Washington University in St. LouisWashington University Open ScholarshipMathematics Faculty Publications Mathematics and Statistics12-2015Non-commutative holomorphic functions onoperator domainsJim AglerJohn E. McCarthyWashington University in St Louis, mccarthy@wustl.eduFollow this and additional works at: https://openscholarship.wustl.edu/math_facpubsPart of the Algebra CommonsThis Article is brought to you for free and open access by the Mathematics and Statistics at Washington University Open Scholarship. It has beenaccepted for inclusion in Mathematics Faculty Publications by an authorized administrator of Washington University Open Scholarship. For moreinformation, please contact digital@wumail.wustl.edu.Recommended CitationAgler, Jim and McCarthy, John E., "Non-commutative holomorphic functions on operator domains" (2015). Mathematics FacultyPublications. 21.https://openscholarship.wustl.edu/math_facpubs/21Non-commutative Holomorphic Functions onOperator DomainsJim Agler ∗U.C. San DiegoLa Jolla, CA 92093John E. McCarthy †Washington UniversitySt. Louis, MO 63130June 4, 2015Abstract: We characterize functions of d-tuples of bounded operatorson a Hilbert space that are uniformly approximable by free polynomials onbalanced open sets.1 IntroductionWhat are non-commutative holomorphic functions in d variables? The ques-tion has been studied since the pioneering work of J.L. Taylor [21], but thereis still no definitive answer. The class should certainly contain the non-commutative, also called free, polynomials, i.e. polynomials defined on dnon-commuting variables. It should be some sort of generalization of thefree polynomials, analogous to how holomorphic functions are generaliza-tions of polynomials in commuting variables. Just as in the commutativecase, the class will depend on the choice of domain. In this note, we shallconsider domains that are sets of d-tuples of operators on a Hilbert space.One approach is to study non-commutative convergent power series ondomains in B(H)d (where B(H)d means d-tuples of bounded operators onsome Hilbert space H). This has been done systematically in G. Popescu’smonograph [20], following on earlier work such as [4, 5, 17–19].∗Partially supported by National Science Foundation Grant DMS 1361720†Partially supported by National Science Foundation Grant DMS 13002801Working with non-commutative power series is natural and appealing,but does present some difficulties. One is that assuming a priori that theseries converges uniformly is a strong assumption, and could be hard to verifyif the function is presented in some other form. On every infinite dimensionalBanach space there is an entire holomorphic function with finite radius ofuniform convergence [6, p. 461]. A second difficulty is dealing with domainsthat are not the domains of convergence of power series.Another approach to non-commutative functions is the theory of nc-functions. Let Mn denote the n-by-n complex matrices, which we shall thinkof as operators on a finite dimensional Hilbert space, and let M[d] = ∪∞n=1Mdn.If x = (x1, . . . , xd) and y = (y1, . . . , yd) are d-tuples of operators on thespaces H and K respectively, we let x⊕y denote the d-tuple (x1⊕y1, . . . , xd⊕yd) on H⊕K; and if s ∈ B(H,K) and t ∈ B(K,H) we let sx and xt denoterespectively (sx1, . . . , sxd) and (x1t, . . . , xdt).Definition 1.1. A function f defined on some set D ⊆M[d] is called gradedif, for each n, f maps D ∩Mdn into Mn.We say f is an nc-function if it is graded and if, whenever x, y ∈ D andthere exists a matrix s such that sx = ys, then sf(x) = f(y)s.The theory of nc-functions has recently become a very active area ofresearch, see e.g. [7–12,15,16]. D. Kaliuzhnyi-Verbovetskyi and V. Vinnikovhave written a monograph [13] which develops the important ideas of thesubject.Nc-functions are a priori defined on matrices, not operators. Certainformulas that represent them (such as (5.5) below) can be naturally extendedto operators. This raises the question of how one can intrinsically characterizefunctions on B(H)d that are in some sense extensions of nc-functions.The purpose of this note is to show that on balanced domains in B(H)dthere is an algebraic property – intertwining preserving – that together withan appropriate continuity is necessary and sufficient for a function to havea convergent power series, which in turn is equivalent to the function beingapproximable by free polynomials on finite sets. Moreover, it is a variationon the idea of an nc-function. On certain domains G]δ defined below, theproperties of intertwining preserving and continuity are equivalent in turn tothe function being the unique extension of a bounded nc-function.Definition 1.2. Let H be an infinite dimensional Hilbert space, let D ⊆B(H)d, and let F : D → B(H). We say that F is intertwining preserving(IP) if:2(i) Whenever x, y ∈ D and there exists some bounded linear operatorT ∈ B(H) such that Tx = yT , then TF (x) = F (y)T .(ii) Whenever (xn) is a bounded sequence in D, and there exists someinvertible bounded linear operator s : H → ⊕H such thats−1 x1 0 · · ·0 x2 · · ·· · · · · · . . . s ∈ D,thenF (s−1 x1 0 · · ·0 x2 · · ·· · · · · · . . . s) = s−1F (x1) 0 · · ·0 F (x2) · · ·· · · · · · . . . s.Note that every free polynomial is IP, and therefore this condition mustbe inherited by any function that is a limit of free polynomials on finite sets.Nc-functions have the property that f(x ⊕ y) = f(x) ⊕ f(y), and we wouldlike to exploit the analogous condition (ii) of IP functions. To do this, wewould like our domains to be closed under direct sums. However, we canonly do this by some identification of H⊕H with H.Definition 1.3. Let H be an infinite dimensional Hilbert space. We say aset D ⊆ B(H)d is closed with respect to countable direct sums if, for everybounded sequence x1, x2, · · · ∈ D, there is a unitary u : H → H ⊕ H ⊕ · · ·such that the d-tuple u∗(x1 ⊕ x2 ⊕ · · · )u ∈ D.Two natural examples are the sets{x ∈ B(H)d : ‖x1‖, . . . , ‖xd‖ < 1} (1.4)and{x ∈ B(H)d : x1(x1)∗ + · · ·+ xd(xd)∗ < I}. (1.5)Definition 1.6. Let F : B(H)d → B(H). We say F is sequentially strongoperator continuous (SSOC) if, whenever xn → x in the strong operatortopology on B(H)d, then F (xn) tends to F (x) in the strong operator topologyon B(H).Since multiplication is sequentially strong operator continuous, it followsthat every free polynomial is SSOC, and this property is also inherited bylimits on sets that are closed w.r.t. direct sums.3Here is our first main result. Recall that a subset B of a complex vectorspace is called balanced if whenever x ∈ B and α is in the closed unit diskD, then αx ∈ B.Theorem 4.1. Let D be a balanced open set in B(H)d that is closed withrespect to countable direct sums, and let F : D → B(H). The following areequivalent:(i) The function F is intertwining preserving and sequentially strong op-erator continuous.(ii) There is a power series expansion∞∑k=0Pk(x)that converges absolutely at each point x ∈ D to F (x), where each Pk is ahomogeneous free polynomial of degree k.(iii) The function F is uniformly approximable on finite subsets of D byfree polynomials.Let δ be an I × J matrix of free polynomials in d variables, where I andJ are any positive integers. ThenGδ := {x ∈M[d] : ‖δ(x)‖ < 1}, (1.7)where if x is a d-tuple of matrices acting on Cn, then we calculate the normof δ(x) as the operator norm from (Cn)J to (Cn)I . Notice that Gδ1 ∩ Gδ2 =Gδ1⊕δ2 , so those sets form a base for a topology; we call this the free topologyon M[d].For the rest of this paper, we shall fix H to be a separable infinite di-mensional Hilbert space, and let B1(H) denote the unit ball in B(H). Let{e1, e2, . . . , } be a fixed orthonormal basis of H, and let Pn denote orthogonalprojection onto ∨{e1, . . . , en}.There is an obvious extension of (1.7) to B(H)d; we shall call this domainG]δ.G]δ := {x ∈ B(H)d : ‖δ(x)‖ < 1}. (1.8)Both (1.4) and (1.5) are of the form (1.8) for an appropriate choice of δ.Note that every G]δ is closed with respect to countable direct sums.By identifying Mn with PnB(H)Pn, we can embed M[d] in B(H)d. If afunction F : G]δ → B(H) satisfies F (x) = PnF (x)Pn whenever x = PnxPn,then F naturally induces a graded function F [ on Gδ.4Here is a slightly simplified version of Theorem 5.7 (the assumption that0 goes to 0 is unnecessary, but without it the statement is more complicated).Theorem 1.9. Assume that G]δ is connected and contains 0. Then everybounded nc-function on Gδ that maps 0 to 0 has a unique extension to anSSOC IP function on G]δ. The extension has a series expansion in free poly-nomials that converges uniformly on G]tδ for each t > 1.2 Intertwining Preserving FunctionsThe normal definition of a an nc-function is a graded function f defined ona set D ⊆ M[d] such that D is closed with respect to direct sums, and suchthat f preserves direct sums and similarities, i.e. f(x ⊕ y) = f(x) ⊕ f(y)and if x = s−1ys then f(x) = s−1f(y)s, whenever x, y ∈ D ∩Mdn and s is aninvertible matrix in Mn. The fact that on such sets D this definition agreeswith our earlier Definition 1.1 is proved in [13, Prop. 2.1].There is a subtle difference between the nc-property and IP, because ofthe roˆle of 0. For an nc-function, f(x⊕0) = f(x)⊕0, but for an IP function,we have f(x⊕ 0) = f(x)⊕ f(0). If f(0) = 0, this presents no difficulty; but0 need not lie in the domain of f , and even if it does, it need not be mappedto 0.Consider, for an illustration, the case d = 1 and the function f(x) = x+1.For each n ∈ N, let Mn be the n-by-n matrix that is 1 in the (1, 1) entry and0 elsewhere. As an nc-function, we havef :1 0 · · · 00 0 · · · 0....... . ....0 0 · · · 0 7→2 0 · · · 00 1 · · · 0....... . ....0 0 · · · 1But now, if we wish to extend f to an IP function on B(H), what is theimage of the diagonal operator T with first entry 1 and the rest 0? We wantto identify T with Mn⊕ 0, and map it to f(Mn)⊕ 0 — but then each n givesa different image.In order to interface with the theory of nc-functions, we shall assume thatall our domains contain 0. To avoid the technical difficulty we just described,we shall compose our functions with Mo¨bius maps to ensure that 0 is mappedto 0.5Lemma 2.1. If F is an IP function on D ⊆ B(H)d, and P ∈ B(H) is aprojection, then, for all c ∈ D satisfying c = cP (or c = Pc) we havea = PaP ⇒ F (a) = PF (a)P + P⊥F (c)P⊥. (2.2)Proof: As Pa = aP , we get PF (a) = F (a)P . As P⊥a = 0 = cP⊥, weget P⊥F (a) = F (c)P⊥. Combining these, we get (2.2). 2We let φα denote the Mo¨bius map on D given byφα(ζ) =ζ − α1− α¯ζ .Lemma 2.3. Let D ⊆ B(H)d contain 0, and assume F is an IP functionfrom D to B1(H). Then:(i) F (0) = αIH.(ii) The map H(x) := φα ◦ F (x) is an IP function on D that maps 0 to0.(iii) For any a ∈ D and any projection P we havea = PaP ⇒ H(a) = PH(a)P.(iv) F = φ−α ◦H.Proof: (i) By Lemma 2.1 applied to a = c = 0, we get that F (0)commutes with every projection P in B(H). Therefore it must be a scalar.(ii) For all z in B1(H), we haveφα(z) = −αIH + (1− |α|2)∞∑n=1α¯n−1zn, (2.4)where the series converges uniformly and absolutely on every ball of radiusless than one. By (i), we have H(0) = φα(αIH) = 0. If Tx = yT , thenTF (x) = F (y)T , and so T [F (x)]n = [F (y)]nT for every n. Letting z be F (x)and F (y) in (2.4) and using the fact that the series converges uniformly, weconclude that Tφα(F (x)) = φα(F (y))T , and hence H is IP.Now (iii) follows from Lemma 2.1 with c = 0. We get (iv) becauseφ−α ◦ φα(z) = z for every z ∈ B1(H). 2By choosing a basis {e1, e2, . . . , } forH, we can identifyMdn with PnB(H)dPn.Let us defineMdn := PnB(H)dPn, M[d] = ∪∞n=1Mdn.Applying Lemma 2.3, we get the following.6Proposition 2.5. Let D ⊆ B(H)d contain 0, and assume F is an IP functionfrom D to B1(H). Let H = φα ◦ F , where α is the scalar such that F (0) =αIH. Then H|D∩M[d] is an nc-function that is bounded by 1 in norm, andmaps 0 in Mdn to the matrix 0 in M1n = PnB(H)Pn.If we let H[ denote H|D∩M[d] , we can askQuestion 2.6 To what extent does H[ determine H?Question 2.7 Does every bounded nc-function from D ∩M[d] to M1extend to a bounded IP function on D?Ifδ(x1, x2) = I − (x1x2 − x2x1),then G]δ is non-empty, but Gδ is empty, and the questions do not makemuch sense. But we do give answers to both questions in Theorem 5.7, inthe special case that D is of the form G]δ and in addition is assumed to bebalanced.3 IP SSOC functions are analyticLet us give a quick summary of what it means for a function to be holomor-phic on a Banach space; we refer the reader to the book [6] by S. Dineen fora comprehensive treatment. Let D be an open subset of a Banach space X,and f : D → Y a map into a Banach space Y . We say f has a Gaˆteauxderivative at x iflimλ→0f(x+ λh)− f(x)λ:= Df(x)[h]exists for all h ∈ X. If f has a Gaˆteaux derivative at every point of Dit is Gaˆteaux holomorphic [6, Lemma 3.3], i.e. holomorphic on each onedimensional slice. If in addition f is locally bounded on D, then it is actuallyFre´chet holomorphic [6, Prop. 3.7], which means that for each x there is aneighborhood G of 0 such that the Taylor seriesf(x+ h) = f(x) +∞∑k=1Dkf(x)[h, . . . , h] ∀h ∈ G, (3.1)7converges uniformly for all h in G. The kth derivative is a continuous linearmap from Xk → Y , which is evaluated on the k-tuple (h, h, . . . , h).The following lemma is the IP version of [8, Prop. 2.5] and [13, Prop 2.2].Lemma 3.2. Let D be an open set in B(H)d that is closed with respect tocountable direct sums, and let F : D → B(H) be intertwining preserving.Then F is bounded on bounded subsets of D, continuous and Gaˆteaux differ-entiable.Proof: (Locally bounded) Suppose there were xn ∈ D such that {‖xn‖}is bounded, but {‖F (xn)‖} is unbounded. Since D is closed with respectto countable direct sums, there exists some unitary u : H → H∞ such thatu∗(⊕xn)u ∈ D. Since F is IP, by Definition 1.2, we have [⊕F (xn)] is bounded,which is a contradiction.(Continuity) Fix a ∈ D and let ε > 0. By hypothesis, there exists aunitary u : H → H2 such thatα := u∗[a 00 a]u ∈ D. (3.3)Choose δ1 > 0 such that B(a, δ1) ⊆ D, B(α, δ1) ⊆ D, and such that onB(α, δ1) the function F is bounded by M . Choose δ2 > 0 such that δ2 0 such that, for all complex numbers t with |t| < ε,u∗[a+ th εh0 a]u ∈ D,8and a+ th ∈ D. Let b = a+ th and λ = εtin (3.4), and as before we concludethatF (u∗[a+ th εh0 a]u) = u∗[F (a+ th) ε F (a+th)−F (a)t0 F (a)]u. (3.5)As F is continuous, when we take the limit as t→ 0 in (3.5), we getF (u∗[a εh0 a]u) = u∗[F (a) εDF (a)[h]0 F (a)]u.Therefore DF (a)[h] exists, so F is Gaˆteaux differentiable, as required. 2When we replace X by a Banach algebra (in our present case, this isB(H)d with coordinate-wise multiplication), we would like something morethan Fre´chet holomorphic: we would like the kth term in (3.1) to be an actualfree polynomial, homogeneous of degree k, in the entries of h.The following result was proved by Kaliuzhnyi-Verbovetskyi and Vinnikov[13, Thm. 6.1] and by I. Klep and S. Spenko [14, Prop. 3.1].Theorem 3.6. Letg : M[d] →M1x 7→ g(x)be an nc-function such that each matrix entry of g(x) is a polynomial ofdegree less than or equal to N in the entries of the matrices xr, 1 ≤ r ≤ d.Then g is a free polynomial of degree less than or equal to N .We extend this result to multilinear SSOC IP maps. Each hj will be ad-tuple of operators, (h1j , . . . , hdj ).Proposition 3.7. LetL : B(H)dN → B(H)(h1, . . . , hN) 7→ L(h1, . . . , hN)be a continuous N-linear map from (B(H)d)N to B(H) that is IP and SSOC.Then L is a homogeneous polynomial of degree N in the variables h11, . . . , hdN .9Proof: By Proposition 2.5, if we restrict L to MdN , we get an nc-function. By Theorem 3.6, there is a free polynomial p of degree N thatagrees with L on MdN . By homogeneity, p must be homogeneous of degreeN . Define∆(h) = L(h)− p(h).Then ∆ vanishes on (Md)N , and is SSOC. Since (Md)N is strong operatortopology dense in (B(H)d)N , it follows that ∆ is identically 0. 2One of the achievements of Kaliuzhnyi-Verbovetskyi and Vinnikov in [13]is the Taylor-Taylor formula, [13, Thm. 4.1]. This comes with a remainderterm, which can be estimated. They show [13, Thm 7.4] that with theassumption of local boundedness, this renders an nc-function analytic. Thefollowing theorem is an IP version of the latter result.Theorem 3.8. Let D be an open neighborhood of 0 in B(H)d, and let F :D → B(H) be a function that is intertwining preserving and sequentiallystrong operator continuous. Then there is an open set U ⊆ D containing 0and homogeneous free polynomials Pk of degree k such thatF (x) = F (0) +∞∑k=1Pk(x) ∀x ∈ U, (3.9)where the convergence is uniform for x ∈ U .Proof: Any open ball centered at 0 is closed with respect to countabledirect sums, so we can assume without loss of generality that D is closed withrespect to countable direct sums and bounded. By Lemma 3.2, F is boundedand Gaˆteaux differentiable on D, and so by [6, Prop. 3.7], F is automaticallyFre´chet holomorphic. Therefore there is some open ball U centered at 0 suchthatF (h) = F (0) +∞∑k=1DkF (0)[h, . . . , h] ∀h ∈ U.We must show that each DkF (0)[h, . . . , h] is actually a free polynomial in h.Claim 1: For each k ∈ N, the functionGk : (h0, . . . , hk) 7→ DkF (h0)[h1, . . . , hk] (3.10)is an IP function on U × (B(H)d)k ⊆ (B(H)d)k+1.10Indeed, when k = 1, we haveDF (h0)[h1] = limt→01t[F (h0 + th1)− F (h0)]. (3.11)As F is IP, so is the right-hand side of (3.11). For k > 1,DkF (h0)[h1, . . . , hk] = limt→01t[Dk−1F (h0 + thk)[h1, . . . , hk−1] (3.12)−Dk−1F (h0)[h1, . . . , hk−1]]. (3.13)By induction, these are all IP.Claim 2: For each k ∈ N, the function Gk from (3.10) is SSOC on U ×(B(H)d)k.Again we do this by induction on k. Let G0 := F , which is SSOC onU ⊆ D by hypothesis. Since Gk−1 is IP on the set Uk, it is locally bounded,and by Lemma 3.2 it is Gaˆteaux differentiable. SupposeSOT limn→∞hjn = hj, 0 ≤ j ≤ k,where each hjn and hj is in U . Let h denote the (k + 1)-tuple (h0, . . . , hk)in Uk+1, and let h˜ denote the k-tuple (h0, . . . , hk−1); similarly, let hn denote(h0n, . . . , hkn) and h˜n denote (h0n, . . . , hk−1n ). There exists some unitary u sothat y = u∗(h˜ ⊕ h˜1 ⊕ h˜2 ⊕ · · · )u is in Uk. Since Gk−1 is differentiable at y,and is IP, we have that the diagonal operator with entries1t[Gk−1(h0 + thk, h1, . . . , hk−1)−Gk−1(h0, h1, . . . , hk−1)],1t[Gk−1(h01 + thk1, h11, . . . , hk−11 )−Gk−1(h01, h11, . . . , hk−11 )],. . . (3.14)has a limit as t→ 0.Let ε > 0, and let v ∈ H have ‖v‖ ≤ 1. Choose t sufficiently close to 0that each of the difference quotients in (3.14) is within ε/3 of its limit (whichis Gk evaluated at the appropriate h or hn). Let n be large enough so that‖[Gk−1(h0 + thk, h1, . . . , hk−1)−Gk−1(h0n + thkn, h1n, . . . , hk−1n )]v‖+ ‖[Gk−1(h0, h1, . . . , hk−1)−Gk−1(h0n, h1n, . . . , hk−1n )]v‖≤ εt3.11Then‖[Gk(h0, . . . , hk)−Gk(h0n, . . . , hkn)]v‖ ≤ ε.So each Gk is SSOC on Uk+1. As Gk is linear in the last k variables, it isSSOC on U × (B(H)d)k as claimed.Therefore for each k, the map(h1, . . . , hk) 7→ DkF (0)[h1, . . . , hk]is a linear IP function that is SSOC in a neighborhood of 0, so by Proposi-tion 3.7 is a free polynomial. 24 Power seriesTheorem 4.1. Let D be a balanced open set in B(H)d that is closed withrespect to countable direct sums, and let F : D → B(H). The following areequivalent:(i) The function F is intertwining preserving and sequentially strong op-erator continuous.(ii) There is a power series expansion∞∑k=0Pk(x) (4.2)that converges absolutely at each point x ∈ D to F (x), where each Pk is ahomogeneous free polynomial of degree k.(iii) The function F is uniformly approximable on finite subsets of D byfree polynomials.Proof: (i) ⇒ (ii). As F is bounded on bounded subsets of D byLemma 3.2, it is Fre´chet holomorphic. By Theorem 3.8, the power seriesat 0 is actually of the form (4.2). We must show the series converges on allof D.Fix x ∈ D. Since D is open and balanced, there exists r > 1 such thatλx ∈ D for every λ ∈ D(0, r). As each Pk is homogeneous, we have that forλ in a neighborhood of 0,F (λx) =∞∑k=0Pk(λx) =∞∑k=0λkPk(x). (4.3)12Therefore the function ψ : λ 7→ F (λx) is analytic on D(0, r), with valuesin B(H), and its power series expansion at 0 is given by (4.3). Let M =sup{‖F (λx)‖ : |λ| < r}.By the Cauchy integral formula, since ‖F‖ is bounded by M , we get that‖dkψdλk(0)‖ ≤ M k!rk. (4.4)Comparing (4.3) and (4.4), we conclude that‖Pk(x)‖ ≤ Mrk,and so the power series in (4.3) converges uniformly and absolutely on theclosed unit disk.(ii) ⇒ (iii). Obvious.(iii) ⇒ (i).(IP (i).) Let x, y ∈ D, and assume there exists T ∈ B(H) such that Tx =yT . Let ε > 0, and choose a free polynomial p such that ‖p(x)− F (x)‖ < εand ‖p(y)− F (y)‖ < ε. Then‖TF (x)− F (y)T‖ = ‖TF (x)− Tp(x) + p(y)T − F (y)T‖≤ 2‖T‖ε.As ε is arbitrary, we conclude that TF (x) = F (y)T .(IP (ii).) Suppose (xn) is a bounded sequence in D, and assume it isinfinite. (The argument for finite sequences is similar). Let z be the diagonald-tuple with entries x1, x2, . . . , and let s : H → H∞ be such that y = s−1zsis in D. For each fixed n, choose a sequence pk of free polynomials thatapproximate F on {y, xn}. ThenF(s−1x1 0 · · ·0 x2 · · ·....... . . s) = limk→∞pk(s−1x1 0 · · ·0 x2 · · ·....... . . s)= s−1 limk→∞pk(x1) 0 · · ·0 pk(x2) · · ·....... . . s.13The nth diagonal entry of the right hand side is F (xn); so we conclude as nis arbitrary thatF(s−1x1 0 · · ·0 x2 · · ·....... . . s) = s−1F (x1) 0 · · ·0 F (x2) · · ·....... . . s.(SSOC). Suppose xn in D converges to x in D in the SOT. As before,by taking direct sums, we can approximate F by free polynomials uniformlyon countable bounded subsets of D. So for any vector v, and any ε > 0, wechoose a free polynomial p so that ‖[F (xn)−p(xn)]v‖ < ε/3, and choose N sothat for n ≥ N , we have ‖[p(x)−p(xn)]v‖ < ε/3. Then ‖[F (x)−F (xn)]v‖ < εfor all n ≥ N . 2In particular, we get the following consequence, which says that boundedIP functions leave closed algebras invariant.In [1, Thm. 7.7] it is shown that for general nc-functions f , it need notbe true that f(x) is in the algebra generated by x.Corollary 4.5. Assume that D is balanced and closed with respect to count-able direct sums, and that F : D → B(H) is SSOC and IP. Then, foreach x ∈ D, the operator F (x) is in the closed unital algebra generated byx1, . . . , xd.5 Free IP functionsRecall the definition of the sets Gδ in (1.7); the topology they generate iscalled the free topology on M[d].Definition 5.1. A free holomorphic function on a free open set D ⊆M[d] isan nc-function that, in the free topology, is locally bounded.Free holomorphic functions are a class of nc-functions studied by theauthors in [2, 3]. In particular, it was shown that there was a representationtheorem for nc-functions that are bounded by 1 on Gδ.Theorem 5.2. [3, Thm. 8.1] Let δ be an I-by-J matrix of free polynomials,and let f be an nc-function on Gδ that is bounded by 1. There exists an14auxiliary Hilbert space L and an isometry[α BC D]: C⊕ LI → C⊕ LJso that for x ∈ Gδ ∩B(K)d,f(x) = αIK+(IK⊗B)(δ(x)⊗IL)[IK⊗ILJ−(IK⊗D)(δ(x)⊗IL)]−1(IK⊗C).(5.3)Obviously one can define a function on G]δ using the right-hand side of(5.3), replacing the finite dimensional space K by the infinite dimensionalspace H. The following theorem gives sufficient conditions on a function toarise this way.Theorem 5.4. Let δ be an I-by-J matrix of free polynomials, and assumethat G]δ is connected and contains 0. Let F : G]δ → B1(H) be sequentiallystrong operator continuous. Then the following are equivalent:(i) The function F is intertwining preserving.(ii) For each t > 1, the function F is uniformly approximable by freepolynomials on G]tδ.(iii) There exists α ∈ D such that if Φ = φα ◦ F , then Φ[ is a freeholomorphic function on Gδ that is bounded by 1 in norm, and that maps 0to 0.(iv) There exists an auxiliary Hilbert space L and an isometry[α BC D]: C⊕ LI → C⊕ LJso that for x ∈ G]δ,F (x) = αIH+(IH⊗B)(δ(x)⊗IL)[IH⊗ILJ−(IH⊗D)(δ(x)⊗IL)]−1(IH⊗C).(5.5)Proof of Theorem 5.4: (i)⇒ (iii). This follows from Proposition 2.5.(iii)⇒ (iv). By Theorem 5.2, we get such a representation for all x ∈ Gδ.The series on the right-hand side of (5.5) that one gets by expanding theNeumann series of [IH ⊗ ILJ − (IH ⊗D)(δ(x)⊗ IL)]−1(5.6)15converges absolutely onG]δ; let us denote this limit byH(x). By Theorem 4.1,since H is a limit of free polynomials, it is IP and SSOC. Moreover, as[α BC D]is an isometry, we get by direct calculation thatIH −H∗(x)H(x) =(IH ⊗ C∗)[IH ⊗ ILJ − (δ(x)∗ ⊗ IL)(IH ⊗D∗)]−1[IH ⊗ ILJ − δ(x)∗δ(x)⊗ IL][IH ⊗ ILJ − (IH ⊗D)(δ(x)⊗ IL)]−1(IH ⊗ C)≥ 0.Indeed, to see the last equality without being deluged by tensors, let us writeH(x) = α˙ + B˙δ˙[I − D˙δ˙]−1C˙,where the dots denote appropriate tensors in (5.5). ThenI −H∗H = I − α˙∗α˙− α˙∗B˙δ˙[I − D˙δ˙]−1C˙ − C˙∗[I − δ˙∗D˙∗]−1δ˙∗B˙∗α˙−C˙∗[I − δ˙∗D˙∗]−1δ˙∗B˙∗B˙δ˙[I − D˙δ˙]−1C˙= C˙∗C˙ + C˙∗D˙δ˙[I − D˙δ˙]−1C˙ + C˙∗[I − δ˙∗D˙∗]−1δ˙∗D˙∗C˙−C˙∗[I − δ˙∗D˙∗]−1δ˙∗[I − D˙∗D˙]δ˙[I − D˙δ˙]−1C˙= C˙∗[I − δ˙∗D˙∗]−1{[I − δ˙∗D˙∗][I − D˙δ˙]+[I − δ˙∗D˙∗]D˙δ˙ + δ˙∗D˙∗[I − D˙δ˙]−δ˙∗[I − D˙∗D˙]δ˙}[I − D˙δ˙]−1C˙= C˙∗[I − δ˙∗D˙∗]−1[I − δ˙∗δ˙][I − D˙δ˙]−1C˙.Therefore ‖H(x)‖ ≤ 1 for all x ∈ G]δ. Let ∆(x) = H(x) − F (x). Then∆ is a bounded IP SSOC Fre´chet holomorphic function on G]δ that vanisheson G]δ ∩M[d] = Gδ. There is a balanced neighborhood U of 0 in G]δ . ByTheorem 4.1, ∆ has a power series expansion ∆(x) =∑Pk(x), and each Pkvanishes on U ∩M[d]. This means each Pk vanishes on a neighborhood ofzero in every Mdn, and hence must be zero. Therefore ∆ is identically zero onU . By analytic continuation, ∆ is identically zero on all of G]δ, and therefore(5.5) holds.(iv) ⇒ (ii). This follows because the Neumann series obtained by ex-panding (5.6) has the kth term bounded by ‖δ(x)‖k. Therefore it convergesuniformly and absolutely on G]tδ for every t > 1.16(ii) ⇒ (i). Repeat the argument of (iii) ⇒ (i) of Theorem 4.1. 2In the notation of the theorem, let F [ = φ−α ◦Φ[. Then the proof of (iii)⇒ (iv) shows that F and F [ determine each other uniquely. So we get:Theorem 5.7. Let δ be an I-by-J matrix of free polynomials, and assumethat G]δ is connected and contains 0. Then every bounded free holomorphicfunction on Gδ has a unique extension to an IP SSOC function on G]δ.References[1] J. Agler and J.E. 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Non-commutative holomorphic functions on Operator domains

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