von Neumann's inequality for row contractive matrix tuples

We prove that for all $n\in \mathbb{N}$, there exists a constant $C_{n}$ such that for all $d \in \mathbb{N}$, for every row contraction $T$ consisting of $d$ commuting $n \times n$ matrices and every polynomial $p$, the following inequality holds: \[ \|p(T)\| \le C_{n} \sup_{z \in \mathbb{B}_d} |p(z)| . \] We apply this result and the considerations involved in the proof to several open problems from the pertinent literature. First, we show that Gleason's problem cannot be solved contractively in $H^\infty(\mathbb{B}_d)$ for $d \ge 2$. Second, we prove that the multiplier algebra $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ of the weighted Dirichlet space $\mathcal{D}_a(\mathbb{B}_d)$ on the ball is not topologically subhomogeneous when $d \ge 2$ and $a \in (0,d)$. In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra $A(\mathcal{D}_a(\mathbb{B}_d))$ of $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball $\mathfrak{C}\mathfrak{B}_d$ that is levelwise uniformly continuous but not globally uniformly continuous.


Introduction
Recall that a tuple T = (T 1 , . . . , T d ) ∈ B(H) d of operators on a Hilbert space H is said to be a row contraction if T = d i=1 T i T * i 1/2 ≤ 1. We say that T is a strict row contraction if T < 1. If in addition T i T j = T j T i for all i, j then we say that T is a commuting row contraction.
The central result of this paper is the following theorem, which answers Question 9.15 in [23]. This result is the content of Theorem 4.7 below. For each d, n ∈ N, we find an explicit upper bound for constants C d,n such that (1) p(T ) ≤ C d,n sup z∈B d |p(z)| holds for every row contraction T consisting of d commuting n × n matrices. The inequality (1) with a constant that possibly depends on d is the main result of this paper. By applying essentially linear algebraic considerations, we will show that the best constants C d,n are bounded in d for fixed n, so that we may define C n = sup d C d,n . However, for fixed d ∈ N, the constants C d,n converge to infinity at a slower rate than the constants C n . Although it is very likely that the expression we find for general d, n is a gross overestimation of the optimal constant, for the special case n = 2 we find the best possible constant; in fact, we show that C d,2 = 1 for all d; see Corollary 3.4. It is well known that when d ≥ 2 no constant can be found that will satisfy the inequality (1) for all n. To see this, recall that the supremum of p(T ) as T ranges over all n × n row contractions and all n is equal to the multiplier norm p Mult(H 2 d ) of p, considered as a multiplier on the Drury-Arveson space H 2 d (see, for example, [23,Section 11]). On the other hand, we have on the right hand side of the inequality the supremum norm p ∞ := sup z∈B d |p(z)| of p on the Euclidean unit ball B d in C d . The incomparability of the multiplier norm and the supremum norm was already observed by Drury [9].
The incomparability of the multiplier and supremum norms notwithstanding, one might guess that Theorem 1.1, at least in the form of inequality (1), can be obtained by a straightforward application of standard techniques, since the row contractions appearing in it are restricted to act on spaces of a fixed finite dimension. However, we found that some new ideas are needed in order to prove the existence of the constants C d,n . It is worth highlighting that as a consequence of Theorem 1.1 and of the techniques used in the proof, we obtain several results that answer other questions in the literature. We now survey these additional results.
To simplify notation, let CB d (n) ⊆ M n (C) d denote the set of all strict row contractions consisting of d commuting n×n matrices, and let CB d (n) denote its closure, the set of all commuting n × n row contractions. The set is called the free commutative ball.
In Section 3, we make a connection between the n-point muliplier norm f Mult(H 2 d ),n , defined in [3], and Theorem 1.1. In Proposition 3.1 we prove that f Mult(H 2 d ),n = sup{ f (T ) : T ∈ CB d (n) diagonalizable}. This is used to prove that C d,2 = 1 for all d.
It is natural to wonder whether C d,n = 1 for other values of d, n (besides the well known C 1,n = 1, which is just von Neumann's inequality). We answer this in Proposition 3.5, which shows that C 2,3 > 1 and hence C d,n > 1 whenever d ≥ 2 and n ≥ 3. This result is used in Section 5 to show that Gleason's problem cannot be solved contractively in For a > 0, let D a (B d ) be the reproducing kernel Hilbert space (RKHS for short) on B d with reproducing kernel If a = 1, then D a (B d ) = H 2 d , the Drury-Arveson space. Section 4 is concerned with the relationship between operator theory and the multiplier algebra of D a (B d ). The main technical achievement is Lemma 4.6, in which we show that if f ∈ H ∞ (B d ) and T is a tuple of commuting n×n matrices whose joint spectrum is contained in B d , then there exists a function g ∈ Mult(D a (B d )) with f (T ) = g(T ) and g Mult(Da(B d )) ≤ C f ∞ , where C is a constant that depends only on n and a. Since the multiplier norms are well behaved with respect to the holomorphic functional calculus, this lemma allows us to control the norm of f (T ) in terms of f ∞ . An immediate consequence of this result is Theorem 4.7, which is a refined version of Theorem 1.1.
In Section 6 we employ the tools from Section 4 to study the representation theory of the multiplier algebras Mult(D a (B d )) and of their norm closed subalgebras A(D a (B d )) generated by the polynomials. We give a complete description of the bounded finite dimensional representations of A(D a (B d )), and we also show that the algebras A(D a (B d )), and hence Mult(D a (B d )), are not topologically subhomogeneous for a < 0 < d, thereby solving an open problem from [3].
We conclude this paper by solving an open problem from [23]: we show, in Proposition 7.2, that there exists a function f ∈ Mult(H 2 d ) that gives rise to a noncommutative function on the closed free commutative unit ball ⊔ ∞ n=1 CB d (n) that is levelwise uniformly continuous, but not globally uniformly continuous (see Section 7 for details). It might be interesting to note that the question behind Theorem 1.1 grew out of an earlier attempt to settle this problem on uniform continuity.
Acknowledgements. The collaboration leading to this paper was spurred by the presentation of the question behind Theorem 1.1 by one of the authors in the "Open Problems Session" that was held at the online conference OTWIA 2020. We wish to thank Meric Augat, the organizer of that session, as well as the organizers of the conference.
We are also grateful to Łukasz Kosiński for helpful comments and for bringing [16] to our attention. Moreover, we are grateful to the editor Mikael de la Salle for bringing to our attention a theorem of Schur, as well [17], where an elementary proof can be found. This led to an improved upper bound in Lemma 4.5.

Preliminaries on multvariable spectral theory
We will require some elementary facts about the spectrum of a commuting tuple of matrices, which we now briefly review. For more information on joint spectra see the monograph [19].
Recall that given a commutative unital Banach algebra B with maximal ideal space ∆(B), and a d-tuple a = (a 1 , . . . , a d ) ∈ B d , the joint spectrum of a with respect to B is the subset of C d defined by When the algebra B is understood we simply write σ(a). Sometimes the joint spectrum is referred to simply as spectrum. This is perhaps the simplest notion of spectrum and it will suffice for our needs.
If T = (T 1 , . . . , T d ) ∈ B(H) d is a tuple of commuting operators on a Hilbert space, then there is also the notion of Taylor spectrum. We shall not define the Taylor spectrum, but we remark that it is contained in σ B (T ) for any commutative unital Banach algebra B ⊆ B(H) that contains T 1 , . . . , T d . In any case, when H is finite dimensional then the spectrum σ(T ) = σ B (T ) is independent of the unital commutative algebra B that contains T , and is given as the set of points where v 1 , . . . , v n is an orthonormal basis for H in which T 1 , . . . , T d are jointly upper triangular. The above set is also equal to the Taylor spectrum as well as to the so-called Waelbroeck spectrum. The equality of all these spectra in the finite dimensional seetting is explained nicely in Section 2.1 in [6]. If T is a commuting row contraction, then the joint spectrum of T with respect to the unital Banach algebra generated by T is contained in the closed unit ball B d . Indeed, this follows from the fact that characters on operator algebras are automatically completely contractive.
We will also require a basic holomorphic functional calculus for commuting tuples of matrices, which can be regarded as a very special case of the Arens-Calderon functional calculus or of the Taylor functional calculus, see [19,Section 30]. Explicitly, we will use that if T is a tuple of commuting matrices whose joint spectrum is contained in B d , then the ordinary polynomial functional calculus p → p(T ) extends to a continuous algebra homomorphism on the algebra O(B d ) of all holomorphic functions on B d ; we denote the extended homomorphism by f → f (T ). If T is jointly diagonalizable, then f (T ) can simply be computed by applying f to the diagonal entries of a diagonal representation of T . As the general constructions of the Arens-Calderon and of the Taylor functional calculus are somewhat involved, we provide an elementary construction that is sufficient for our needs in Theorem A.1 in the appendix.

Small matrices and the n-point norm
First, we observe that the question behind Theorem 1.1 is closely related to the relationship between the n-point multiplier norm on the Drury-Arveson space and the sup norm. To recall the definition of the n-point multiplier norm, let H be a reproducing kernel Hilbert space of functions on B d . For background on reproducing kernel Hilbert spaces, see [1,21]. For F ⊂ B d , we denote by H F the reproducing kernel Hilbert space on F whose reproducing kernel is the restriction of the reproducing kernel of H to F × F . For n ∈ N with n ≥ 1, the n-point multiplier norm of a function f : B d → C is defined as Clearly, the condition |F | ≤ n can be replaced with |F | = n. See [3] for background on the n-point muliplier norm.
Taking the supremum over all subsets F of B d with |F | = n, we therefore find that We wish to show that f (T ) ≤ 1. An approximation argument shows that we may assume that T ∈ CB d (n). Let λ 1 , . . . , λ n ∈ B d be the joint eigenvalues of T and set F = {λ 1 , . . . , λ n }. Since T is diagonalizable, if g is another holomorphic function on B d that agrees with f on F , then f (T ) = g(T ).
,n , we obtain the following corollary.
The fact that : n ∈ N and T ∈ CB d (n)} has been already observed in the literature on nc functions; see, e.g., [23,Remark 11.3]. This raises the question whether We have not been able to answer this question.
We next use Proposition 3.1 to show that the two point norm on the Drury-Arveson space is simply the supremum norm.
be the inclusion and let P : B d → D be the projection onto the first coordinate. Then ) satisfies the conclusions of the lemma.
If always holds. Conversely, if |F | = 2, then we apply the first statement to find that As a consequence, we obtain a von Neumann-type inequality with constant 1 for 2 × 2 row contractions.
In other words, we may choose C 2 = 1 in Theorem 1.1.
Proof. Suppose initially that T is jointly diagonalizable. Applying Proposition 3.1 and Lemma 3.3, we find that In general, it is known that any tuple T of commuting 2 × 2 matrices can be approximated by a sequence (T n ) of commuting diagonalizable 2 × 2 matrices (see the remarks on page 133 of [14]). The row norm of (T n ) converges to the row norm of T , so by applying (2) to r n T n for a suitable sequence r n ∈ (0, 1) tending to 1, the general result follows.
The following result shows that the last corollary does not extend to 3×3 matrices.
Proposition 3.5. There exists a polynomial p so that p Mult(H 2 2 ),3 > p ∞ . In particular, there exists a pair of commuting 3 × 3 matrices that is a row contraction such that p(T ) > p ∞ . Consequently, C d,n > 1 for all d and n such that d ≥ 2 and n ≥ 3.
One possible approach to showing that C d,3 < ∞, extending the basic idea behind the proof of Lemma 3.3, is to use the special structure of solutions to extremal 3-point Pick problems on the ball obtained by Kosińsksi and Zwonek [16]. It is conceivable that the numerical value of the constant C d,3 could be determined in this way. In the next section, we use a somewhat different method, which very likely does not give optimal constants, but will yield that C d,n < ∞ for any d, n ≥ 1.

von Neumann's inequality up to a constant
Our next goal is to prove Theorem 1.1 in general. To this end, we use a variant of the Schur algorithm, somewhat similar to the proof of the main result in [13].
We require the solution of Gleason's problem in H ∞ (B d ), which we state as a lemma for easier reference. See [22, Section 6.6] for a proof.
Our arguments do not just apply to the Drury-Arveson space, but to standard weighted spaces on the ball. For a > 0, let D a (B d ) be the RKHS on B d with reproducing kernel Proof. Let c > 0. We have to show that the row which is positive if and only if every coefficient of z, w n is non-negative (see, e.g. [12, Corollary 6.3]), which happens if and only if −a n , this happens if and only if If 0 < a ≤ 1, then the function t → a+t t+1 is increasing, so (3) holds if and only if it holds for n = 0, that is, if and only if c 2 ≤ a. If a ≥ 1, then the right-hand side of (3) is at least 1 and tends to 1 as n → ∞, so (3) holds if and only if c 2 ≤ 1. Unfortunately, it is in general not true that commuting diagonalizable matrices are dense in the set of commuting matrices, so consideration of the n-point norm alone is not sufficient. Instead, we will work directly with the n × n matrices. The following result is the key lemma in the proof of Theorem 1.1 and some of the later results.
, let T be a d-tuple of commuting n × n matrices whose joint spectrum is contained in B d and let a > 0. Then there exists a function g ∈ Mult(D a (B d )) with f (T ) = g(T ) and Proof. The proof is by induction on n. If n = 1, we may choose g to be a constant function. Suppose that n ≥ 2 and that the statement has been shown for (n − 1) × (n − 1) row contractions.
Let T be a tuple of commuting n × n matrices whose spectrum is contained in B d . By a unitary change of basis, we may assume that each T i is upper triangular, say where a i is a scalar, b i is a row of length n − 1, and A i is an (n − 1) × (n − 1) matrix.
The assumption on the spectrum of T implies that (a 1 , . . . , a d ) Note that the joint spectrum of the tuple , and suppose that f ∞ ≤ ε. We will show that there exists g ∈ Mult(D a (B d )) with f (T ) = g(T ) and g Mult(Da(B d )) ≤ 1. Let c = f (0) and let ψ be an automorphism of D that maps c to 0 and 0 to c. Define h = ψ • f . Using a standard estimate for holomorphic self-maps of D [7, Corollary 2.40], we see that (4), we infer that By the inductive hypothesis, there exist u 1 , . . . , u d ∈ Mult(D a (B d )) with u i (A) = h i (A) and The following lemma is useful for improving estimates if the number of variables d is significantly larger than the size of the matrix n. It is inspired by the result in [16] that solutions to extremal 3-point Pick problems in any number of variables only depend on two variables up to automorphisms.
By a theorem of Schur, a commutative subalgebra of M n has dimension at most ⌊n 2 /4⌋ + 1. It follows that ker(Φ) has codimension at most ⌊n 2 /4⌋ + 1. Therefore, there exists an orthonormal basis (u j ) d j=1 of C d such that u j ∈ ker(Φ) for j > ⌊n 2 /4⌋ + 1. Define a unitary matrix and let θ be the automorphism given by U . By definition, the j-th entry of the d-tuple θ(T ) is given by Φ(u j ), which is zero if j > ⌊n 2 /4⌋ + 1.
With the help of the preceding lemma, we can improve the constant in Lemma 4.4 in the case when d is siginificantly larger than n. Proof. Let T = (T 1 , . . . , T d ). In view of Lemma 4.4, we may assume that d > n ′ = ⌊n 2 /4⌋ + 1. In this case, part (b) of Lemma 4.5 shows that there exists a biholomorphic automorphism θ of B d such that at most the first n ′ entries of θ(T ) are non-zero. Thus, by replacing T with θ(T ) and using automorphism invariance of Mult(D a (B d )) as in the proof of Lemma 4.4, we may assume that T j = 0 for j ≥ n ′ + 1.
Let P : C d → C n ′ be the projection onto the first n ′ coordinates and let i : as desired.
The following theorem is a refinement of Theorem 1.1. Proof. As noted in Section 2, we have that σ(T ) ⊆ B d . Replacing T with rT for 0 < r < 1, we may assume that the spectrum of T is contained in B d . By Lemma 4.6, there exists g ∈ Mult(H 2 d ) with g(T ) = p(T ) and g Mult( Hence, by the von Neumann inequality for H 2 d of Drury [9], Müller-Vasilescu [18] and Arveson [4], we obtain the estimate Notice that the constant in Theorem 4.7 may be bounded above by which is independent of d and only depends on n. However, for fixed d and large n, the estimate in Theorem 4.7 is better.
where n = dk+d d is the dimension of the space of polynomials of degree at most k in d variables. In particular, for sufficiently large n, we have Unfortunately, the gap between these lower bounds and the upper bounds given by Theorem 4.7 is huge, and we are not able to determine whether there is a strict inequality C 2,n < C d,n for any n > 2.
We also easily obtain a completely bounded version of Theorem 4.7.

Gleason's problem
Recall that Gleason's problem for H ∞ (B d ) is the question of whether every function f ∈ H ∞ (B d ) with f (0) = 0 can we written as Work of Leibenson and of Ahern and Schneider shows that this question has a positive answer (see [22,Section 6.6] and Lemma 4.1). Thus, it is natural to ask about the minimal possible norm of a tuple (f 1 , . . . , f d ) of solutions. In [8], Doubtsov studied linear operators solving Gleason's problem; he showed that the solution of Leibenson and Ahern-Schneider gives the minimal norm among all such operators. Moreover, he determined the minimal norm when H ∞ (B d ) is replaced with H 2 (B d ).
Then g Mult(H 2 d ) ≤ 1 and g θ(F ) = f θ(F ) , because f and g agree on F ′ and at 0. Set u = ψ −1 • g • θ. By automorphism invariance of Mult(H 2 d ) and by the classical von Neumann inequality, we have u Mult(H 2 d ) ≤ 1. Moreover, since f = ψ • p • θ −1 , we find that u agrees with p on F . This contradicts (7) and hence finishes the proof.

Applications to RKHS on the ball
In this section, we show that our methods can also be used to answer Question 10.3 of [3]. It was shown in [3] that for 0 ≤ a < d+1 2 , the n-point multiplier norm on D a (B d ) is not comparable to the full multiplier norm, and in fact Mult (D a (B d )) is not topologically subhomogeneous. This means that there do not exist constants c and C and an integer N such that for all f ∈ Mult(D a (B d )), where the supremum is taken over all unital homomorphisms π : A → M k with π ≤ C and all k ≤ N .
If a ≥ d, then Mult(D a (B d )) = H ∞ (B d ) completely isometrically, so it was asked whether Mult(D a (B d )) is topologically subhomogeneous for d+1 2 ≤ a < d. To answer this question, we require the following consequence of Lemma 4.6. We let A (D a (B d )) denote the norm closure of the polynomials in Mult(D a (B d )). Let a > 0. There exists a constant C(a, n) so that for any unital bounded homomorphism π : A(D a (B d )) → M n , we have Proof. Let π : A(D a (B d )) → M n be a unital bounded homomorphism and let T = (T 1 , . . . , T d ) = (π(M z 1 ), . . . , π(M z d )), so that π(p) = p(T ) for all polynomials p.
Recall that the spectrum of M z in the Banach algebra A(D a (B d )) is contained in B d (for a ≤ 1, this follows for instance from the discussion following Lemma 5.3 in [5]; for a ≥ 1, it holds since M z is a row contraction). Hence the joint spectrum of T is also contained in B d .
If f ∈ O(B d ) and 0 < r < 1, let f r (z) = f (rz). It is well known that the map f → f r is a unital completely contractive homomorphism from Mult(D a (B d )) into A (D a (B d )). Defining π r (f ) = π(f r ), we obtain a unital homomorphism π r : Mult(D a (B d )) → M n with π r ≤ π and π r (M z ) = rT . Given a polyonomial p and 0 < r < 1, Lemma 4.6 yields a constant C(a, n) and a function g ∈ Mult(D a (B d )) so that g(rT ) = p(rT ) and g Mult(Da(B d )) ≤ C(a, n) p ∞ . Moreover, since g r ∈ A(D a (B d )) and π is continuous, π r (p) = p(rT ) = g(rT ) = π(g r ) = π r (g).

Thus
π r (p) = π r (g) ≤ π r g Mult(Da(B d )) ≤ C(a, n) π p ∞ . Taking the limit r → 1, we obtain the desired conclusion for polynomials. By continuity, the result follows for all f ∈ A(D a (D d )).
Theorem 6.2. Let 0 < a < d. Then the algebras A(D a (B d )) and Mult(D a (B d )) are not topologically subhomogeneous.
Proof. If 0 < a < d, then the multiplier norm on D a (B d ) is not dominated by a constant times the the supremum norm for polynomials; see [3, Proposition 9.7] and its proof. Thus, it follows from Lemma 6.1 that A (D a (B d )) is not topologically subhomogeneous. Hence, the larger algebra Mult(D a (B d )) is not topologically subhomogeneous either.
We also see that for a > 0, the n-point multiplier norm on D a (B d ) is comparable to the supremum norm. If d = 1, this was shown in [3, Corollary 3.3]. Corollary 6.3. Let a > 0. Then there exists a constant C(a, n) so that Proof. The first inequality is trivial. For the second inequality, let f ∈ H ∞ (B d ) and suppose that F ⊂ B d with |F | ≤ n. Applying Lemma 4.6 to the diagonal tuple T = (T 1 , . . . , T d ) ∈ M d n whose entries are the points of F , we obtain a constant C(a, n) and g ∈ Mult(D a (B d )) with g F = f F and g Mult(Da(B d )) ≤ C(a, n) f ∞ . Hence Hence the second inequality holds.
We now use Theorem 4.7 and Lemma 6. Proof. Suppose that π : A(D a (B d )) → M n is a bounded unital homomorphism. By the first part of the proof of Lemma 6.1, there is a d-tuple T with σ(T ) ⊆ B d , such that π(p) = p(T ) for every polynomial. By Lemma 6.1, this extends uniquely to a continuous unital representation of A(B d ). Indeed, when can simply define f (T ) := π(f ) := lim π(p n ) where p n are polynomials that converge to f uniformly on the closed ball.
Conversly, if π : A(B d ) → M n is a bounded unital homomorphism, then it restricts to the subalgebra A(D a (B d )), and because the multiplier norm is always bigger than the supremum norm, π A(Da(B d )) is also bounded.
Finally, since the bounded and unital n-dimensional representations of all the algebras A(D a (B d )) coincide, and since a representation is clearly determined by the images of the coordinate functions T 1 = π(z 1 ), . . . , T d = π(z d ), it suffices to identify the representations of A d := A(D 1 (B d )). However, by [24,Proposition 10.1], the bounded representations of A d are in one to one correspondence with the d-tuples T which are jointly similar to a row contraction.
Remark 6.5. It is natural to ask whether one may replace the condition for the d-tuple T to be similar to a row contraction, with the condition that σ(T ) ⊆ B d . However, this is not true. For example, consider the case d = 1 and T = 1 1 0 1 . Then σ(T ) = {1} ⊆ D, but T is not similar to a contraction. Indeed, T does not define a bounded representation of A(D), as T is not power bounded.

An application to uniform continuity of noncommutative functions
In this section we use our previous results to answer an open question regarding uniform continuity of noncommutative (nc) functions on the nc unit ball [23]. For a thorough introduction to the theory of nc functions, see [15]; the beginner might prefer to start with the expository paper [2].
The multiplier algebra Mult(H 2 d ) can be identified (via the functional calculus) with the algebra H ∞ (CB d ) of bounded nc holomorphic functions on the nc variety ; for a discussion of this point of view see [23] (in particular Section 11).
For brevity and to be compatible with other parts of the literature, let us write A d for the norm closure of the polynomials in Mult(H 2 d ), that is A d will be just shorthand for A (D 1 (B d )). By [23,Corollary 9.4], A d equals the subalgebra of H ∞ (CB d ) consisting of all bounded nc functions that extend uniformly continuously is said to be uniformly continuous if for every ǫ > 0, there exists a δ > 0 such that for all n and all X, Y ∈ CB d (n), By Theorem 6.4, every row contraction T ∈ CB d (n) gives rise to a a bounded unital representation of A(B d ), which we denote f → f (T ).
is uniformly continuous, in the sense that for all n, and for every ǫ > 0, there exists a δ > 0 such that X, Y ∈ CB d (n) and Proof. It is clear that every p ∈ C[z 1 , . . . , z d ] can be evaluated at every T ∈ CB d (n), and that p is uniformly continuous on CB d (n). Moreover, A(B d ) is the closure of C[z 1 , . . . , z d ] with respect to the supremum norm. Therefore, if f ∈ A(B d ) and p n ∈ C[z 1 , . . . , z d ] is a sequence of polynomials that converges in the supremum norm to f , then Theorem 4.7 implies that p n (T ) converges in norm to f (T ), and the convergence is uniform in T ∈ CB d (n). As the uniform limit of the uniformly continuous functions p n : CB d (n) → M n , the function f : CB d (n) → M n is also uniformly continuous.
By the proposition, every f ∈ A(B d ) extends to a function on CB d , and since f (T ) is given by the functional calculus (see the appendix) it is not hard to see that f is actually an nc function. Moreover, f is levelwise bounded and also levelwise uniformly continuous, in the obvious sense. However, there are functions in A(B d ) which are not multipliers, and hence not uniformly bounded on CB d (see Section 3.7 "The strict containment M d H ∞ (B d )" in [25]).
Since bounded noncommutative functions have some remarkable regularity properties, it might seem plausible that an nc function that is both globally bounded and uniformly continuous on every level CB d (n), will be forced somehow to be uniformly continuous on CB d . Question 9.16 in [23] asked whether there exist functions in H ∞ (CB) that are levelwise uniformly continuous but not uniformly continuous. We can now show that the answer to this question is positive.
The existence of such a function was established in Section 5.2 of [25] ("Continuous multipliers versus A d "), where it was explained how this follows from the methods of [11]. Since σ(a) ⊂ B d is compact, the Cauchy-Schwarz inequality in C d implies that σ( a, ζ ) ⊂ {λ ∈ C : |λ| ≤ s} for some 0 < s < 1 for all ζ ∈ ∂B d . Using continuity of the inverse in B, we may therefore define where dσ is the normalized surface measure on ∂B d . It is clear that Φ is linear. Moreover, where the last factor is finite by compactness of ∂B d and continuity of the inverse in B. Thus, Φ is continuous. We finish the proof by showing that Φ(z β ) = a β for every monomial z β . Since the polynomial functional calculus is a homomorphism, it then follows that Φ is a homomorphism as well. As σ( a, ζ ) ⊂ {λ ∈ C : |λ| ≤ s} for all ζ ∈ ∂B d , the spectrum of ζ → a, ζ in the Banach algebra C(∂B d , B) of all continuous functions from ∂B d into B is contained in {λ ∈ C : |λ| ≤ s} as well. Applying the spectral radius formula in C(∂B d , B), we find that lim n→∞ sup ζ∈∂B d a, ζ n 1/n ≤ s < 1. Consequently, we may expand (1 − a, ζ ) −d into a binomial series that converges uniformly in ζ ∈ ∂B d , so A basic orthogonality relation for the surface integral (Propositions 1.4.8 and 1.4.9 in [22]) shows that so Φ(z β ) = a β , as desired.
Alternatively, in the case of a tuple T = (T 1 , . . . , T d ) of commuting Hilbert space operators, it is possible to define the holomorphic functional calculus Φ of Theorem A.1 with the help of convergent power series. This uses one inequality of the multivariable spectral radius formula [20, Theorem 1], namely lim sup n→∞ |α|=n n α (T * ) α T α 1/2n ≤ sup{|λ| : λ ∈ σ(T )}, which can be proved in an elementary fashion. Using this, one shows that if σ(T ) ⊂ B d , then for each f ∈ H 2 d with homogeneous expansion f = ∞ n=0 f n , the series ∞ n=0 f n (T ) converges absolutely, from which the holomorphic functional calculus can be easily deduced. We omit the details.
As is customary, we usually write f (a) for Φ(f ) in the setting of Theorem A.1. The superposition principle for the functional calculus in our particular setting can also be obtained by elementary means. Proof. (a) Let a = (a 1 , . . . , a d ). If χ is a character on B, then for every polnomial p, we have χ(p(a)) = p(χ(a 1 ), . . . , χ(a d )), hence χ(f j (a)) = f j (χ(a 1 ), . . . , χ(a d )) for j = 1, . . . , d by an approximation argument. The definition of joint spectrum then yields σ B (f (a)) = f (σ B (a)).
(b) Since the functional calculus for the tuple a is a homomorphism, we have p(f (a)) = (p • f )(a) for every polynomial p. The general case then follows from an approximation argument.