B}_n$ for some $n \leq d$. Now, $Aut(B)$ is well understood (see Chapter 2 in Rudin’s book “Function Theory in the Unit Ball of $\mathbb{C}^n$“). For every $a \in B$, then the following map defines an automorphism of $B$:

$\phi_a(z) = \frac{a - P_a z - s_a Q_a z}{1 - \langle z, a \rangle}$,

where $P_a$ is the orthogonal projection onto $span(a)$, $Q_a = I- P_a$ is the orthogonal projection on the complement, and $s_a = \sqrt{1 - \|a\|^2}$. (It is interesting to spell out what this formula says when $n =1$.) Note that $\phi_a(0) = a$ and that $\phi_a(a) = 0$. This immediately implies that $Aut(B)$ is transitive. In particular, if $f : B \to B$ is an automorphism and $f(0) = a$, then $g:= \phi_a \circ f$ is an automorphism that sends zero to zero. This essentially proves Theorem A for the case where $V$ and $W$ are both balls.

We note in passing – and this will be used below – that every automorphism of $B$ has the form $\phi_a \circ U$ for some $a \in B$ and a unitary $U \in \mathcal{U}_n$. Indeed, if $\phi \in Aut(B)$ and $a = \phi(0)$, then $\phi_a \circ \phi$ is an automorphism mapping $0$ to $0$. Hence by Cartan’s Uniqueness Theorem $\phi_a \circ \phi$ is equal to a unitary $U$, so $\phi = \phi_a \circ U$.

3. General case, first observation. For a general $W$, we do not know much about the group $Aut(W)$. If we knew that $f(0) = a$ was in the orbit of $0$ under $Aut(W)$, then we’d be done, because if there was an automorphism $\phi \in Aut(W)$ such that $\phi(0) = a$, then $g:=\phi^{-1} \circ f : 0 \mapsto 0$. Our theorem will imply, as expected, that for every point $a$ in $W$ such that there exists a biholomorphism $f : V \to W$ such that $f(0) = a$, there does exist an automorphism mapping $0$ to $a$, but we do not know this in advance.

4. A few more examples of homogeneous varieties. Suppose that $V$ and $W$ are both a union of lines. (Draw two varieties, each one a union of two lines.) We see that $0$ is a special point, from a topological point of view (also in the complex case, removing it leaves a disconnected topological space). Since $f$ is a homeomorphism, it must send zero to zero.

Suppose that $V$ is a cone $V = \{(x,y,z) : x^2 + y^2 = z^2\}$. (Draw a cone in $\mathbb{R}^3$). We see that $0$ is a special point, a singular point. In fact it is the only singular point in $V$ (it is hard to imagine what happens in the complex three dimensional space, but we can simply check that $\nabla (x^2 + y^2 - z^2) = 2(x,y,-z)$ vanishes only at the origin). Since $f$ preserves smoothness and tangent spaces (remember that $f$ is defined in a neighborhood of the variety), it must send $0$ to a singular point of $W$. So if $W$ also has $0$ as a unique singular point, we are done.

We see, that when the varieties both have a unique singular point at the origin, the biholomorphism must preserve this. Not every homogeneous variety has a singular point. A linear subspace is a homogeneous variety, and it has no singular points. The singular locus (set of singular points) of a (complex) homogeneous variety that is not a subspace is never empty (always contains $0$) is also a homogeneous variety. However, it can be as complicated as any homogeneous variety can be. So things are not as easy as in the above two paragraphs.

For the simplest example, consider the union of two subspaces $W = W_1 \cup W_2$. The singular locus of $W$ is $S(W) = W_1 \cap W_2$. This might be a point, but it might be a higher dimensional subspaces (that is, a higher dimensional ball). Didn’t we solve this case in step 2 above? No! We know that the orbit of $0$ under $Aut(W)$ must be contained in $S(W)$, but we don’t know that it is equal to all of it.

5. The disc trick. Above we saw examples in which it followed from topological or geometric reasons that any map $f : V \to W$ must apriori preserve the origin. Let us consider the next simplest case, in which there exist discs $D_V \subseteq V$ and $D_W \subseteq W$, such that $f(D_V) = D_W$.

(Draw this.) If $f(0) = 0$ then we are done. Otherwise, let $a = f(0)$. Consider the set

$\mathcal{O}_W = \{w \in W : w = g(0)$ for some biholo $g : V \to W\}$.

Since $\gamma_\lambda \circ f$ is a biholomorphism $V \to W$ for all $\lambda \in T$, we have that $\mathcal{O}_W$ contains the circle $C$ of radius $|a|$ centered at $0$ (inside $D_W$), which is given by $C = \{\gamma_t(a) : t \in T\}$.

Now let us define

$\mathcal{O}_V = \{v \in V : v = h(0)$ for some automorphism $h : V \to V\}$.

Since for every biholomorphism $g : V \to W$, we have that $f^{-1} \circ g$ is an automorphism of $V$, it must hold that $\mathcal{O}_V$ contains $C' :=f^{-1}(C)$, which is a simple closed loop (in fact, a circle) passing through $0$. Now, the point $f^{-1}(0)$ is contained in the interior of $C'$. But $\gamma_\lambda(C') \subseteq \mathcal{O}_V$ for all $\lambda \in T$. In other words, we can rotate $C'$ until we hit $f^{-1}(0)$ with a point of the form $\gamma_s \circ f^{-1} \circ \gamma_t(a)$. Since $f^{-1}(0) \in \mathcal{O}_V$, it follows that $0 = f(f^{-1}(0)) \in \mathcal{O}_W$, and we are done.

(Explicitly, we found that there are $s,t \in T$ such that $0 = f\circ \gamma_s \circ f^{-1} \circ \gamma_t \circ f(0)$. )

6. Handling the general case (the singular nucleus). We now finish the proof by showing that whenever there exists a biholomorphism $f : V \to W$, then either we can show that $f$ has to map zero to zero, or we can show that there exist discs $D_V \subseteq V$ and $D_W \subseteq W$, such that $f(D_V) = D_W$, in which case we can apply the disc trick and find another biholomorphism $g : V \to W$, such that $g$ is zero preserving.

As we started discussing above, a homogeneous variety $V$ is either a linear subspace (or ball, if we restrict to the unit ball), or it has a singular locus $S(V)$ which is again a homogeneous variety (the case $V = \{0\}$ is a special case of a linear subspace). Therefore the singular locus $S(V)$, if it is not a subspace (or $\{0\}$) also has a singular locus $S(S(V))$. We define the singular nucleus $N(V)$ of a homogeneous variety $V$ to be the singular locus of the singular locus of …. of the singular locus of $V$, applied until we get a linear subspace (or $\{0\}$). In case $V$ is a subspace, then we define $N(V) = V$. Note that $N(V)$ always contains $0$.

Now, $f : V \to W$ is a biholomorphism, then it maps $S(V)$ onto $S(W)$ (if one of them exists), and hence it maps $N(V)$ onto $N(W)$. If $N(V) = \{0\}$ then we are done, because then $N(W) = \{0\}$, and we must have that $f(0) = 0$.

Otherwise, $f$ maps the ball $N(V)$ onto the ball $N(W)$. They must be of the same positive dimension, and so $f \big|_{N(V)}$ is essentially an automorphism of a ball $B$. But for any automorphism of $B$, there exists a disc $D_1$ which it mapped onto another disc $D_2$ (indeed, up to a unitary an automorphism has the form $\phi_a$, which preserves the disc obtained as the span of $a$ intersect the ball). Thus, we can apply the disc trick, and we are done.

#### 3. An application

He is a nice application due to Michael Hartz.

Theorem B: The group of unitaries $\mathcal{U}_d$ is a maximal subgroup of the group $Aut(\mathbb{B}_d)$ of conformal automorphisms of the unit ball. In fact, it is a maxial subsemigroup.

Proof: If I’ll have time I’ll do this in the talk, but it looks like it’s going to take enough time. So. Exercise! You have to prove that every (not necessarily closed) semigroup of $S \subseteq Aut(\mathbb{B}_d)$ which contains $\mathcal{U}$ is all of $Aut(\mathbb{B}_d)$. If you don’t feel like solving the exercise, you can find the proof in Hartz’s paper which I link to below.

#### 4. References and historical remarks

Since it came up again and again in different but similar situations (isometric isomorphisms between operator algebras, bounded isomorphisms between operator algebras, bi-Lipschitz biholomorphisms between noncommutative varieties), we once tried to formulate it as a general theorem, so that it could be invoked in all possible situations, but we ended up with a very boring theorem. It seems that if ever a situation calling for the disc trick arises in the future, the easiest thing would be to simply use the trick again.

Here is a list of places where it has appeared.

The disc trick appeared first in the paper Subproduct systems, by Baruch Solel and myself (Section 11). It was used to classify, up to isometric isomorphism, the operator algebras in a rather limited class (all the tensor algebras that come from a subproduct systems $X$ with $\dim X(1) = 2$ and $X(2) = 3$). Later, in the paper The isomorphism problem for some universal operator algebras, by Davidson, Ramsey and myself, we figured out how we can use the singular nucleus and the properties of automorphisms of the ball, in order to find invariant discs (or points) in the maximal ideal spaces, therefore leading to the classification of all tensor algebras of subproduct systems with finite dimensional Hilbert space fibers. In the same paper, we also applied the idea to obtain the classification of tensor algebras of commutative subproduct systems up to isomorphism (which turn out to be multiplier algebras of complete Pick spaces, and this result also opened up the area of research “the isomorphism problem for multiplier algebras of complete Pick spaces”).

The trick was used by Adam Dor-On and Daniel Markiewicz in their beautiful paper Operator algebras and subproduct systems arising from stochastic matrices (Theorem 7.24 there), where operator algebras of subproduct systems with fibers that are of C*-correspondences over a commutative von Neumann algebra were treated.

The trick was also used in the important paper Classification of noncommutative domain algebras by Arias and Latremoliere. This paper was the first example where the trick was used for operator algebras not arising from subproduct systems.

The trick was also used by Michael Hartz, in his lovely paper On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces, where the classification results I got with Davidson and Ramsey were extended to a significantly larger class of multiplier algebras. It is in this paper that Hartz presented the neat proof to the fact that the unitary group is a maximal subsemigroup in $Aut(\mathbb{B}_d)$.

The trick reappeared in the paper Operator algebras of monomial ideals in noncommuting variables by Kakariadis and myself, where it was used to finish off the completely bounded isomorphism problem for operator algebras of subproduct systems with finite dimensional Hilbert space fibers, and also in these two papers (one, two) by Salomon, Shamovich and myself, on the classification of operator algebras of bounded noncommutative analytic functions.