Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem
Abstract.
This paper continues our study of algebras of bounded, noncommutative (nc) holomorphic functions on nc subvarieties of the nc unit ball (where ). Given a nc variety in the nc unit ball , we consider the algebra of bounded nc holomorphic functions on . We find that the finite dimensional and weak continuous representations are parametrized by the similarity envelope of . Therefore, can be considered as a certain subalgebra of nc holomorphic functions on . We investigate the problem of when two algebras and are isomorphic, and we obtain various results in different settings. In full generality, we prove that these algebras are weak continuously isomorphic if and only if there is a nc biholomorphism between the similarity envelopes that is biLipschitz with respect to the free pseudohyperbolic metric. Moreover, such an isomorphism always has the form , where is a nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras studied by DavidsonPitts and by Popescu. In particular, we find that is a proper subgroup of .
When and the varieties are homogeneous, we remove the weak continuity assumption, showing that two such algebras are boundedly isomorphic if and only if there is a biLipschitz nc biholomorphism between the similarity envelopes of the nc varieties; further, we show that such a biholomorphism can be replaced by a linear map. We provide two proofs: in the noncommutative setting, our main tool is the noncommutative spectral radius; in the free commutative case, we use a new free commutative Nullstellensatz that allows us to bootstrap techniques from the fully commutative case.
We discuss completely bounded versions of the above classification results, and it turns out that in the homogeneous case, two algebras are boundedly isomorphic if and only if they are completely boundedly isomorphic.
We also briefly treat the algebras of bounded nc holomorphic functions on that extend uniformly continuously to . In the case of homogeneous varieties, we find the same classification results as for the algebras of bounded nc holomorphic functions.
2010 Mathematics Subject Classification:
47L80, 46L07,47L251. Introduction
1.1. Background
We study noncommutative (nc) function theory in complex variables, where or . Let denote the set of all matrices over , and let be the set of all tuples of matrices, such that the row determines a bounded operator from to (of course, this specification matters only when ). We norm with the row operator norm , and endow with the induced topology. We define (the nc universe)
and (the commutative nc universe)
A set is said to be a nc set if it is closed under direct sums. If is a nc set, we denote . A subset is said to be open/closed if for all , is open/closed. This collection of open sets gives rise to a topology on and its subsets, called the disjoint union topology. The boundary of , denoted , is defined to be . The principle nc open set that we shall consider is the (dimensional) open matrix unit ball , which is defined to be
Let be a vector space. A function from a nc set to is said to be a nc function (with values in ) if

is graded: ;

respects direct sums: ; and

respects similarities: if and is invertible, and if , then .
A nc function with values in is said to be a scalarvalued nc function. In this paper, we shall deal only with scalarvalued nc functions.
A function defined on a nc open set is said to be nc holomorphic if it is a nc function and, in addition, it is locally bounded. It turns out that a nc holomorphic function is really a holomorphic function when considered as a function , for all , and moreover it has a “Taylor series” at every point (see [23]). We let
denote the algebra of all bounded nc holomorphic functions on the nc unit ball. This algebra (with the norm) can be shown to be the algebra of multipliers of the noncommutative DruryArveson space, and as an operator algebra it is unitarily equivalent to the noncommutative analytic Toeplitz algebra studied by Davidson and Pitts, which was also studied by Popescu under the name noncommutative Hardy algebra (see [34, Section 3]).
A noncommutative (nc) variety in the unit ball is a set of the form
where is a set of bounded nc holomorphic functions. We emphasize that although it makes sense to consider varieties determined by arbitrary nc holomorphic functions, such generality is beyond our scope, and we will assume that every variety is given by . Note that nc varieties are nc sets.
A free polynomial is an element in (the free algebra in noncommuting variables). Let be the free monoid on generators . If (in which case we write ), we define the free monomial . A polynomial can be written in a unique way as where
The polynomial is called the homogeneous component of degree of . Every free polynomial is a (scalarvalued) nc function on in a natural way, by evaluation; moreover, this nc function is bounded on every uniformly bounded subset of . The inclusion of in is injective (this might not be entirely obvious, but follows from the known fact the matrix algebra satisfies no polynomial identity of degree less than ; see [2]).
Given a nc variety , we define to be the algebra of bounded nc functions on , and to be the algebra of bounded nc functions that extend to uniformly continuous functions on (the closure is taken in the disjoint union topology). We give and the obvious operator algebra structure, where the matrix norm of is given by
It follows from [4, Corollary 5.6] (see also [34, Theorem 4.7]) that if , and if is a nc function that is bounded on , then there exists a bounded nc holomorphic function on such that and .
The algebra can be identified with the multiplier algebra of a nc reproducing kernel Hilbert space [34, Theorem 5.4]. Given a nc variety , we define
and
Then we have that is completely isometrically isomorphic to [34, Theorems 5.2 and 5.4], and when is homogeneous, is completely isometrically isomorphic to [34, Proposition 9.7].
This paper continues the work [34]. Our goal is to understand the structure of the algebras of the form and , and in particular to classify them up to ((completely) bounded) isomorphisms in terms of the nc complex analytic geometry of the underlying variety . Before we state our main results — for motivation as well as perspective — let us discuss a similar problem in the purely algebraic case.
1.2. Motivation – the purely algebraic case
As motivation for our main investigations, we consider the purely algebraic analogues of our problems. Let denote the algebra of complex polynomials in commuting variables (here ). With every ideal one naturally associates the corresponding affine variety
Together with this geometric object, there are two natural algebraic objects: the quotient — which is the universal unital algebra generated by commuting elements satisfying the relations in — and the algebra of regular functions:
Consider two ideals . One may ask when are the quotients and isomorphic, as algebras. When and are radical, then it follows from Hilbert’s Nullstellensatz that and , and it is then not hard to show that and are isomorphic if and only if there exist polynomial maps that restrict to mutually inverse bijections between and .
What happens if and are not radical ideals? The concrete and geometric object is no longer a complete invariant for the quotient algebra . Algebraic geometry offers some elaborate but opaque “geometric” replacements for the variety. A more simpleminded (and perhaps more satisfying) alternative is suggested to us by nc function theory.
We can consider as an algebra of nc functions on (recall that is the set of all commuting tuples of matrices, of all sizes). Given an ideal , let
Points in correspond bijectively to all finite dimensional representations of , via the map that sends every finite dimensional representation to its image on the coordinate functions in the quotient:
The inverse of is given by
for all , where is evaluation at :
Suppose we are given a homomorphism . Then gives rise to a map between the spaces of representations, by . Now, as
there exists such that for every . Then, if ,
Thus, we see that a homomorphism gives rise to a polynomial map mapping into .
On the other hand, suppose we are given a polynomial map that restricts to a map from into . Let
Then clearly gives rise, via precomposition, to a homomorphism from to .
We therefore see that a homomorphism always gives rise to a polynomial map that restricts to a map from into , and such a map always gives rise to a homomorphism . To close the loop, we need a link, a Nullstellensatz, between the quotient and the function algebra .
Given a set , let
In [34, Corollary 11.7] we obtained that for every . We call this the commutative free Nullstellensatz, and it has been known to algebraists in one form or another (see [15]). The commutative free Nullstellensatz implies at once that is isomorphic to the function algebra .
This shows, additionally, that homomorphisms are necessarily precomposition with a polynomial map mapping into . Indeed, after identifying , we saw before that the relation between and is given by for all . Applying to this equality, we obtain that , and therefore
We summarize the conclusion of the above discussion, in the case of an isomorphism, as follows.
Theorem 1.1.
Let and be two ideals in . The algebras and are isomorphic if and only if and are isomorphic, in the sense that there exists polynomial maps and that restrict to bijections between and . Moreover, every homomorphism from to is implemented by a polynomial map .
One can consider ideals inside the algebra of free polynomials in noncommuting variables, and given such an ideal , one can consider the noncommutative variety
If is a homogeneous ideal, then there is an appropriate noncommutative homogeneous Nullstellensatz [34, Theorem 7.3], which says that (with obvious notation)
If one replaces the commutative free Nullstellensatz with the noncommutative homogeneous Nullstellensatz, then the same argument as above (where polynomials are replaced by free polynomials) gives the following theorem:
Theorem 1.2.
Let and be two homogeneous ideals in . The algebras and are isomorphic if and only if and are isomorphic, in the sense that there exists free polynomial maps and that restrict to bijections between and . Moreover, every homomorphism and is implemented by a free polynomial map .
Our main goal in this paper (and in several earlier ones) is to understand the analogue of the above results in an operator algebraic setting. As such algebras are not finitely generated in an algebraic sense by the coordinate functions, there are interesting technical issues to overcome. In passing, we are happy to note that it was by considering the operator algebraic problems that the above purely algebraic results crystallized for us, and they seem to have been overlooked. It is worth noting that Theorems 1.1 and 1.2 fail for general ideals in (consider the ideal generated by the nc polynomial ).
1.3. Main results
In Section 2 we lay the foundations of the paper by providing the necessary preliminaries and obtaining the technical tools used throughout the paper. In particular, we give two alternative descriptions of , the similarity envelope of the free ball. We prove that the similarity envelope of the free ball coincides with the set of all pure matrix tuples, that are also precisely the tuples that have joint spectral radius (in the sense of Popescu) strictly less than . We then proceed to prove a version of the Schwartz lemma for the joint spectral radius of tuples. The proof is based on ideas and results of Vesentini and is thus referred to as the Vesentini–Schwartz lemma. Additionally, we provide a proof for a joint spectral version of the Cartan uniqueness theorem in the case of the similarity envelope of the free ball.
In Section 3 we give a brief description of the other necessary ingredients, namely the nc Szego kernel and the free Drury–Arveson space. We then recall the theory of finite dimensional representation of , which can be identified as the multiplier algebra of the free Drury–Arveson space, developed in [10], and the consequent theory of finite dimensional representations of , for free subvarieties . We prove that in the case , the weak continuous finite dimensional representations are precisely parametrized by the similarity envelope of the variety . Moreover, for every point , the weak representation given by evaluation is the unique bounded representation in the fiber over .
Section 4 is dedicated to two versions of the pseudohyperbolic metric on the similarity envelope of the free ball, the completely bounded and the bounded. We prove that every nc biholomorphism of the similarity envelopes of two varieties and , that is biLipschitz with respect to the completely bounded (resp. bounded) pseudohyperbolic metric, induces a weak continuous completely bounded (resp. bounded) isomorphism of with . In the concluding remarks of this section, we discuss the connection of the pseudohyperbolic metrics to the notions of free hyperbolic geometry that appear in the literature.
In Section 5 we establish the converse of the results of Section 4. We prove that every completely bounded (resp. bounded) unital homomorphism , that maps the weak continuous finite dimensional representations of one algebra to the weak continuous finite dimensional representation of the other, is implemented as a composition with a nc map , that is Lipschitz with respect to the completely bounded (resp. bounded) pseudohyperbolic metric. We conclude the section by discussing automorphisms of the free ball and . We provide an example of a nc automorphism of , that does not induce a bounded automorphism of . We also use our spectral version of the Cartan uniqueness theorem to show that a nc automorphism of induced from a quasiinner automorphism of preserves the similarity orbits of irreducible finite dimensional representations.
Section 6 is concerned with the homogeneous case, in which we obtain sharper results. We show that when and the varieties at hand are homogeneous, then every homomorphism arises as a composition with a nc biholomorphism of their similarity envelopes. In particular, this implies that every bounded isomorphism between and is weak continuous, generalizing the result of Davidson and Pitts for [10, Theorem 4.6]. We then proceed to show that if and are homogeneous varieties, such that and are boundedly isomorphic, then they are completely boundedly isomorphic and the isomorphism is implemented by composition with a linear map. Section 6 also contains an example of a noncommutative variety, for which we can give a complete answer to the isomorphism question.
Section 7 contains a detailed study of the family of nc varieties determined by the commutation relations, and we illustrate our classification results by solving the isomorphism problem in certain cases.
In Section 8 the results of the previous section are derived again, this time in the case of commutative varieties, using commutative techniques. In particular, we prove a homogeneous free commutative Nullstellensatz and classify homogeneous free commutative nc varieties. This allows us to deal with nonradical varieties, due to the fact that the scheme structure is encoded in the higher matricial strata.
In Section 9 we extend our results to the norm closed (instead of wotclosed) algebras generated by the free polynomial functions on homogeneous nc varieties. We prove that every completely bounded (resp. bounded) unital homomorphism is implemented by composition with a nc map , such that is Lipschitz with respect to the relevant pseudohyperbolic metric (recall that denotes the algebra of multipliers that are uniformly continuous on ). Consequently, we deduce that and are boundedly isomorphic if and only if they are completely boundedly isomorphic, and moreover in this case an isomorphism is given by composition with a linear map which maps one variety onto the other.
2. The similarity invariant envelope and the joint spectral radius
Recall that , the set of all tuples of all matrices, of all sizes. Unless we make the explicit assumption that is finite, we shall treat .
2.1. The similarity invariant envelope
Let . The similarity invariant envelope (or simply the similarity envelope) of is defined to be the set given by
Clearly, if is open, then so is . In the appendix of [23], it is shown that is a nc set if is, and that every nc function on extends uniquely to a nc function on .
Every extends uniquely to a nc holomorphic function on , and likewise, if is a nc variety, then every extends uniquely to a nc holomorphic function on . Of course, need not be bounded. The nc holomorphic functions on which are extensions of functions in are precisely those whose restriction to is bounded, or, equivalently, whose restriction to every set bounded in the completely bounded norm, is bounded.
The similarity envelope of will be of particular interest in this paper and thus we will provide two descriptions of this set. For every and we have the associated completely positive map on , given by
A point is in if and only if . Recall that a point is called pure if . We will first show that every pure point is in fact similar to a strict row contraction. To prove this claim we need two lemmas.
Lemma 2.1.
For every and , is similar to a strict row contraction if and only if there exists a strictly positive , such that .
Proof.
First let us assume that such an exists. If is the unique positive square root of , then by assumption we have
In other words is a strict row contraction.
Conversely, if is similar to a strict row contraction, then there exists invertible, such that is a strict row contraction. Using the same consideration as above we deduce that
Now set . ∎
Lemma 2.2.
Suppose that and are similar to strict row contractions. Then, for every tuple of matrices we have that the point is similar to a strict row contraction.
Proof.
If and are strict row contractions, then by conjugating by we may assume that in fact and are strict row contractions, since will be replaced by .
Let and consider the conjugation of our point by
Note that . Since is open, there exists such that implements the similarity of our matrix to a strict row contraction. ∎
A tuple is called irreducible (or sometimes generic) if the only nontrivial subspace satisfying is , or equivalently, if the have no joint nontrivial proper invariant subspace. If is not irreducible, then it is called reducible.
It is not hard to see that every is similar to a block upper triangular tuple whose diagonal blocks are all irreducible. Indeed, if is reducible, let be a joint invariant subspace of the . Then there exists an invertible matrix such that where both and are of size smaller than that of . An induction argument now completes the proof.
Alternatively, since it is finite dimensional, the module is both Noetherian and Artinian. Such modules (i.e., Nothereian and Artinian modules) are exactly the modules that have composition series [21, Proposition 3.2.2].
One way or another, the Jordan–Hölder theorem [21, Theorem 3.2.1], implies that if
are two “decompositions” as above of — namely, both are similar to and each of the diagonal blocks is irreducible — then , and for every we have that is similar to one of the s. Thus, up to similarity are unique. We call them the Jordan–Hölder components of .
Proposition 2.3.
A point is pure if and only if it is similar to a strict row contraction.
Proof.
Assume first that is similar to a strict row contraction, i.e., there exists an invertible , such that is a strict row contraction. Thus and we conclude that . One checks that for , we have the identity
It follows that . Since is strictly positive, we can choose , such that and applying we conclude that or in other words that is pure.
Now assume that is pure. First, let us assume that is also irreducible. Let denote the spectral radius of . By [17, Theorem 2] the map is irreducible in the sense of [16]. By Theorems 2.3 and 2.5 in [16], there is a strictly positive , such that . Since is completely positive, and thus in norm. We can conclude that , so . Thus by Lemma 2.1 we are done in the case that is irreducible.
To prove the general case we proceed by induction on the number of Jordan–Hölder components of . If the number is , then is irreducible and we have proved it. Now if the number is , then we can write , where is irreducible and has Jordan–Hölder components. Note that since is pure, and must also be pure. By induction is similar to a strict row contraction and so is , since it is irreducible. It remains to apply Lemma 2.2. ∎
2.2. The joint spectral radius
Let . Given , write for the row comprising of the free monomials of degree in the entries of . Following Popescu, we define the joint spectral radius of the tuple as
Since is a completely positive map we have that and thus is the spectral radius of . Note that ; in particular this notion of spectral radius reduces to the usual one when .
The joint spectral radius will be crucial tool for showing that every bounded isomorphism gives rise to a nc biholomorphism between the similarity envelopes of varieties. The following lemma lists some properties of the joint spectral radius.
Lemma 2.4.
For every we have that:

for every , ;

if is irreducible, then ; and

if is reducible and are its Jordan–Hölder components, then , and thus it is the same as the joint spectral radius of the semisimple elements in the closure of the similarity orbit of .
Proof.
Let and let . Observe that for every , . Therefore, . Applying the same consideration we see that . Hence we have that similarities preserve the joint spectral radius. In particular we have that for every .
On the other hand, if is irreducible, then — as noted in the proof of Proposition 2.3 — there exists a strictly positive , such that . Letting , we put , and we find that . On the other hand, , and . We therefore get that and additionally .
For the last claim observe that applying a similarity we may assume that is block upper triangular, or in other words, that is similar to a sum of block diagonal tuple and jointly nilpotent tuple . Now we can choose , such that . Since , and since the spectral radius of an operator on a finite dimensional space is a continuous function, we get that . It is obvious that . ∎
Lemma 2.5.
Let and . Then, if and only if , and this happens if and only if .
Proof.
If , then for sufficiently large , and thus . If , then , so is pure. By Proposition 2.3, . Finally, let be a strict row contraction. Then is a strict contraction, so . Since similar tuples have the same joint spectral radius, the proof is complete. ∎
The goal of the following discussion is to obtain a Schwarz type lemma for the joint spectral radius, by proving that it is “subharmonic on discs”. We will follow the ideas of Vesentini [38] and [39].
Lemma 2.6.
If is an analytic function, then the function is subharmonic.
Proof.
It is obvious that the function is holomorphic. Therefore, since the norm of a holomorphic Banachvalued function is subharmonic (this follows from Cauchy’s formula) the function is continuous and subharmonic. As in the proof of [38, Theorem 1] we use the fact that a function is subharmonic in if and only if for every the function is subharmonic. Therefore is subharmonic.
Let us write , then and thus . Therefore, for every . Thus the sequence of functions monotonically pointwise decreases to . As in part B of the proof of [38, Theorem 1] we conclude that is subharmonic.
∎
Corollary 2.7.
The function is subharmonic for every analytic.
The following lemma is a version of the Schwarz lemma for the joint spectral radius.
Lemma 2.8 (Vesentini–Schwarz Lemma).
If is an analytic function and , then for every and .
Proof.
Note that for every , . Since the function is analytic, by Corollary 2.7 we get that is subharmonic of the disc. Now for every and every , since the joint spectral radius is homogeneous we get that . Passing to the limit we see that . Thus for every .
Finally, note that and thus .
∎
We would like to obtain a spectral counterpart of the statement in the classical Schwartz lemma that if is such that and , then . We will break the preparation for this result into several lemmas.
Lemma 2.9.
Let be an irreducible point, such that . Then is similar to a coisometry.
Proof.
Lemma 2.10.
Let be an analytic function, such that is a coisometry for some . Then is constant.
Proof.
Consider every as a linear map . Let be a unit vector and define . Note that is an analytic function on . Furthermore, by Cauchy–Schwarz on and , since is a coisometry. Conclude that is the constant function . Since this is true for every we see that for every , the numerical range of the operator is the singleton , and thus this operator is the identity. Thus,
Therefore, . ∎
Lemma 2.11.
Let be an analytic function, such that is irreducible and similar to a coisometry for every , then there exists an irreducible coisometry , such that is similar to , for every .
Proof.
Since for every , is irreducible and we know that is a simple eigenvalue of with a positive eigenvector. Furthermore, since is always similar to a coisometry we can consider the matrixvalued function . By the above consideration the kernel of this matrix is onedimensional for every . Choose a positive . By the implicit function theorem, we can find and a function defined and smooth on , such that for all and . Since , by shrinking further if necessary, we can assume that . Since the eigenvector associated to the spectral radius of is a scalar multiple of a positive definite matrix, we conclude that , for every .
Now fix , where is always strictly positive. Since the minimal eigenvalue of is bounded away from on the circle , by [8, Corollary III.2.1] we can find a holomorphic function on the disc and continuous on the circle, such that for every . There are two things to note regarding the result cited. Firstly, the result discusses right factorization, but it is equivalent to the left factorization by just factoring as Clancey and Gohberg indicate. Secondly, it is immediate that has entries in the Wiener algebra since it is smooth.
Define