2.1 Noncommutative functions and the noncommutative unit ball

We now recall the rudiments of the theory of noncommutative (nc) functions that are relevant to our main problem. The basic general theory is developed in [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov39] and a very rapid introduction to the subject can be found in [Reference Agler and McCarthy2] (see also [Reference Salomon, Shalit and Shamovich31] for an introduction that is geared toward the kind of non-self-adjoint operator algebras that we are studying). Let H be a Hilbert space of dimension d. We fix throughout an index set $\Lambda $ with $|\Lambda | = d$ and an orthonormal basis $\{e_i\}_{i \in \Lambda }$ for H. For us, the most interesting case is $d = \aleph _0$ . We note that in the literature the case $d> \aleph _0$ is hardly considered, but at least some of the basics can be developed in this generality as well (in essence, each individual function in $d>\aleph _0$ variables will only involve at most $\aleph _0$ variables).

For every n, we let $M_n(\mathbb {C})^d := M_n(\mathbb {C})^\Lambda $ denote the space of all d-tuples $T = (T_i)_{i\in \Lambda }$ that define a bounded operator from the direct sum $\bigoplus _{i\in \Lambda } \mathbb {C}^n$ to $\mathbb {C}^n$ . We let $\|T\| = \|\sum _i T_iT_i^*\|$ denote the norm of this operator, referred to also as the row norm of the tuple T. The set $\mathbb {M}_d = \sqcup _n M_n(\mathbb {C})^d$ is our noncommutative universe, by which we mean the big set that contains all the “things” that we will plug as arguments into nc functions. Elements of $\mathbb {M}_d$ can be considered either as d-tuples $T = (T_i)_{i \in \Lambda }$ or as (convergent) sums $\sum _i T_i \otimes e_i$ in the $n \times n$ matrices over the row operator space $H_r$ . On $\mathbb {M}_d$ , there are natural operations of direct sum $T\oplus U = (T_i \oplus U_i)_{i \in \Lambda }$ and left/right multiplication by matrices: $S \cdot (T_i)_{i \in \Lambda } = (ST_i)_{i \in \Lambda }$ and $T \cdot S = (T_i S)_{i \in \Lambda }$ .

For us, the most interesting nc set will be the noncommutative (nc) open unit ball ${\mathfrak {B}}_d$ defined by

$$\begin{align*}{\mathfrak{B}}_d = \bigsqcup_{n=1}^\infty \left\{(T_i)_{i \in \Lambda} \in M_n(\mathbb{C})^d : \left \|\sum_{i \in \Lambda} T_i T_i^* \right\| < 1\right\}. \end{align*}$$

In other words, ${\mathfrak {B}}_d$ is the set of all strict row contractions consisting of matrices.

On ${\mathfrak {B}}_d$ , we have the natural tuple of coordinate functions $z = (z_i)_{i\in \Lambda }$ defined by $z_j(T) = T_j$ . The functions $z_i, i\in \Lambda $ , together with the unit $1$ , generate the algebra $\mathbb {F}_d$ of free polynomials in d noncommuting variables. When treating finite d, ideals in the algebra $\mathbb {F}_d$ were enough to determine all subproduct systems X with $\dim X_1 = d$ . However, when $d = \infty $ , the ideal structure of $\mathbb {F}_d$ is not rich enough, and we will need to work with function algebras. The issue is, for example, that a function of the form $f(z) = \sum _{k=1}^\infty a_k z_k$ for a given sequence $(a_k) \in \ell ^2$ is not a polynomial according to the above definition, but it is a homogeneous nc function of degree one that is contained in either one of the Banach algebras of nc functions that we consider below (some authors do indeed consider bounded linear functionals and their products as polynomials).

The nc Drury–Arveson space $\mathcal {H}_d^2$ is defined to be the space of all nc functions f on ${\mathfrak {B}}_d$ that have a Taylor series $\sum _\alpha a_\alpha z^\alpha $ for which $\|f\|^2_{\mathcal {H}^2_d} = \sum |a_\alpha |^2 < \infty $ .

The algebra of all bounded nc holomorphic functions on ${\mathfrak {B}}_d$ is denoted $H^\infty ({\mathfrak {B}}_d)$ ; this algebra is a Banach algebra with respect to the supremum norm $\|f\|_\infty := \sup \{\|f(X)\| : X \in {\mathfrak {B}}_d\}$ . It can be shown that $H^\infty ({\mathfrak {B}}_d)$ is in fact equal to the algebra of left multipliers of $\mathcal {H}^2_d$ (see [Reference Salomon, Shalit and Shamovich31] for details).

Every element in $A_d$ extends uniquely to a uniformly continuous nc function on $\overline {{\mathfrak {B}}}_d$ (defined as the levelwise closure of ${\mathfrak {B}}_d$ , which is equal to the set of all row contractions acting on finite-dimensional spaces).

In [Reference Salomon, Shalit and Shamovich31, Section 9], it was shown that $A_d$ is equal to the norm closure of the polynomials with respect to the sup norm in $H^\infty ({\mathfrak {B}}_d)$ (although there it was implicitly assumed that $d\leq \aleph _0$ , the argument works for any d). This norm closure of polynomials can be identified with the tensor algebra ${\mathfrak {A}}_d$ corresponding to the full product system (see Example 1.4), by the natural map that sends the coordinate function $z_i$ to the operator $S(e_i)$ (see [Reference Salomon, Shalit and Shamovich31, Section 3]). We shall henceforth identify ${\mathfrak {A}}_d$ with $A_d$ and identify elements in ${\mathfrak {A}}_d$ as nc functions in $A_d$ and vice versa, as convenient. By Proposition 1.3, we can also identify $A_d$ as a linear and dense subspace of the full Fock space $\mathcal {F}(H)$ for a d-dimensional Hilbert space H, via

$$\begin{align*}H^{\otimes n} \ni x \mapsto S(x) \in {\mathfrak{A}}_d \cong A_d \end{align*}$$

and

(2.1) $$ \begin{align} {\mathfrak{A}}_d \cong A_d \ni f \mapsto f((S(e_i))_{i\in \Lambda}) 1 \cong f\cdot 1 \in \mathcal{F}(H). \end{align} $$

2.2 Homogeneous ideals and varieties

Every $f \in \mathcal {H}^2_d$ can be written as a norm convergent sum $f = \sum f_n$ where $f_n$ is the norm limit of n-homogeneous polynomials. The function $f_n$ is said to be an n-homogeneous function. We shall let $\mathcal {H}^2_d(n)$ denote the space of all n-homogeneous functions in $\mathcal {H}^2_d$ . Every function $f \in A_d$ has a Cesàro convergent decomposition

$$\begin{align*}f = \sum f_n \end{align*}$$

into homogeneous components (see [Reference Salomon, Shalit and Shamovich31, Lemma 7.9]). The n-homogeneous component $f_n$ can be obtained from f by the completely contractive projection

$$\begin{align*}f_n(X) = \phi_n(f)(X) = \frac{1}{2\pi} \int_0^{2\pi} f(e^{i\theta} X)e^{-in\theta} d\theta \,\, , \,\, X \in {\mathfrak{B}}_d. \end{align*}$$

Note that under the identification $A_d \cong {\mathfrak {A}}_d$ , the projection $\phi _n$ corresponds to $\Phi _n$ from Proposition 1.3.

Definition 2.4 An ideal $J \subseteq A_d$ is said to be homogeneous, if whenever $f \in J$ , then for all n, the homogeneous component $f_n$ is also in J. A subset ${\mathfrak {V}} \subseteq {\mathfrak {B}}_d$ is said to be a homogeneous variety if it is the joint zero locus (in ${\mathfrak {B}}_d$ ) of all functions in a homogeneous ideal J:

$$\begin{align*}{\mathfrak{V}} = V(J) := \{X \in {\mathfrak{B}}_d : f(X) = 0 \,\, \textrm{ for all } \,\, f \in J\}. \end{align*}$$

For any subset $\Omega \subseteq {\mathfrak {B}}_d$ , we denote

$$\begin{align*}I(\Omega) := \{f \in A_d : f(X) = 0 \,\, \textrm{ for all } \,\, X \in \Omega \}. \end{align*}$$

One can show that if ${\mathfrak {V}} \subseteq {\mathfrak {B}}_d$ is a homogeneous variety, then it is a homogeneous set in the sense that $\mathbb {C} X \cap {\mathfrak {B}}_d \subseteq {\mathfrak {V}}$ for all $X \in {\mathfrak {V}}$ , and from this, it follows that $I({\mathfrak {V}})$ is a homogeneous ideal.

By [Reference Salomon, Shalit and Shamovich31, Section 9], for every homogeneous nc variety ${\mathfrak {V}} \subseteq {\mathfrak {B}}_d$ , the algebra $A({\mathfrak {V}})$ is the closure in the supremum norm of the polynomials, i.e., the closure of the algebra generated by the restriction of coordinate functions to ${\mathfrak {V}}$ . Note that every nc function in $A({\mathfrak {V}})$ extends to an nc function on $\overline {{\mathfrak {V}}}$ and we may identify $A({\mathfrak {V}})$ with $A(\overline {{\mathfrak {V}}})$ .

The operators $I(\cdot )$ and $V(\cdot )$ are inclusion-reversing maps between closed ideals and nc varieties and vice versa, but they are not mutual inverses. We have the following tautological fact that is well known in many analogous circumstances.

Lemma 2.5 For every $\Omega \subseteq {\mathfrak {B}}_d$ and every $E \subseteq A_d$ , we have

$$\begin{align*}I(\Omega) = I(V(I(\Omega)) \end{align*}$$

and

$$\begin{align*}V(I(V(E))) = V(E). \end{align*}$$

Definition 2.6 Let J be a closed ideal in ${\mathfrak {A}}_d$ . We say that J satisfies the Nullstellensatz if

$$\begin{align*}J = I(V(J)) = \{f \in {\mathfrak{A}}_d : f(X) = 0 \text{ for all } X \in V(J)\}. \end{align*}$$

By Lemma 2.5, every ideal that is the annihilating ideal of a prescribed zero set satisfies the Nullstellensatz. In particular, for every homogeneous $J \triangleleft A_d$ , the ideal $I(V(J))$ is a closed homogeneous ideal in $A_d$ that satisfies the Nullstellensatz. On the other hand, we shall see in Section 3.3 that there are closed homogeneous ideals that do not satisfy the Nullstellensatz.

2.3 Subproduct systems and closed homogeneous ideals

Given $d < \infty $ and an orthonormal basis $\{e_1, e_2, \ldots , e_d\}$ for a d-dimensional Hilbert space H, there is a $1$ – $1$ correspondence between homogeneous ideals $I \triangleleft \mathbb {F}_d$ and subproduct systems $X = (X_n)_{n=0}^\infty $ with $X_1 \subseteq H$ given by

$$\begin{align*}I \longleftrightarrow X^I \end{align*}$$

where the nth fiber of $X^I$ is given by

$$\begin{align*}X^I_n = X_1^{\otimes n} \ominus \{ p(e) : p \in I \textrm{ is \(n\)-homogeneous}\} \end{align*}$$

and $p(e) = \sum _{|\alpha |=n} a_\alpha e_{\alpha _1} \otimes \cdots \otimes e_{\alpha _n}$ where $p(z) = \sum _{|\alpha |=n} a_\alpha z_{\alpha _1} \cdots z_{\alpha _d}$ (see [Reference Shalit and Solel36, Proposition 7.2]). When $d = \infty $ , this correspondence ceases to be surjective. However, if we replace homogeneous ideals in $\mathbb {F}_d$ with closed homogeneous ideals in $A_d$ , we recover a bijective correspondence, as we will show below.

Letting now H be a Hilbert space of dimension d, and given a homogeneous closed ideal $J \subseteq A_d$ , the identification of $A_d$ as a subspace of the full Fock $\mathcal {F}(H)$ space given in (2.1) allows us to define $X^J = (X^J_n)_{n=0}^\infty $ by

$$\begin{align*}X^J_n = H^{\otimes n} \ominus \{f_n \in J : f_n \textrm{ is } n\textrm{-homogeneous} \}. \end{align*}$$

Conversely, given a subproduct system $X = (X_n)_{n=0}^\infty $ with $X_1 = H$ , we can define $J_X \subseteq A_d$ with the help of the orthogonal complement

$$\begin{align*}J_X = \overline{\sum_{n=0}^\infty H^{\otimes n} \ominus X_n}^{\|\cdot\|_\infty}. \end{align*}$$

Proof Indeed, if J is a homogeneous ideal, then for all homogeneous functions $f \in J$ and $g \in A_d$ , we have $f g \in J$ and $g f \in J$ . Using closedness of J, this translates to

$$\begin{align*}(X^J_m)^\perp \otimes H^{\otimes n} \cup H^{\otimes m} \otimes (X^J_n)^\perp \subseteq (X^J_{m+n})^\perp = J_{m+n}, \end{align*}$$

which implies

$$\begin{align*}X^J_{m+n} \subseteq X^J_m \otimes X^J_n. \end{align*}$$

Conversely, if X is a subproduct system, then every n-homogeneous element $f \in J_X$ satisfies, for every m-homogeneous polynomial $g \in \mathbb {F}_d$ ,

$$\begin{align*}gf \cong g \otimes f \in H^{\otimes m} \otimes X_n^\perp \subseteq X_{m+n}^\perp \subseteq J_X\end{align*}$$

(and likewise from the right) and an approximation argument gives that $J_X$ is an ideal.

Using Proposition 1.3, these two operations are seen to be mutual inverses of each other.

3.1 Universality of tensor algebras

The noncommutative disk algebra ${\mathfrak {A}}_d$ [Reference Popescu28] is the tensor algebra of the full subproduct system that is the full product system over a Hilbert space H with $\dim H = d$ (see Example 1.4). For this algebra, we have a noncommutative functional calculus [Reference Popescu27]: for every row contraction $T = (T_i)_{i \in \Lambda }$ with $|\Lambda |=d$ , there exists a unital completely contractive homomorphism $f \mapsto f(T)$ from ${\mathfrak {A}}_d$ into the unital closed operator algebra $\overline {\operatorname {alg}}(T)$ generated by T, that maps $L_i = S(e_i)$ to $T_i$ . When thinking of ${\mathfrak {A}}_d$ as $A_d = A({\mathfrak {B}}_d)$ , the noncommutative functional calculus when applied to $T \in \overline {{\mathfrak {B}}}_d$ corresponds to evaluating f at the point T, and it allows us to think of ${\mathfrak {A}}_d$ (or $A_d$ ) as an algebra of functions defined on row contractions in all dimensions. This functional calculus gives the full shift L the role of a universal row contraction.

The functional calculus allows us to say that a row contraction T annihilates an ideal $J \triangleleft {\mathfrak {A}}_d$ if $f(T) = 0$ for all $f \in J$ . The X-shift of a subproduct system X is a universal row contraction that annihilates $J_X$ . The following is known, and can be dug out of the literature (e.g., [Reference Popescu29] or [Reference Shalit and Solel36]), but unfortunately it does not appear precisely in the form we need.

Alternatively, the above theorem could also be proved by using a commutant lifting approach or the distance formula $d(T,J) = \|P_{\mathcal {F}_X} T \big |_{\mathcal {F}_X}\|$ and its matrix-valued variant (see [Reference Arias and Popescu3, Reference Davidson and Pitts11]; cf. [Reference Salomon, Shalit and Shamovich31, Proposition 9.7]), and then one could deduce Proposition 3.1 from the theorem.

3.2 Finite-dimensional representability of tensor algebras

For an operator algebra, we let $\operatorname {Rep}_{\mathrm {fin}}(\mathcal {B})$ denote the space of all unital completely contractive finite-dimensional representations of $\mathcal {B}$ . Proposition 3.1 implies that $\operatorname {Rep}_{\mathrm {fin}}(\mathcal {A}_X)$ can be identified with the homogeneous nc variety

$$\begin{align*}\overline{V}(J_X) := \{X \in \overline{{\mathfrak{B}}}_d : f(X) = 0 \text{ for all } f \in J_X\}, \end{align*}$$

where the identification is given by

$$\begin{align*}\operatorname{Rep}_{\mathrm{fin}}(\mathcal{A}_X) \ni \rho \longleftrightarrow (\rho(S_i))_{i \in \Lambda} \in \overline{V}(J_X) \end{align*}$$

and

$$\begin{align*}\overline{V}(J_X) \ni X \longleftrightarrow \rho_X \in \operatorname{Rep}_{\mathrm{fin}}(\mathcal{A}_X), \end{align*}$$

where $\rho _X$ is the representation mapping $S_i$ to $X_i$ . Every element $a \in \mathcal {A}_X$ can be considered as a function on $\operatorname {Rep}_{\mathrm {fin}}(\mathcal {A}_X)$ in the usual way $\hat {a}(\rho ) = \rho (a)$ , and the above identification gives a as a function, which is also denoted by $\hat {a}$ , on $\overline {V}(J_X)$ . Let us call the map $a \mapsto \hat {a}$ the restriction map. Note that $\hat {a} \in A(\overline {V}(J_X))$ , that is, $\hat {a}$ is a uniformly continuous nc function on $\overline {V}(J_X)$ . Our goal in this section is to understand the extent to which the restriction map

$$ \begin{align*} \mathcal{A}_X \to A(\overline{V}(J_X)), \quad a \mapsto \hat{a}, \end{align*} $$

is faithful.

Definition 3.3 An operator algebra $\mathcal {B}$ is said to be residually finite dimensional (RFD) if for every $b \in M_n(\mathcal {B})$

$$\begin{align*}\|b\| = \sup\left\{\left\|\pi^{(n)}(b)\right\| : \pi \in \operatorname{Rep}_{\mathrm{fin}}(\mathcal{B})\right\}. \end{align*}$$

Proof By Theorem 3.2, the compression map induces a completely isometric isomorphism $\mathfrak {A}_d / J \cong \mathcal {A}_X$ . On the other hand, [Reference Salomon, Shalit and Shamovich31, Proposition 9.7] shows that restriction to $\overline {V}(J)$ induces a completely isometric isomorphism $\mathfrak {A}_d / I(\overline {V}(J)) \cong A(\overline {V}(J))$ . Moreover, we have a commuting diagram

where the top row is the restriction map and the bottom row is the natural quotient map. In particular, we see that the restriction map $\mathcal {A}_X \to A(\overline {V}(J))$ is always a complete quotient map.

As for the equivalent conditions, we first observe that $I(V(J)) = I(\overline {V}(J))$ . Thus, J satisfies the Nullstellensatz if and only if $J = I(\overline {V}(J))$ . By the previous paragraph, this happens if and only if the restriction map is injective. Furthermore, since the restriction map is always a complete quotient map, injectivity is equivalent to being completely isometric. Hence, (1) $\Longleftrightarrow $ (2) $\Longleftrightarrow $ (3). Finally, the equivalence $(3) \Longleftrightarrow (4)$ follows immediately from the identification $\operatorname {Rep}_{\mathrm {fin}}(\mathcal {A}_X) \cong \overline {V}(J_X)$ .

In [Reference Salomon, Shalit and Shamovich31, Theorem 9.5], it was shown that under the assumption that $d< \infty $ , every homogeneous closed ideal satisfies the Nullstellensatz. In the next theorem, we record the fact that if an ideal J is generated by homogeneous polynomials involving only finitely many variables, then J satisfies the Nullstellensatz. The next theorem also gives another sufficient condition that works in the case $d = \infty $ , namely that the ideal is generated by monomials. Monomial ideals are a special class of ideals, but it is worth recalling that subproduct systems associated with monomial ideals give rise to many operator algebras including Cuntz–Krieger and subshift C*-algebras [Reference Kakariadis and Shalit22].

Theorem 3.6 If a closed homogeneous ideal in $J \triangleleft A_d$ is generated by monomials or if there exists a finite subset $\Lambda ' \subseteq \Lambda $ such that J is generated by homogeneous polynomials in the variables $(z_i)_{i \in \Lambda '}$ , then

$$\begin{align*}J = I(V(J)). \end{align*}$$

Proof Clearly, $J \subseteq I(V(J))$ . On the other hand, if $f \in A_d \setminus J$ , then at least one of the homogeneous components $f_n \notin J$ . It suffices to show that $f_n \notin I(V(J))$ , and we now relabel $f_n$ as f and remember that it is homogeneous of degree n. Now, we consider the subproduct system $X = X_J$ and we form the space $\mathcal {E} = X_0 \oplus X_1 \oplus \cdots \oplus X_n$ . Let $1$ denote the copy of $1$ in $X_0 = \mathbb {C}$ . For any $r<1$ , the compression Y of $(r S(e_i))_{i \in \Lambda }$ to $\mathcal {E}$ is a strict row contraction such that $f(Y) 1 \neq 0$ , while $g(Y) = 0$ for all $g \in J$ . Now, if $\mathcal {E}$ happened to be finite-dimensional, then we have $Y \in V(J)$ , while $f(Y) \neq 0$ , and thus $f \notin I(V(J))$ and the proof would be complete (this is precisely how the proof for the case $d < \infty $ works). In general, we cannot yet conclude that $Y \in V(J)$ , because the space $\mathcal {E}$ that Y is acting on might be infinite-dimensional.

We overcome this difficulty as follows. The function $f \notin J$ that we are considering might not be a polynomial, but it is the limit in the norm of homogeneous polynomials of degree n. Now, let $\epsilon> 0$ be such that $\epsilon < \frac {\|f(Y)1\|}{3}$ and find a homogeneous polynomial p such that $\|p-f\|_\infty <\epsilon $ .

Assume first that J is generated by monomials. Let $(z_i)_{i \in \Lambda '}$ be the coordinate functions that appear in p, and let $\mathcal {F} \subseteq \mathcal {E}$ be the subspace spanned by $q(Y)1$ for all noncommutative polynomials q in the variables $\{z_i\}_{i \in \Lambda '}$ of degree less than or equal to n. Let $Z = (Z_i)_{i \in \Lambda } $ be the tuple defined such that $Z_i$ is equal to the compression of $Y_i$ to $\mathcal {F}$ for $i \in \Lambda '$ and $Z_i = 0_{\mathcal {F}}$ for $i \in \Lambda \setminus \Lambda '$ .

We claim that $Z \in V(J)$ . First, Z is a strict row contraction and it is acting on the finite-dimensional space $\mathcal {F}$ . Next, for every monomial $g \in J$ , either g involves only the variables $(z_i)_{i \in \Lambda '}$ , in which case $g(Z) = g(Y)\big |_{\mathcal {F}} = 0$ , or else g involves one of the variables $z_i$ for $i \in \Lambda \setminus \Lambda '$ , in which case $g(Z) = 0$ .

Finally, we show that $f(Z) \neq 0$ , establishing that $f \notin I(V(J))$ . For this, note that

$$ \begin{align*} \|f(Z)1\| &\geq \|p(Z)1\| - \|p(Z)1 - f(Z)1\| \\ &> \|p(Y)1\| - \epsilon \\ &\geq \|f(Y)1\| - \|f(Y)1-p(Y)1\| - \epsilon \\ &> \frac{\|f(Y)1\|}{3} > 0, \end{align*} $$

as required.

If J is generated by homogeneous polynomials in the variables $(z_i)_{i \in \Lambda '}$ for a finite set $\Lambda ' \subseteq \Lambda $ , enlarge $\Lambda '$ so that p is also a polynomial in $(z_i)_{i \in \Lambda '}$ and define $\mathcal {F}$ and Z as above. A similar argument then shows that $Z \in V(J)$ , but $f(Z) \neq 0$ , so that $f \notin I(V(J))$ .

3.3 Failure of the Nullstellensatz

In this section, we exhibit a closed homogeneous ideal $J \triangleleft \, {\mathfrak {A}}_d$ that does not satisfy the Nullstellensatz. Consequently, if $X = X^J$ is the corresponding subproduct system, then the tensor algebra $\mathcal {A}_X$ is not RFD and cannot be represented as an algebra of uniformly continuous nc functions on $V(J)$ .

We will work with $d = \aleph _0$ , and simply write $d = \infty $ . It will be natural to work in the setting of nc functions; thus, we identify the Fock space $\mathcal {F}(H)$ with the nc Drury-Arveson space $\mathcal {H}^2_\infty $ , and we identify ${\mathfrak {A}}_\infty $ with the subspace $A_\infty $ of $\mathcal {H}^2_\infty $ consisting of uniformly continuous functions on ${\mathfrak {B}}_\infty $ .

Proof Let $f(z) = \sum _{n=1}^\infty 2^{-n} z_n^2$ , let $M = \mathcal {H}^2_\infty (2) \ominus \mathbb {C} f$ , and let J be the closed ideal of ${\mathfrak {A}}_\infty $ generated by M. Notice that the degree $2$ part of J is simply $J_2 = M$ . In particular, $f \notin J$ .

We will show that $\mathcal {H}^2_\infty (2) \subseteq I(V(J))$ , which in particular implies that $f \in I(V(J))$ . To this end, we observe that:

1. (1) $z_i z_j \in J$ whenever $i \neq j$ and

2. (2) $2 z_{n+1}^2 - z_n^2 \in J$ for all $n \in \mathbb {N}$ .

Suppose now that $T = (T_1,T_2,\ldots ) \in V(J)$ . Since the entries of T are elements of a finite-dimensional vector space, they must be linearly dependent. Thus, there exist $k \in \mathbb {N}$ and scalars $\lambda _j \in \mathbb {C}$ , all but finitely many of which are zero, such that

$$ \begin{align*} T_k = \sum_{j \neq k} \lambda_j T_j. \end{align*} $$

Multiplying this relation with $T_k$ from the left and using (1) above, we find that

$$ \begin{align*} T_k^2 = \sum_{j \neq k} \lambda_j T_k T_j = 0. \end{align*} $$

Relation (2) above then implies that $T_n^2 = 0$ for all $n \in \mathbb {N}$ . In combination with (1), this implies that $\mathcal {H}^2_\infty (2) \subseteq I(V(J))$ , as desired.

4.1 Reduction of the isomorphism problem to existence of graded maps

In this subsection, we recall known results about the isomorphism problem for tensor algebras of subproduct systems, results which allow one to restrict attention to graded maps.

Recall the maps $\Phi _n : \mathcal {A}_X \to \mathcal {A}_X^{(n)}$ from Proposition 1.3.

In [Reference Dor-On and Markiewicz14, Section 6], Dor-On and Markiewicz reduced a general form of Problem 1.7 to the problem of whether the existence of an isomorphism implies the existence of a vacuum preserving isomorphism.

Let us say a few words about the proof of Proposition 4.2. We indicated right after Definition 1.6 how a similarity/isomorphism between the subproduct systems gives rise to a completely bounded/isometric isomorphism between the tensor algebras. Conversely, assume first that we have a graded bounded isomorphism $\varphi \colon \mathcal {A}_X \to \mathcal {A}_Y$ . In this case, one can check that the “restriction” of $\varphi $ to $\mathcal {A}_X^{(n)} \cong X_n$ defines maps $V_n : X_n \to Y_n$ , in other words

$$\begin{align*}V_n \colon x \in X_n \mapsto S^{-1}(\varphi(S(x)) \in Y_n , \end{align*}$$

and that these maps assemble to form a similarity of subproduct systems [Reference Dor-On and Markiewicz14, Proposition 6.12]. Furthermore, by Proposition 1.3, if $\varphi $ is isometric, then V is an isomorphism of subproduct systems.

If $\varphi $ is not necessarily graded, but merely vacuum-preserving, then one can show that $\varphi $ can be modified to a graded isomorphism $\tilde {\varphi }$ by “dropping higher-order terms,” that is,

$$\begin{align*}\tilde{\varphi}\big|_{\mathcal{A}_X^{(n)}} = \Phi_n \circ \varphi\big|_{\mathcal{A}_X^{(n)}}. \end{align*}$$

See Propositions 6.18 and 6.19 in [Reference Dor-On and Markiewicz14] (this modification is needed for the case of bounded isomorphism; a vacuum-preserving isometric isomorphism is already graded; see [Reference Shalit and Solel36, Theorem 9.7]).

Thus, the isomorphism problem will be settled once we show that the existence of an isomorphism implies the existence of a vacuum-preserving isomorphism.

4.2 The disk trick

Let X and Y be two subproduct systems. We will show that the tensor algebras $\mathcal {A}_X$ and $\mathcal {A}_Y$ are isometrically isomorphic if and only if X is isomorphic to Y, and that $\mathcal {A}_X$ and $\mathcal {A}_Y$ are completely boundedly isomorphic if and only if X is similar to Y. As explained in Section 4.1, Proposition 4.2 reduces our task to showing that if there exists an isometric or a completely bounded isomorphism $\varphi : \mathcal {A}_X \to \mathcal {A}_Y$ , then there is a vacuum-preserving one. Showing how isomorphisms give rise to vacuum-preserving ones can be achieved using a technique that has come to be known as “the disk trick” (see [Reference Shalit34]), which becomes available after one shows that the induced map $\varphi ^*$ between the character spaces sends a disk to a disk. In [Reference Davidson, Ramsey and Shalit13], this was shown using ideas from elementary algebraic geometry, but it is not clear whether these methods extend to the infinite-dimensional case. A different approach for showing that $\varphi ^*$ maps a disk to a disk was introduced in [Reference Hartz20] to treat certain weighted tensor algebras with finite-dimensional fibers; this approach is very general and can be adapted to the infinite-dimensional setting.

Since the following two lemmas are of independent interest and are applicable in a wider scope, it will be convenient for us to somewhat change our notation as follows. If $\mathcal {E}$ is a Hilbert space, we let $B_1(\mathcal {E})$ denote the open unit ball of $\mathcal {E}$ . A nonempty subset $A \subseteq B_1(\mathcal {E})$ is said to be homogeneous if $x \in A$ implies ${\mathbb {C} x \cap B_1(\mathcal {E}) \subseteq A}$ (cf. the definition of homogeneous set right after Definition 2.4). Clearly, every homogeneous set contains the origin. For every $\lambda \in \mathbb {T}$ , let $U_\lambda : \mathcal {E} \to \mathcal {E}$ denote the gauge transformation $z \mapsto \lambda z$ . Clearly, a homogeneous subset $A \subseteq B_1(\mathcal {E})$ is invariant under the gauge transformations. We require the following ad hoc definition.

Definition 4.4 Let $\mathcal {E},\mathcal {F}$ be Hilbert spaces, and let $A \subseteq \mathcal {E}$ be a homogeneous set. A map $F: A \to \mathcal {F}$ is said to be holomorphic if for all $x \in A \setminus \{0\}$ and all $y \in \mathcal {F}$ , the function

$$ \begin{align*} \mathbb{D} \to \mathbb{C}, \quad t \mapsto \Big \langle F \Big( t \frac{x}{\|x\|} \Big) , y \Big\rangle, \end{align*} $$

is holomorphic. A biholomorphism is a bijective holomorphic map with holomorphic inverse.

We begin with the following standard adaptation of the maximum modulus principle and the Schwarz lemma, which is a slight generalization of Lemmas 9.1 and 9.2 in [Reference Hartz20]. The proof carries over almost verbatim.

Proof In both cases, we may assume that $A \neq \{0\}$ .

(a) Suppose that $\|F(w)\| = 1$ for some $w \in A$ . We first show that $F(0) = F(w)$ , for which we may assume that $w \neq 0$ . Then the holomorphic function

$$ \begin{align*} \mathbb{D} \to \overline{\mathbb{D}}, \quad t \mapsto \Big \langle F \Big( t \frac{w}{\|w\|} \Big), F(w) \Big \rangle, \end{align*} $$

maps $\|w\|$ to $1$ and hence is identically $1$ by the maximum modulus principle. Thus, equality holds in the Cauchy–Schwarz inequality, so $F( t \frac {w}{\|w\|}) = F(w)$ for all $t \in \mathbb {D}$ , and in particular $F(0) = F(w)$ .

Next, if $z \in A \setminus \{0\}$ is arbitrary, we apply the maximum modulus principle and the equality case of the Cauchy–Schwarz inequality to

$$ \begin{align*} \mathbb{D} \to \overline{\mathbb{D}}, \quad t \mapsto \Big \langle F \Big( t \frac{z}{\|z\|} \Big), F(0) \Big \rangle, \end{align*} $$

to find that $F(t \frac {z}{\|z\|}) = F(0)$ for all $t \in \mathbb {D}$ ; hence $F(z) = F(0)$ . Therefore, F is constant.

(b) Let $z \in A \setminus \{0\}$ and assume that $F(z) \neq 0$ . The Schwarz lemma shows that the function

$$ \begin{align*} f: \mathbb{D} \to \overline{\mathbb{D}}, \quad t \mapsto \Big \langle F \Big( t \frac{z}{\|z\|} \Big), \frac{F(z)}{\|F(z)\|} \Big\rangle, \end{align*} $$

satisfies $|f(t)| \le |t|$ for all $t \in \mathbb {D}$ . The first statement follows by taking $t = \|z\|$ .

If $\|F(z)\| = \|z\|$ , then $f(\|z\|) = \|z\|$ ; hence, f is the identity on $\mathbb {D}$ by the equality case in the Schwarz lemma. Since $\|F(t \frac {z}{\|z\|})\| \le |t|$ for all $t \in \mathbb {D}$ by the first part, we also have equality in the Cauchy–Schwarz inequality, so $F(t \frac {z}{\|z\|}) = t \frac {F(z)}{\|F(z)\|}$ for all $t \in \mathbb {D}$ .

The following result will allow the reduction to the vacuum preserving case.

Lemma 4.6 Let $\mathcal {E},\mathcal {F}$ be Hilbert spaces, let $A \subseteq B_1(\mathcal {E})$ and $B \subseteq B_1(\mathcal {F})$ be homogeneous sets, and let $F: A \to B$ be a biholomorphism. Then there exist $\lambda , \mu \in \mathbb {T}$ such that

$$ \begin{align*} F \circ U_\lambda \circ F^{-1} \circ U_\mu \circ F : A \to B \end{align*} $$

maps $0$ to $0$ .

Proof We may assume that $F(0) \neq 0$ , and define $b = F(0)$ and $a = F^{-1}(0)$ . In the first step, we show that F maps the disk $(\mathbb {C} a) \cap A \subseteq B_1(\mathcal {E})$ onto the disk $(\mathbb {C} b) \cap B \subseteq B_1(\mathcal {F})$ . To this end, let

$$ \begin{align*} f: \mathbb{D} \to B, \quad t \mapsto F \Bigg( t \frac{a}{\|a\|} \Bigg), \end{align*} $$

and let $\theta $ be a biholomorphic automorphism of $\mathbb {D}$ mapping $0$ to $\|a\|$ and vice versa. Lemma 4.5 (b), applied to $h = f \circ \theta $ , shows that

$$ \begin{align*} \|b\| = \|h(\|a\|)\| \le \|a\|. \end{align*} $$

By symmetry, $\|a\| \le \|b\|$ , so equality holds. The second part of Lemma 4.5 (b) now shows that h maps $\mathbb {D}$ onto $(\mathbb {C} b) \cap B_1(\mathcal {F})$ . Hence, F maps $(\mathbb {C} a) \cap B_1(\mathcal {E})$ onto $(\mathbb {C} b) \cap B_1(\mathcal {F})$ , which completes the proof of the first step.

Now, that we know that F maps the disk $D_1 = (\mathbb {C} a) \cap A$ onto the disk $D_2 = (\mathbb {C} b) \cap B$ , the known versions of the disk trick yield the result (see [Reference Davidson, Ramsey and Shalit13, Proposition 4.7], [Reference Hartz20, Lemmas 9.5 and 9.6], or [Reference Shalit34]). For completeness, we repeat the argument. Assume that $F(0) \neq 0$ , for otherwise there is nothing to prove. Define

$$\begin{align*}C = \{U_\mu \circ F (0) : \mu \in \mathbb{T}\} \subseteq D_2, \end{align*}$$

a circle centered at $0$ with radius $\|F(0)\|$ . Then $F^{-1}(C) \subseteq D_1$ is a circle the passes through $0$ , with $F^{-1}(0)$ in its interior. Rotate $F^{-1}(C)$ until its boundary hits $F^{-1}(0)$ and voilà – we have found $\mu , \lambda \in \mathbb {T}$ such that $F^{-1}(0) = U_\lambda \circ F^{-1} \circ U_\mu \circ F(0)$ , or

$$\begin{align*}F \circ U_\lambda \circ F^{-1} \circ U_\mu \circ F(0) = 0. \end{align*}$$

Since A and B are homogeneous and $F : A \to B$ a biholomorphism, we have that $F \circ U_\lambda \circ F^{-1} \circ U_\mu \circ F : A \to B$ is a biholomorphism mapping $0$ to $0$ , as required.

4.3 Classification

We let $\overline {\mathbb {B}}_d$ denote the closed unit ball of a Hilbert space H with $\dim H = d$ . We shall consider some fixed orthonormal basis $\{e_i\}_{i \in \Lambda }$ for H, and use the identification $h \leftrightarrow \left ( \langle h, e_i \rangle \right )_{i \in \Lambda }$ to think of elements in $\overline {\mathbb {B}}_d$ as d-tuples of scalar row contractions. Thus, $\overline {\mathbb {B}}_d$ can be considered as the first level $\overline {{\mathfrak {B}}}_d(1)$ of the closed nc unit ball $\overline {{\mathfrak {B}}}_d$ . Likewise, we let $\mathbb {B}_d$ denote the corresponding open unit ball.

If $J \subseteq A_d$ is an ideal, we let $\overline {Z}(J)$ denote the first level of the nc set $\overline {V}(J)$ , that is,

$$ \begin{align*} \overline{Z}(J) = \{ z \in \overline{\mathbb{B}}_d: f(z) = 0 \text{ for all } f \in J \} \end{align*} $$

is the scalar vanishing locus of J in the closed ball $\overline {\mathbb {B}}_d$ . We also write $Z(J) = \overline {Z}(J) \cap \mathbb {B}_d$ . We endow $\overline {Z}(J)$ with the (relative topology of the) weak topology.

If $\mathcal {B}$ is a Banach algebra, we let $\mathcal {M}(\mathcal {B})$ denote the character space of $\mathcal {B}$ , that is, the space of all nonzero homomorphisms of $\mathcal {B}$ into $\mathbb {C}$ .

Proposition 4.7 Let $J \triangleleft A_d$ be a homogeneous ideal, let $X = X^J$ be the associated subproduct system, and let $\mathcal {A}_X$ be the tensor algebra of X. Then

$$ \begin{align*} \mathcal{M}(\mathcal{A}_X) \to \overline{Z}(J), \quad \chi \mapsto (\chi(S_i))_{i \in \Lambda}, \end{align*} $$

is a homeomorphism.

We now prove that the existence of an isomorphism of a certain type between the tensor algebras implies the existence of a vacuum-preserving isomorphism of the same type. As explained in Section 4.1, this will conclude the proof of Theorem 1.8.

Proof Let $I,J$ be the homogeneous ideals associated with $X,Y$ , respectively. Let $\varphi : \mathcal {A}_X \to \mathcal {A}_Y$ be an isomorphism. By Proposition 4.7, the adjoint map $\varphi ^* : \chi \mapsto \chi \circ \varphi $ can be regarded as a homeomorphism $\varphi ^*: \overline {Z}(J) \to \overline {Z}(I)$ . Modulo the identification in Proposition 4.7, the adjoint is given by

$$ \begin{align*} \varphi^*(\xi) = (\varphi(S^X_i)(\xi))_{i \in \Lambda}. \end{align*} $$

Indeed, the ith component of $\varphi ^*(\xi )$ is $S^X_i(\varphi ^*(\xi )) = \varphi (S^X_i)(\xi )$ , where we have written $S^X_i = S^X(e_i)$ .

From Section 3.2, we have the continuous restriction map $\mathcal {A}_Y \to A(\overline {V}(J))$ . At the level of scalars, this shows that each element of $\mathcal {A}_Y$ restricts to a function on $Z(J)$ that is a uniform limit of polynomials, hence holomorphic in each disk contained in $Z(J)$ centered at the origin. Hence, $\varphi ^*$ is holomorphic on the homogeneous set $Z(J)$ . In this setting, Lemma 4.5(a) implies that $\varphi ^*$ maps $Z(J)$ onto $Z(I)$ . By symmetry, we find that $\varphi ^*$ is a biholomorphism from $Z(J)$ onto $Z(I)$ . Applying Lemma 4.6 with $A = Z(J), B = Z(I)$ , and $F = \varphi ^*$ , we obtain $\lambda ,\mu \in \mathbb {T}$ so that

(4.1) $$ \begin{align} \varphi^* \circ U_\lambda \circ (\varphi^{-1})^* \circ U_\mu \circ \varphi^* \end{align} $$

maps $0$ to $0$ .

Given $\lambda \in \mathbb {T}$ , let $\Gamma _\lambda $ be the gauge automorphism on $\mathcal {A}_X$ determined uniquely by $\Gamma _\lambda (S^X_i) = \lambda S^X_i$ (the existence of $\Gamma _\lambda $ is guaranteed by Proposition 3.1). We use the same notation for the gauge automorphism on $\mathcal {A}_Y$ . Note that

$$\begin{align*}\Gamma_\lambda^* (\xi) (S^X_i) = \lambda S^X_i (\xi) = \lambda \xi_i = S^X_i (\lambda\xi), \end{align*}$$

whence $\Gamma _\lambda ^* = U_\lambda $ . We find that (4.1) is the adjoint of the map

$$ \begin{align*} \psi := \varphi \circ \Gamma_\mu \circ \varphi^{-1} \circ \Gamma_\lambda \circ \varphi: \mathcal{A}_X \to \mathcal{A}_Y , \end{align*} $$

which is therefore a vacuum-preserving isomorphism. Finally, $\psi $ is isometric if $\varphi $ is, and $\psi $ is a complete isomorphism (completely bounded isomorphism with a completely bounded inverse) if $\varphi $ is.