The isomorphism problem for complete Pick algebras: a survey
aa r X i v : . [ m a t h . OA ] D ec THE ISOMORPHISM PROBLEM FOR COMPLETE PICK ALGEBRAS:A SURVEY
GUY SALOMON AND ORR MOSHE SHALIT
Abstract.
Complete Pick algebras — these are, roughly, the multiplier algebras in whichPick’s interpolation theorem holds true — have been the focus of much research in the lasttwenty years or so. All (irreducible) complete Pick algebras may be realized concretely asthe algebras obtained by restricting multipliers on Drury-Arveson space to a subvariety ofthe unit ball; to be precise: every irreducible complete Pick algebra has the form M V = { f (cid:12)(cid:12) V : f ∈ M d } , where M d denotes the multiplier algebra of the Drury-Arveson space H d ,and V is the joint zero set of some functions in M d . In recent years several works weredevoted to the classification of complete Pick algebras in terms of the complex geometryof the varieties with which they are associated. The purpose of this survey is to give anaccount of this research in a comprehensive and unified way. We describe the array of toolsand methods that were developed for this program, and take the opportunity to clarify,improve, and correct some parts of the literature. Introduction
Motivation and background.
Consider the following two classical theorems.
Theorem A (Gelfand, [18]) . Let X and Y be two compact Hausdorff spaces. The alge-bras of continuous functions C ( X ) and C ( Y ) are isomorphic if and only if X and Y arehomeomorphic. Theorem B (Bers, [7]) . Let U and V be open subsets of C . The algebras of holomorphicfunctions Hol( U ) and Hol( V ) are isomorphic if and only if U and V are biholomorphic The common theme of these two theorems is that an appropriate algebra of functionson a space encapsulates in its algebraic structure every aspect of the topological/complex-geometric structure of the space. The problem that we are concerned with in this paper hasa very similar flavour. Let M d denote the algebra of multipliers on Drury-Arveson space —precise definitions will be given in the next section, for now it suffices to say that M d is acertain algebra of bounded analytic functions on the unit ball B d ⊆ C d . For every analyticvariety V ⊆ B d one may define the algebra M V = { f (cid:12)(cid:12) V : f ∈ M d } . The natural question to ask is: in what ways does the variety V determine the algebra M V , and vice versa? In other words, if M V and M W are algebraically isomorphic, can weconclude that V and W are “isomorphic” in some sense? Conversely, if V and W are, say,biholomorphic, can we conclude that the algebras are isomorphic? The second author was partially supported by ISF Grant no. 474/12, by EU FP7/2007-2013 Grant no.321749, and by GIF Grant no. 2297-2282.6/20.1. s we shall explain below, M V is also an operator algebra: it is the multiplier algebra ofa certain reproducing kernel Hilbert space on V , and it is generated by the multiplicationoperators [ M z i h ]( z ) = z i h ( z ) (it will be convenient to denote henceforth Z i = M z i ). Thusone can ask: do the Banach algebraic or operator algebraic structures of M V encode finercomplex-geometric aspects of V ?These questions in themselves are interesting, natural, nontrivial, and studying theminvolves a collection of tools combining function theory, complex geometry and operatortheory. However, it is worth noting that there are routes, other than analogy with TheoremsA and B, that lead one to study the structure and classify the algebras M V described above.One path that leads to considering the algebras M V comes from non-selfadjoint operatoralgebras: it is the study of operator algebras universal with respect to some polynomialrelations. For simplicity consider the case in which V = Z B d ( I ) is the zero set of a radicaland homogeneous polynomial ideal I ⊳ C [ z , . . . , z d ], where Z B d ( I ) = { λ ∈ B d | p ( λ ) = 0 for all p ∈ I} . Then M V is the universal wot -closed unital operator algebra, that is generated by a purecommuting row contraction T = ( T , . . . , T d ) satisfying the relations in I (see [26, 30]). Thismeans that(1) The d -tuple of operators ( Z , . . . , Z d ), given by multiplication by the coordinate func-tions, is a pure, commuting row contraction satisfying the relations in I , and itgenerates M V ;(2) For any such tuple T , there is a unital, completely contractive and wot -continuoushomomorphism from M V into Alg wot (1 , T ) determined by Z i T i .In general (when V is not necessarily the variety of a homogeneous polynomial ideal) it is alittle more complicated to explain the universal property of M V . Roughly, M V is universalfor tuples “satisfying the relations” in J V = { f ∈ M d | f ( λ ) = 0 for all λ ∈ V } .Thus the algebras M V are an operator algebraic version of the coordinate ring on analgebraic variety, and studying the relations between the structure of M V and the geometryof V can be considered as rudimentary steps in developing “operator algebraic geometry”.A different road that leads one to consider the collection of algebras M V runs from functiontheory, in particular from the theory of Pick interpolation. Let H be a reproducing kernelHilbert space on a set X with kernel k . If x , . . . , x n ∈ X and A , . . . , A n ∈ M k ( C ), thenone may consider the problem of finding a matrix valued multiplier F : X → M k ( C ) whichhas multiplier norm 1 and satisfies F ( x i ) = A i , i = 1 , . . . , d. This is called the
Pick interpolation problem . It is not hard to show that a necessary conditionfor the existence of such a multiplier is that the following matrix inequality hold:(1.1) [(1 − F ( x i ) F ( x j ) ∗ ) K ( x i , x j )] ni,j =1 ≥ . G. Pick showed that for the Szeg˝o kernel k ( z, w ) = (1 − z ¯ w ) − the condition (1.1) is alsoa sufficient condition for the existence of a solution to this problem [25]. Kernels for whichcondition (1.1) is a sufficient condition for the existence of a solution to the Pick interpolationproblem have come to be called complete Pick kernels , and their multiplier algebras completePick algebras . We refer the reader to the monograph [2] for thorough introduction to Pickinterpolation and complete Pick kernels. The connection to our problem is the following heorem, which states that under a harmless irreducibility assumption all complete Pickalgebras are completely isometrically isomorphic to one of the algebras M V described above. Theorem C (Agler-McCarthy, [1]) . Let H be a reproducing kernel Hilbert space with anirreducible complete kernel k . Then there exists d ∈ N ∪ {∞} and there is an analyticsubvariety V ⊆ B d such that the multiplier algebra Mult ( H ) of H is unitarily equivalent to M V . In fact the theorem of Agler-McCarthy says much more: the Hilbert space H can (up tosome rescaling) be considered as a Hilbert space of functions on V , which is a subspace ofthe Drury-Arveson space. Since we require this result only for motivation, we do not go intofurther detail.Thus, by studying the algebras M V in terms of the complex-geometric structure of V onemay hope to obtain a structure theory of irreducible complete Pick algebras. In particular,we may hope to use the varieties as complete invariants of irreducible complete Pick algebrasup to isomorphism — be it algebraic, isometric or spatial. This is why we call this study The Isomorphism Problem for Complete Pick Algebras .1.2.
About this survey.
The goal of this survey is to present in a unified way the mainresults on the isomorphism problem for complete Pick algebras obtained in recent years. Wedo not provide all the proofs, but we do give proofs (or at least an outline) to most keyresults, in order to highlight the techniques involved. We give precise references so that allomitted details can be readily found by the interested reader. We also had to omit someresults, but all results directly related to this survey may be found in the cited references.Although one may treat the case where V ⊆ B d and W ⊆ B d ′ where d and d ′ might bedifferent, we will only treat the case where d = d ′ . It is easy to see that this simplificationresults in no real loss.This paper also contains some modest improvements to the results appearing in the lit-erature. In some cases we unify, in others we simplify the proof somewhat, in one case wewere able to extend a result from d < ∞ to d = ∞ (see Theorem 4.8). There is also onecase where we correct a mistake that appeared in an earlier paper (see Remark 4.4).Furthermore, we take this opportunity to call to attention a little mess that resides in theliterature, and try to set it right. (The reader may skip the following paragraph and returnto it after reading Section 2.5.) The results we review in this survey are based directly onresults in the papers [4, 5, 10, 15, 16, 20, 23]. The papers [10, 16] relied in a significant wayon many earlier results of Davidson and Pitts [12, 13, 14], and in particular on [12, Theorem3.2]. The content of that theorem, phrased in the language of this survey, is that over everypoint of V there lies a unique character in the maximal ideal space M ( M V ), and moreoverthat there are no characters over points of B d \ V . Unfortunately, at the time that the papers[10, 16] were in press it was observed by Michael Hartz that [12, Theorem 3.2] is true onlyunder the assumption d < ∞ , a counter example shows that it is false for d = ∞ (see theexample on the first page of [11], or Example 2.4 in the arXiv version of [10]).Luckily, the main results of [10, 16] survived this disaster, but significant changes in thearguments were required, and some of the results survived in a weaker form. The paper [10]has an erratum [11], and [16] contains some corrections made in proof. However, thoroughrevisions of the papers [10, 16] appeared on the arXiv, and when we refer to these papers werefer to the arXiv versions. We direct the interested reader to the arXiv versions. .3. Overview of main results.
Sections 2 and 3 contain some basic results which areused in all of the classification schemes. The main results are presented in Sections 4, 5and 6, which can be read independently after Sections 2 and 3. Some open problems arediscussed in the final section.The following table summarizes what is known and what is not known regarding theisomorphism problem of the algebras M V , where V is a variety in a finite dimensional ball.(In several cases the result also holds for d = ∞ , see caption). onditions on V , W Type of isomorphism M V ∼ = M W Type of isomorphism V ∼ = W ⇒ ⇐ Reference
Weak- ∗ continuous Multiplier biholomorphic √ × Corollary 3.4Example 5.7Isometric There is F ∈ Aut( B d ) s.t. F ( W ) = V √ √ Proposition 4.8Theorem 4.6Completely isometric There is F ∈ Aut( B d ) s.t. F ( W ) = V √ √ Theorem 4.6Unitary equivalence There is F ∈ Aut( B d ) s.t. F ( W ) = V √ √ Theorem 4.6Finite union of irreduciblevarieties and a discrete va-riety Algebraic Multiplier biholomorphic √ ×
Theorem 5.5Example 5.7Irreducible Algebraic Multiplier biholomorphic √ ? Theorem 5.5Subsection 7.1Homogeneous Algebraic There is A ∈ GL d ( C ) s.t. A ( W ) = V √ √ Theorem 5.14Homogeneous Algebraic Biholomorphic √ √
Theorem 5.14Images of finite Riemannsurfaces under a holomapthat extends to be a 1-to-1 C -map on the boundary Algebraic Biholomorphism that ex-teneds to be a 1-to-1 C -map on the boundary ? √ Corollary 5.18Embedded discs Algebraic Biholomorphic √ ×
Example 5.21
Table 1.
Isomorphisms of varieties in B d for d < ∞ corresponding to isomorphisms of the associated multiplieralgebras. The first four lines also hold for d = ∞ with minor adjustments. . Notation and preliminaries
Basic notation.
It this survey, d always stands for a positive integer or ∞ = ℵ . The d -dimpensional Hilbert space over C is denoted by C d (when d = ∞ , C d stands for ℓ ), and B d denotes the open unit ball of C d . When d = 1, we usually write D instead of B d .2.2. The Drury-Arveson space.
Let H d be the Drury-Arveson space (see [29]). H d is thereproducing Hilbert space on B d , the unit ball of C d , with kernel functions k λ ( z ) = 11 − h z, λ i for z, λ ∈ B d . We denote by M d the multiplier algebra Mult( H d ) of H d .2.3. Varieties and their reproducing kernel Hilbert spaces.
We will use the term analytic variety (or just a variety ) to refer to the common zero set of a family of H d -functions.If E is a set of functions on B d which is contained in H d , let V ( E ) := { λ ∈ B d : f ( λ ) = 0 for all f ∈ E } . On the contrary, if S is a subset of B d let H S := { f ∈ H d : f ( λ ) = 0 for all λ ∈ S } , and J S := { f ∈ M d : f ( λ ) = 0 for all λ ∈ S } . Proposition 2.1 ([16], Proposition 2.1) . Let E be a subset of H d , and let V = V ( E ) . Then V = V ( J V ) . Given an analytic variety V , we also define F V := span { k λ : λ ∈ V } . This Hilbert space is naturally a reproducing kernel Hilbert space of functions living on thevariety V . Proposition 2.2 ([16], Proposition 2.3) . Let S ⊆ B d . Then F S := span { k λ : λ ∈ S } = F V ( H S ) = F V ( J S ) . The multiplier algebra of a variety.
The reproducing kernel Hilbert space F V comeswith its multiplier algebra M V = Mult( F V ). This is the algebra of all functions f on V suchthat f h ∈ F V for all h ∈ F V . A standard argument shows that each multiplier determinesa bounded linear operator M f ∈ B ( F V ) given by M f h := f h . We will usually identifythe function f with its multiplication operator M f . We will also identify the subalgebra of B ( F V ) consisting of the M f ’s and the algebra of functions M V (endowed with the samenorm). We let Z i denote both the multiplier corresponding to the i th coordinate function z z i , as well as the multiplication operator it gives rise to. In some cases, for emphasis,we write Z i (cid:12)(cid:12) V instead of Z i .Now consider the map from M d into B ( F V ) sending each multiplier f to P F V M f | F V .One verifies that this map coincides with the map f f | V and therefore its kernel is J V .Thus, the multiplier norm of f | V , for f ∈ M d , is k f + J V k = k P F V M f | F V k . The completeNevanlinna-Pick property then implies that this map is completely isometric onto M V . Thisgives rise to the following proposition. roposition 2.3 ([16], Proposition 2.6) . Let V be an analytic variety in B d . Then M V = { f | V : f ∈ M d } . Moreover the mapping ϕ : M d → M V given by ϕ ( f ) = f | V induces a completely isometricisomorphism and weak- ∗ continuous homeomorphism of M d /J V onto M V . For any g ∈ M V and any f ∈ M d such that f | V = g , we have M g = P F V M f | F V . Given any F ∈ M k ( M V ) ,one can choose e F ∈ M k ( M d ) so that e F | V = F and k e F k = k F k . In the above proposition we referred to the weak- ∗ topology in M V ; this is the weak- ∗ topology which M V naturally inherits from B ( F V ) by virtue of being a wot -closed (henceweak- ∗ closed) subspace. The fact that M V is a dual space has significant consequences forus. It is also useful to know the following. Proposition 2.4 ([16], Lemma 3.1) . Let V be a variety in B d . Then then weak- ∗ and theweak-operator topologies on M V coincide. The character space of M V . Let A be a unital Banach algebra. A character on A isa nonzero multiplicative linear functional. The set of all characters on A , endowed with theweak- ∗ topology, is called the character space of A , and will be denoted by M ( A ). It is easyto check that a character is automatically unital and continuous with norm 1. If furthermore A is an operator algebra, then its characters are automatically completely contractive [24,Proposition 3.8].The algebras we consider are semi-simple commutative Banach algebras, thus one mightexpect that the maximal ideal space will be a central part of the classification. However,these algebras are not uniform algebras; moreover, the topological space M ( M V ) can berather wild. Thus the classification does not use M ( M V ) directly, but rather a subset ofcharacters that can be identified with a subset of B d and can be endowed with additionalstructure.Let V be a variety in B d . Since ( Z , . . . , Z d ) is a row contraction, it holds that k ( ρ ( Z ) , . . . , ρ ( Z d )) k ≤ ρ ∈ M ( M V ) . The map π : M ( M V ) → B d , given by π ( ρ ) = ( ρ ( Z ) , . . . , ρ ( Z d )) , is continuous as a map from M ( M V ), with the weak- ∗ topology, into B d (endowed with theweak topology, in case d = ∞ ). Since π is continuous, π ( M ( M V )) is a compact subset ofthe closed unit ball. For every λ ∈ π ( M ( M V )), the set π − { λ } ⊆ M ( M V ) is called the fiber over λ .For every λ ∈ V , the fiber over λ contains the evaluation functional ρ λ , which is given by ρ λ ( f ) = f ( λ ) , f ∈ M V . The following two results are crucial for much of the analysis of the algebras M V . Proposition 2.5 ([16], Proposition 3.2) . V can be identified with the wot -continuous char-acters of M V via the correspondence λ ↔ ρ λ . Proposition 2.6 ([16], Proposition 3.2) . If d < ∞ , then π ( M ( M V )) ∩ B d = V, and for every λ ∈ V the fiber over λ , that is π − { λ } , is a singleton. .6. Metric structure in M ( M V ) . Let ν ∈ B d , and let Φ ν be the automorphism of theball that exchanges ν and 0 (see [28, p. 25]):Φ ν ( z ) := ν − P ν z − s ν Q ν z − h z, ν i , where P ν = ( h z,ν ih ν,ν i ν if ν = 0 , ν = 0 , Q ν = I − P ν , and s ν = (1 − k ν k ) . If µ ∈ B d is another point, the pseudohyperbolic distance between µ and ν is defined to be d ph ( µ, ν ) := k Φ ν ( µ ) k = k Φ µ ( ν ) k . One can check that the pseudohyperbolic distance defines a metric on the open ball.The following proposition will be useful in the sequel. Among other things it will implythat the metric structure induced on V by the pseudohyperbolic metric is an invariant of M V . Proposition 2.7 ([16], Lemma 5.3) . Let V be a variety in B d .(a) Let µ ∈ ∂ B d and let ϕ ∈ π − ( µ ) . Suppose that ψ ∈ M ( M V ) satisfies k ψ − ϕ k < . Then ψ ∈ π − ( µ ) .(b) If µ, ν ∈ V , then d ph ( µ, ν ) = k ρ µ − ρ ν k sup k f k≤ (cid:12)(cid:12)(cid:12) − f ( µ ) f ( ν ) (cid:12)(cid:12)(cid:12) . As a result, d ph ( µ, ν ) ≤ k ρ µ − ρ ν k ≤ d ph ( µ, ν ) . Weak- ∗ continuous isomorphisms Let V and W be two varieties in B d . We say that V and W are biholomorphic if thereexist holomorphic maps F : B d → C d and G : B d → C d such that G ◦ F | V = id V and F ◦ G | W = id W . If furthermore the coordinate functions of F are multipliers, then we saythat V and W are multiplier biholomorphic .In this section we will see that in the finite dimensional case, if there is a weak- ∗ continuousisomorphism between two multiplier algebras M V and M W , then V and W are multiplierbiholomorphic. We start with the following proposition, which is a basic tool in the theory. Proposition 3.1 ([16], Proposition 3.4) . Let V and W be two varieties in B d , and let ϕ : M V → M W be a unital homomorphism. Then ϕ gives rise to a function F ϕ : W → B d by F ϕ = π ◦ ϕ ∗ | W . Moreover, there exist multipliers F , F , . . . , F d ∈ M such that F ϕ = ( F | W , F | W , . . . , F d | W ) . Furthermore, if ϕ is completely bounded or d < ∞ , then F ϕ extends to a holomorphic functiondefined on B d . Here and below ϕ ∗ is the map from M ( M W ) into M ( M V ) given by ϕ ∗ ( ρ ) = ρ ◦ ϕ for all ρ ∈ M W . roof. Proposition 2.5 gives rise to the following commuting diagram (cid:8) wot -continuouscharacters of M W (cid:9) M ( M W ) M ( M V ) (cid:8) wot -continuouscharacters of M V (cid:9) W π ( M ( M W )) π ( M ( M V )) Vϕ ∗ λ ↔ ρ λ π π λ ↔ ρ λ and the composition of the thick arrows from W to π ( M ( M V )) ⊆ B d yields the map F ϕ .Now since ϕ ( Z i ) ∈ M W = { f | W : f ∈ M} , there is an element F i ∈ M such that ϕ ( Z i ) = F i | W and k F i k = k ϕ ( Z i ) k . Thus, for every λ ∈ W , F ϕ ( λ ) = π ( ϕ ∗ ( ρ λ ))= ( ϕ ∗ ( ρ λ )( Z ) , ϕ ∗ ( ρ λ )( Z ) , . . . , ϕ ∗ ( ρ λ )( Z d ))= ( ϕ ( Z )( λ ) , ϕ ( Z )( λ ) , . . . , ϕ ( Z d )( λ ))= ( F | W ( λ ) , F | W ( λ ) , . . . , F d | W ( λ )) . It remains to show that if ϕ is completely bounded or d < ∞ then ( F , . . . , F d ) defines afunction B d → C d . If d < ∞ it is of course clear. If d = ∞ and ϕ is completely boundedthen the norm of ( ϕ ( Z ) , ϕ ( Z ) , . . . ) is finite, and the F i ’s could have been chosen such that k ( M F , M F , . . . ) k = k ( ϕ ( Z ) , ϕ ( Z ) , . . . ) k . Hence, with this choice of the F i ’s, ( F , F , . . . )defines a function B ∞ → ℓ . (cid:3) Remark . When d = ∞ and ϕ is not completely bounded, we cannot even say that themap F ϕ : W → B d , in the above proposition, is a holomorphic map. The reason is thatby definition a holomorphic function on a variety should be extendable to a holomorphicfunction on an open neighborhood of the variety. However, it is not clear whether thereexists a choice of the F i ’s and a neighborhood of W such that for any λ in this neighborhood( F ( λ ) , F ( λ ) , . . . ) belongs to ℓ .Chasing the diagram in the proof of Proposition 3.1 shows that whenever ϕ ∗ takes weak- ∗ continuous characters of M W to weak- ∗ continuous characters of M V , F ϕ maps W into V . Therefore, if ϕ is a weak- ∗ continuous unital homomorphism, then F ϕ ( W ) ⊆ V . This,together with the observation that the inverse of a weak- ∗ continuous isomorphism is weak- ∗ continuous, gives rise to the following corollary. Corollary 3.3 ([16], Corollary 3.6) . Let V and W be varieties in B d . If ϕ : M V → M W isa unital homomorphism that preserves weak- ∗ continuous characters, then F ϕ ( W ) ⊆ V and ϕ is given by (3.2) ϕ ( F ) = f ◦ F ϕ , f ∈ M V . Moreover, if there exists a weak- ∗ continuous isomorphism ϕ : M V → M W , then F ϕ ( W ) = V , F ϕ − ( V ) = W , and there are multipliers F , . . . , F d , G , . . . , G d ∈ M such that F ϕ = ( F | W , . . . , F d | W ) , and F ϕ − = ( G | V , . . . , G d | V ) . Proof.
It remains only to verify (3.2), the rest follows from the discussion above. If f ∈ M V and λ ∈ W , we find ϕ ( f )( λ ) = ϕ ∗ ( ρ λ )( f ) = ρ F ϕ ( λ ) ( f ) = f ◦ F ϕ ( λ ) , s required. (cid:3) When d < ∞ , we obtain the following result. Corollary 3.4 ([16], Corollary 3.8) . Let V and W be varieties in B d for d < ∞ . If thereexists a weak- ∗ continuous isomorphism ϕ : M V → M W , then V and W are multiplierbiholomorphic. The converse does not hold; see Example 5.7 (see also Corollary 6.9). We conclude thissection with the following assertion which is a direct result of Proposition 2.7(b) togetherwith the fact that isomorphisms are automatically bounded.
Corollary 3.5 ([10], Theorem 6.2) . Suppose F : W → V is a biholomorphism which induces(by composition) an isomorphism ϕ : M V → M W . Then F must be bi-Lipschitz with respectto the pseudohyperbolic metric, i.e., there is a constant c > such that c − d ph ( µ, ν ) ≤ d ph ( F ( µ ) , F ( ν )) ≤ cd ph ( µ, ν ) . The converse does not hold; see [10, Example 6.6].4.
Isometric, completely isometric, and unitarily implemented isomorphisms
Let V and W be two varieties in B d . We say that V and W are conformally equivalent if there exists an automorphism of B d (that is, a biholomorphism from B d into itself) whichmaps V onto W . In this section we will see that if V and W are conformally equivalentthen M V and M W are (completely) isometrically isomorphic (in fact, unitarily equivalent).When d < ∞ the converse also holds, and morally speaking it also holds for d = ∞ . In fact,when d = ∞ it may happen that M V and M W are unitarily equivalent but V and W arenot conformally equivalent. This, however, can only be the result of an unlucky embeddingof V and W into B ∞ , and is easily fixed.4.1. Completely isometric and unitarily implemented isomorphisms.Proposition 4.1 ([16], Proposition 4.1) . Let V and W be varieties in B d . Let F be anautomorphism of B d that maps W onto V . Then f f ◦ F is a unitarily implementedcompletely isometric isomorphism of M V onto M W ; i.e. M f ◦ F = U M f U ∗ . The unitary U ∗ is the linear extension of the map U ∗ k w = c w k F ( w ) for w ∈ W, where c w = (1 − k F − (0) k ) k F − (0) ( w ) . The proof in [16] relies on Theorem 9.2 of [15], which uses Voiculescu’s construction ofautomorphisms of the Cuntz algebra. For the convenience of the reader we give here aslightly different proof.
Proof.
Let F be such an automorphism, and set α = F − (0). We first show that the lineartransformation defined on reproducing kernels by k w c w k F ( w ) extends to be a boundedoperator of norm 1. First note that c − w = (1 − k α k ) − (1 − h w, α i ), so c − w (as a function of w ) is a multiplier. The transformation formula for ball automorphisms [28, Theorem 2.2.5],shows that k F ( w ) ( F ( z )) = c − w c − z k w ( z ) for w, z ∈ B d . ow, h c w k F ( w ) , c z k F ( z ) i = c w c z k F ( w ) ( F ( z )) = k w ( z ) = h k w , k z i . Thus, the linear transformation k w c w k F ( w ) extends to an isometry. We denote by U itsadjoint. A short calculation shows that U h = (1 − k α k ) k α · ( h ◦ F ) for h ∈ H d . We have already noted that U ∗ is an isometry, and since its range is evidently dense weconclude that U is a unitary.Finally, we show that conjugation by U implements the isomorphism between M V and M W given by composition with F . For f ∈ M V and w ∈ W , U M ∗ f U ∗ k w = U M ∗ f c w k F ( w ) = f ( F ( w )) U c w k F ( w ) = ( f ◦ F )( w ) k w . Therefore, M f ◦ F is a multiplier on F W and M f ◦ F = U M f U ∗ . (cid:3) Before discussing the converse direction, we recall a few definitions on affine sets. The affine span (or affine hull ) of a set S ⊆ C d is the set aff( S ) := λ + span( S − λ ) for λ ∈ S .This is independent of the choice of λ . An affine set is a is a set A with A = aff( A ). Thedimension dim( A ) of an affine set A is the dimension of the subspace A − λ for λ ∈ A ,and the codimension codim( A ) is the dimension of the quotient space C d /A − λ for λ ∈ A .Both definitions, again, are independent of the choice of λ . By the affine dimension (resp. codimension ) of a subset S ⊆ C d we mean the dimension (resp. codimension) of aff( S ).Furthermore, we use the term affine subset of B d for any intersection A ∩ B d , where A isaffine in C d . By [28, Proposition 2.4.2], automorphisms of the ball map affine subsets of theball to affine subsets of the ball. Therefore, we obtain the following lemma. Lemma 4.2.
Let V and W be varieties in B d and let F be an automorphism of B d thatmaps W onto V . Then, F (aff( V ) ∩ B d ) = aff( W ) ∩ B d . In particular, aff( V ) and aff( W ) have the same dimension and the same codimension.Proof. The first argument is clear, so it suffices to show that an automorphism of the ballpreserves dimensions and codimensions of affine subsets. Indeed, as F is a diffeomorphism,its differential at any point of the ball is an invertible linear transformation. Let A be anaffine subset of B d and let λ ∈ A . Let T λ B d ∼ = C d be the tangent space of B d at λ , and let T λ A ∼ = A − λ be the tangent space of A at λ . As A is a submanifold of B d , we may thinkof T λ A as a subspace of T λ B d . Hence, the invertible linear transformation dF λ maps thesubspace T λ A onto T F ( λ ) F ( A ). We conclude that T λ A and T F ( λ ) F ( A ) must have the samedimension and the same codimension. (cid:3) Proposition 4.1 and Lemma 4.2 imply, in particular, that if there is an automorphism ofthe ball which sends W onto V , then V and W must have the same affine codimension,and this automorphism gives rise to a completely isometric isomorphism of M V onto M W (by precomposing this automorphism). The converse is also true: any completely isometricisomorphism of M V onto M W , for V and W varieties in the ball having the same affinecodimension, arises in this way. Proposition 4.3.
Let V and W be varieties in B d , with the same affine codimension or with d < ∞ . Then every completely isometric isomorphism ϕ : M V → M W arises as composition ϕ ( f ) = f ◦ F where F is an automorphism of B d mapping W onto V . roof. Recall that Proposition 3.1 assures the existence of a holomorphic map F : B d → B d representing ϕ ∗ (cid:12)(cid:12) W . A deep result of Kennedy and Yang [22, Corollary 6.4] asserts that M V and M W have strongly unique preduals. It then follows that every isometric isomorphismbetween these algebras, is also a weak- ∗ homeomorphism. Thus, by Corollary 3.3, F ( W ) ⊆ V and ϕ ( f ) = f ◦ F . (We note that if d < ∞ , then we may argue differently: firstone shows using the injectivity of ϕ that F ( B d ) ⊆ B d , and then one uses the argument V = π ( M ( M V )) ∩ B d of Proposition 2.6 to obtain that ϕ preserves weak- ∗ continuouscharacters.) Similarly, ϕ − : M W → M V gives rise to a holomorphic map G : B d → B d suchthat G ( V ) ⊆ W and ϕ − ( g ) = g ◦ G . It is clear that F ◦ G | V = id | V and G ◦ F | W = id | W ,and so F ( W ) = V .By Proposition 4.1 and Lemma 4.2, we may assume that V and W both contain 0, and that F (0) = 0. Some technical several-complex-variables arguments, which we will not presenthere, now show that F | span W ∩ B d is an isometric linear transformation that maps span W ∩ B d onto span V ∩ B d (see [16, Lemma 4.4]). In particular, span W and span V have the samedimension. Since they also have the same codimension, we may extend the definition of F | span W ∩ B d to a unitary map on C d . This yields the desired automorphism. (cid:3) Remark . The original statement of Proposition 4.3 (which appears in [16, Theorem4.5]) does not include the requirement that V and W have the same affine codimension.Example 4.5 below shows that this requirement is indeed necessary (for the case d = ∞ ).Nonetheless, it is clear that up to an isometric embedding of the original infinite ball intoa “larger” one, the original statement does hold. For example, if we replace V and W withtheir images under the embedding U : ( z , z , . . . ) ( z , , z , , . . . ), then both V and W have an infinite affine codimension, and it is now true that M V and M W are completelyisometrically isomorphic if and only if V and W are conformally equivalent. Example 4.5.
Let V = B ∞ and W = { ( z , z , z , . . . ) ∈ B ∞ : z = 0 } . Let F : W → V bedefined by F (0 , z , z , . . . ) = ( z , z , . . . ) . Then F is a biholomorphism which cannot be extended to an automorphism of B ∞ . Let ϕ : M V → M W be defined by ϕ ( f ) = f ◦ F . Then ϕ is a completely isometric isomorphismof M V onto M W , which does not arise as a precomposition with an automorphism of theball. The reason is of course that V has an affine codimension 0 while W has an affinecodimension 1.Combining Propositions 4.1 and 4.3 yields the following result. Theorem 4.6 ([16], Theorem 4.5) . Let V and W be varieties in B d , with the same affinecodimension or with d < ∞ . Then M V is completely isometrically isomorphic to M W ifand only if there exists an automorphism F of B d such that F ( W ) = V . In fact, under theseassumptions, every completely isometric isomorphism ϕ : M V → M W arises as composition ϕ ( f ) = f ◦ F where F is such an automorphism. In this case, ϕ is unitarily implemented bythe unitary sending the kernel function k w ∈ F W to a scalar multiple of the kernel function k F ( w ) ∈ F V .If V and W are not assumed to have the same affine codimension, then every completelyisometric isomorphism ϕ : M V → M W arises as composition with U ∗ ◦ F ◦ U , where F ∈ Aut ( B d ) and U is the isometry from Remark 4.4, and is unitarily implemented. .2. Isometric isomorphisms.
By Theorem 4.6 the conformal geometry of V is completelyencoded by the operator algebraic structure M V (and vice versa). It is natural to ask whetherthe Banach algebraic structure M V also encodes some geometrical aspect of V . It turns outthat within the family of irreducible complete Pick algebras, every isometric isomorphism of M V and M W is actually a completely isometric isomorphism, and the results of the previoussection apply. Lemma 4.7.
Let V and W be varieties in B d , and suppose that ϕ : M V → M W is an iso-metric isomorphism. Then ϕ ∗ maps W onto V and preserves the pseudohyperbolic distance.Proof. The first assertion was obtained in the proof of Proposition 4.3. It then follows that ϕ is implemented by composition with ϕ ∗ (cid:12)(cid:12) W . Using this together with Proposition 2.7 (b),one obtains the second assertion. (cid:3) The following theorem appears in [16, Proposition 5.9] with the additional assumptionthat d < ∞ . Here we remove this restriction. Theorem 4.8 ([16], Proposition 5.9) . Let V and W be varieties in B d . Then every isometricisomorphism of M V onto M W is completely isometric, and thus is unitarily implemented.Proof. Without the loss of generality we may assume that V and W have the same affinecodimension by embedding the original ball in a larger one, if needed (see Remark 4.4). Let ϕ be an isometric isomorphism of M V onto M W . By Lemma 4.7, ϕ ∗ maps W onto V andpreserves the pseudohyperbolic distance. Let F = F ϕ .As above, we may assume that 0 belongs to both V and W , and that F (0) = 0. Let w , w , . . . ∈ W be a sequnce spanning a dense subset of span W . For every p ≥ v p = F ( w p ) = ϕ ∗ ( w p ). Put r p := k w p k = d ph ( w p , k v p k = d ph ( v p ,
0) = r p . For every p let h p ( z ) := h z, v p r p i . This is a continuous linear functional (restricted to V ), and thus liesin M V . Furthermore, since ( Z , Z , . . . , Z d ) is a row contraction it follows that k h p k M V ≤ k ϕ ( h p ) k M W ≤ w be an arbitrary point in W , set v = F ( w ) ∈ V , and fix p ≥
1. Since, ϕ ( h p )is a multiplier of norm at most 1 which satisfies ϕ ( h p )(0) = 0, ϕ ( h p )( w p ) = h p ( v p ) and ϕ ( h p )( w ) = h p ( v ), we have by a standard necessary condition for interpolation [2, Theorem5.2] that −h v,v p i −h w p ,w i −h v,v p i −h w,w p i −|h v,v p /r p i| −h w,w i ≥ . Examining the determinant we find that −h v,v p i −h w,w p i = 1. Therefore, h v, v p i = h w, w p i for all p. In particular, we obtain h v i , v j i = h w i , w j i for all i, j . Therefore, there is a unitary operator U : span W → span V such that U w i = v i for all 1 ≤ i ≤ k . Since codim(span W ) =codim(span V ), it can be extended to a unitary operator U on C d . From here one shows that F agrees with the unitary U , and hence ϕ is implemented by an automorphism of the ball.Thus, by Proposition 4.1, ϕ is completely isometric and is unitarily implemented. (cid:3) . Algebraic isomorphisms
We now turn to study the algebraic isomorphism problem. It is remarkable that, underreasonable assumptions, purely algebraic isomorphism implies multiplier biholomorphism.Throughout this section we will assume that d < ∞ .5.1. Varieties which are unions of finitely many irreducible varieties and a discretevariety.
Let V be a variety in the ball. We say that V is irreducible if for any regular point λ ∈ V , the intersection of zero sets of all multipliers vanishing on a small neighborhood V ∩ B ǫ ( λ ) is exactly V . We say that V is discrete if it has no accumulation points in B d . Wewill see that if V and W are two varieties in B d ( d < ∞ ), which are the union of finitely manyirreducible varieties and a discrete variety, then whenever M V and M W are algebraicallyisomorphic, V and W are multiplier biholomorphic. Remark . The definition of irreducibility given in the previous paragraph is not to beconfused with the classical notion of irreducibility (that is, that there is not non-trivial de-composition of the variety into subvarieties). Nonetheless, whenever a variety V is irreduciblein the classical sense, it is also irreducible in our sense (see e.g. [19, Theorem, H1]).We open this section with two observations. The first is that every homomorphism betweenmultiplier algebras is norm continuous. A general result in the theory of commutative Banachalgebras, says that every homomorphism from a Banach algebra into a commutative semi-simple Banach algebra is norm continuous [9, Proposition 4.2]. As M W is easily seen to besemi-simple, it holds that every homomorphism from M V to M W is norm continuous.The second observation relates to isolated characters of a multiplier algebra. Suppose that ρ is an isolated point in M ( M V ). By Shilov’s idempotent theorem [8, Theorem 5], there isa function 0 = f ∈ M V such that every character except ρ annihilates f . As f = 0, there is λ ∈ V such that f ( λ ) = 0. And so, ρ ∈ π − ( V ). Thus, when d < ∞ any isolated characterof a multiplier algebra is an evaluation. This gives rise to the following proposition. Proposition 5.2 ([16], Lemma 5.2) . Let V and W be varieties in B d , with d < ∞ . Let ϕ : M V → M W be an algebra isomorphism. Suppose that λ is an isolated point in W . Then ϕ ∗ ( ρ λ ) is an evaluation functional at an isolated point in V . From the first observation above, together with Proposition 2.7, we obtain:
Proposition 5.3.
Let V and W be a varieties in B d , with d < ∞ , and let ϕ : M V → M W bea homomorphism. Let U be a connected subset of W . Then ϕ ∗ ( π − ( U )) is either a connectedsubset of π − ( V ) (with respect to the norm topology induced by M ∗ V ) or contained in a singlefiber of the corona M ( M V ) \ π − ( V ) . Proposition 5.4 ([16], Corollary 5.4) . Let V and W be varieties in B d , d < ∞ , and assumethat each one is the union of a discrete variety and a finite union of irreducible varieties.Suppose that ϕ is an algebra isomorphism of M V onto M W . Then ϕ ∗ must map W onto V .Proof. Let us write V = D V ∪ V · · · ∪ V m and W = D W ∪ W ∪ · · · ∪ W n , where D V and D W are the discrete parts of V and W , and V i , W j are all irreducible varieties of dimensionat least 1. By Proposition 5.2 ϕ ∗ maps D W onto D V .First let us show that if W , say, is not mapped entirely into V then it is mapped intoa single fiber of the corona M ( M V ) \ π − ( V ). Suppose that λ is some regular point of W apped to a fiber of the corona. Without loss of generality, we may assume it is the fiberover (1 , , ..., λ in W is mapped into the same fiber,by the previous proposition. If there exists another point µ ∈ W which is mapped into V or into another fiber in the corona, then by the previous proposition, the whole connectedcomponent of µ is mapped into V or into the other fiber. The function h = ϕ ( Z | V ) − | W vanishes on the component of λ , but does not vanish on the component containing µ . Thiscontradicts the fact that W is irreducible.Thus, to show that W is mapped into V we must rule out the possibility that it ismapped into a single fiber of the corona. Fix λ ∈ W \ S ni =2 W i . For each 2 ≤ i ≤ n ,there is a multiplier h i ∈ M d vanishing on W i and satisfying h i ( λ ) = 0. Moreover, since D W is a variety, there is a multiplier k vanishing on D W and satisfying k ( λ ) = 0. Hence, h := k Q ni =2 h i belongs to M W and vanishes on D W ∪ S ni =2 W i but not on W . Therefore ϕ − ( h ) is a non-zero element of M V .Now suppose that ϕ ∗ ( W ) is contained in a fiber over a point in ∂ B d , say (1 , , . . . , Z − | V is never zero, we see that ( Z − | V ϕ − ( h ) is not the zero function. However,( Z − | V ϕ − ( h ) vanishes on ϕ ∗ ( W ). Therefore, ϕ (( Z − | V ϕ − ( h )) vanishes on W andon D W ∪ S ni =2 W i , contradicting the injectivity of ϕ . We deduced that W is mapped into V . Replacing the roles of V and W shows that ϕ ∗ must map W onto V . (cid:3) From Proposition and 5.4 and Corollary 3.3 we obtain the following.
Theorem 5.5 ([16], Theorem 5.6) . Let V and W be varieties in B d , with d < ∞ , which areeach a union of finitely many irreducible varieties and a discrete variety. Let ϕ be an algebraisomorphism of M V onto M W . Then there exist holomorphic maps F and G from B d into C d with coefficients in M d such that(a) F | W = ϕ ∗ | W and G | V = ( ϕ − ) ∗ | V ,(b) G ◦ F | W = id W and F ◦ G | V = id V ,(c) ϕ ( f ) = f ◦ F for f ∈ M V , and(d) ϕ − ( g ) = g ◦ G for g ∈ M W . Theorem 5.5 shows in particular that every automorphism of M d = M B d is implementedas composition by a biholomorphic map of B d onto itself, i.e. a conformal automorphismof B d . Proposition 4.1 shows that these automorphisms are unitarily implemented (hence,completely isometric). Thus, we obtain the following corollary. Corollary 5.6 ([16], Corollary 5.8) . Every algebraic automorphism of M d for d finite iscompletely isometric, and is unitarily implemented. The converse of Theorem 5.5 does not hold.
Example 5.7.
Let V = (cid:26) − n : n ∈ N (cid:27) and W = n − e − n : n ∈ N o . Since they both satisfy the Blaschke condition, they are analytic varieties in D (recall that { a n ∈ C : n ∈ N } satisfies the Blaschke condition if P (1 − | a n | ) < ∞ ). Let B ( z ) be theBlaschke product with simple zeros at points in W . Define h ( z ) = 1 − e z − and g ( z ) = log(1 − z ) + 1log(1 − z ) (cid:18) − B ( z ) B (0) (cid:19) . hen g, h ∈ H ∞ = M D and they satisfy h ◦ g | W = id W and g ◦ h | V = id V . However, by the corollary in [21, p. 204], W is an interpolating sequence and V is not. Thisimplies that M W is algebraically isomorphic to ℓ ∞ while V is not (see [16, Theorem 6.3]).Thus, M V and M W cannot be isomorphic.5.2. Homogeneous varieties.
Let V be a variety in the ball. We say that V is a homoge-neous variety if it is the common vanishing locus of homogeneous polynomials.We wish to apply Theorem 5.5 to homogeneous varieties in B d , d < ∞ . It is well knownthat every algebraic variety can be decomposed into a finite union of irreducible varieties, butcaution is required, since the well known result is concerned with irreducibility in anothersense than the one we used in Section 5.1. However, one may show that a homogeneousalgebraic variety which is irreducible (in the sense of algebraic varieties) is also irreduciblein our sense. Proposition 5.8.
Every homogeneous variety in the ball is a union of finitely many irre-ducible varieties.Proof.
Let V be a homogeneous variety and let V = V ∪ · · · ∪ V n be its decompositioninto algebraic irreducible homogeneous varieties (in the sense of algebraic varieties). We willshow that every V i is irreducible in our sense. By [19, Theorem E19, Corollary E20], oncewe remove the set of singular points S ( V i ), the connected components of V i \ S ( V i ) is suchthat their closures are varieties. Since S ( V i ) is a homogeneous variety, these connected com-ponents are invariant under nonzero scalar multiplication so their closures are homogeneousvarieties. Thus, if there was more than one connected component we would obtain an alge-braic decomposition of the variety V i , so V i \ S ( V i ) is connected. By the identity principle[19, Theorem, H1], the V i ’s are irreducible in our sense. (cid:3) Thus we obtain the following theorem (the original proof of this theorem was somewhatdifferent — see [16, Section 11]).
Theorem 5.9 ([15], Theorem 11.7(2)) . Let V and W be homogeneous varieties in B d , d < ∞ .If M V and M W are algebraically isomorphic, then there is a multiplier biholomorphismmapping W onto V . The rest of this subsection is devoted towards the converse direction. Remarkably, astronger result than the converse holds: it turns out that the existence of a biholomorphismfrom W onto V implies that the algebras are isomorphic.We will start by showing that whenever a homogeneous variety W ⊆ B d is the imageof homogeneous variety V ⊆ B d under a biholomorphism, then it is also the image of V under an invertible linear transformation. To see this, we first need to present the notionof the singular nucleus of a homogeneous variety. Lemma 4.5 of [15] and its proof say thata homogeneous variety V in C d is either a linear subspace, or has singular points, and thatwhenever it is not a linear subspace, the set of singular points S ( V ) (also known as the singular locus ) of V is a homogeneous variety. Since the dimension of S ( V ) must be strictlyless than the dimension of V , there exists a smallest integer n such that S ( . . . ( S ( S ( V ))) . . . ) n times) is empty. The set N ( V ) := S ( . . . ( S ( S ( V ))) . . . ) | {z } n − is called the singular nucleus of V . By the above discussion, it is a subspace of C d . By basiccomplex differential geometry, a biholomorphism of V onto W must map N ( V ) onto N ( W ).The following lemma — which seems to be of independent interest — was used implicitlyin [15], but in fact does not appear anywhere in the literature. The proof follows closely theproof of [15, Proposition 4.7]. Lemma 5.10.
Let V and W be two biholomorphically equivalent homogeneous varieties in B d . Then there exists a biholomorphism F of V onto W that maps to .Proof. Let G be a biholomorphism of V onto W . If N ( V ) = N ( W ) = { } , then G (0) = 0,and we are done. Otherwise, N ( V ) ∩ B d and N ( W ) ∩ B d are both complex balls of thesame dimension, say d ′ ≤ d . As G takes N ( V ) ∩ B d onto N ( W ) ∩ B d , we may think of G as an automorphism of B d ′ . We can find two discs D ⊆ N ( V ) and D ⊆ N ( W ) such that G ( D ) = D (see [15, Lemma 4.6]). Define O (0; V ) := { z ∈ D : z = F (0) for some automorphism F of V } and O (0; V, W ) := n z ∈ D : z = F (0) for some biholomorphism F of V onto W o . Since homogeneous varieties are invariant under multiplication by complex numbers, it iseasy to check that these sets are circular, that is, for every µ ∈ O (0; V ) and ν ∈ O (0; V, W ),it holds that C µ,D := { z ∈ D : | z | = | µ |} ⊆ O (0; V ) and C ν,D := { z ∈ D : | z | = | ν |} ⊆O (0; V, W ).Now, as G (0) belongs to O (0; V, W ), we obtain that C := C G (0) ,D ⊆ O (0; V, W ). There-fore, the circle G − ( C ) is a subset of O (0; V ). As O (0; V ) is circular, every point of theinterior of the circle G − ( C ) is a subset of O (0; V ). Thus, the interior of the circle C mustbe a subset of O (0; V, W ). We conclude that 0 ∈ O (0;
V, W ). (cid:3) Proposition 5.11.
Let V and W be two biholomorphically equivalent homogeneous varietiesin B d . Then there is a linear map on C d which maps V onto W .Sketch of proof. By Lemma 5.10, V and W are biholomorphically equivalent via a 0 pre-serving biholomorphism; i.e. there exist two holomorphic maps F and G from B d into C d such that G ◦ F | V = id V and F ◦ G | W = id W . Cartan’s uniqueness theorem says that ifthere exists a 0 preserving biholomorphism between two bounded circular regions , then itmust be a restriction of a linear transformation; see [28, Theorem 2.1.3]. Now, V and W areindeed circular (since they are homogeneous varieties) and bounded, but do not have to be“regions” (their interior might be empty). Nevertheless, it turns out that adapting the proofof Cartan’s uniqueness theorem to the setting of varieties, rather than regions, does work(see [15, Theorem 7.4]). Thus, there exists a linear map A : C d → C d which agrees with F on V . (cid:3) Up to now we have seen that if M V and M W are isomorphic, then V and W are biholo-morphically equivalent; and we have seen that if V and W are biholomorphically equivalent,then there is a linear map sending V onto W , and it is not hard to see that this map can be aken to be invertible. To close the circle, one needs to show that whenever there is an in-vertible linear transformation mapping a homogeneous variety W ⊆ B d onto a homogeneousvariety V ⊆ B d , we have that M V and M W are similar. In [15, Section 7], this statementwas proved for a class of varieties which satisfy some extra assumptions (e.g., irreduciblevarieties, union of two irreducible components, hypersurfaces, and for the case d ≤ Lemma 5.12 ([20]) . Let V and W be homogeneous varieties in B d , d < ∞ , If there is a lineartransformation A : C d → C d that maps W bijectively onto V , then the map C A ∗ : F W → F V ,given by C A ∗ k λ = k Aλ for λ ∈ W, is a bounded linear transformation from F W into F V . We omit the proof of Lemma 5.12. The crucial step in its proof is to show that whenever V , . . . , V n are subspaces of C d , the algebraic sum of the associated Fock spaces F ( V ) + · · · + F ( V n ) ⊆ F ( C d )is closed. In fact, most of [20] is devoted for proving this crucial step. Theorem 5.13.
Let V and W be homogeneous varieties in B d , d < ∞ . If there is aninvertible linear transformation A ∈ GL d ( C ) that maps W onto V , then the map ϕ : M V →M W , given by ϕ ( f ) = f ◦ A for f ∈ M V , is a completely bounded isomorphism, and when regarding M V and M W as operator algebrasacting on F V and F W , respectively, ϕ is given by ϕ ( M f ) = ( C A ∗ ) ∗ M f ( C − A ∗ ) ∗ for f ∈ M V . Thus, M V and M W are similar.Proof. By Lemma 5.12, both C A ∗ and C ( A − ) ∗ are bounded, and it is clear that C ( A − ) ∗ =( C A ∗ ) − . A calculation shows that M f ◦ A = ( C A ∗ ) ∗ M f ( C − A ∗ ) ∗ . (cid:3) We sum up the results of Theorems 5.11, 5.9 and 5.13 as follows.
Theorem 5.14 ([15, 20]) . Let V and W be homogeneous varieties in B d with d < ∞ . Thenthe following are equivalent:(a) M V and M W are similar.(b) M V and M W are algebraically isomorphic.(c) V and W are biholomorphically equivalent.(d) There is an invertible linear map on C d which maps W onto V . If a linear map A maps V onto W this means that A is length preserving on the homo-geneous varieties ˜ V and ˜ W , where ˜ V is the homogeneous variety such that V = ˜ V ∩ B d ,and likewise ˜ W . This does not mean that A is isometric (as Example 5.16 shows), but it istrue that A is isometric on the span of every irreducible component of W [15, Proposition7.6]. Combining this fact with Proposition 4.1 we obtain the following result, which sharpensCorollary 5.6 substantially. heorem 5.15 ([15], Theorem 8.7) . Let V and W be homogeneous varieties in B d , d < ∞ , such that W is either irreducible or a non-linear hypersurface. If M V and M W areisomorphic, then they are unitarily equivalent. Example 5.16.
Suppose that V and W are each given as the union of two (complex)lines. There is always a linear map mapping W onto V that is length preserving on W ,thus M V and M W are algebraically isomorphic. On the other hand, these algebras will beisometrically isomorphic if and only if the angle between the two lines is the same in eachvariety.The case of three lines is also illuminating: it reveals how the algebra Alg(1 , Z ) andits wot -closure, the algebra M V , each encodes different geometrical information. Indeed,suppose that V = span { v }∪ span { v }∪ span { v } and W = span { w }∪ span { w }∪ span { w } ,where v i , w j are all unit vectors in C spanning distinct lines. There always exists a bijectivelinear map from W onto V : indeed, define A : w a v , w a v , and choose a , a so that w = b w + b w is mapped to v . One only has to choose a , a such that a b v + a b v = v . It follows that the algebras Alg(1 , Z (cid:12)(cid:12) V ) and Alg(1 , Z (cid:12)(cid:12) W ) areisomorphic (the latter two algebras are easily seen to be isomorphic to the coordinate ringsof the varieties).On the other hand, if we require the linear map A to be length preserving on W , then | a | = | a | = 1. If v = c v + c v , then for such a map to exist we will need a b = c and a b = c . This is possible if and only if | b | = | c | and | b | = | c | . Thus the algebras M V and M W in this setup are rarely isomorphic.5.3. Finite Riemann surfaces.
In seeking a the converse of Theorem 5.5, it is natural torestrict attention to certain well behaved classes of varieties. In the previous subsection itwas shown that the converse of Theorem 5.5 holds within the class of homogeneous varieties.In this subsection we concentrate on generic one-dimensional subvarieites of B d , d < ∞ .A connected finite Riemann surface Σ is a connected open proper subset of some compactRiemann surface such that the boundary ∂ Σ is also the boundary of the closure and isthe union of finitely many disjoint simple closed analytic curves. A general finite Riemannsurface is a finite disjoint union of connected ones.Let Σ be a connected finite Riemann surface and let a ∈ Σ be some base-point. Let ω bethe harmonic measure with respect to a , i.e. the measure on ∂ Σ with the property that u ( a ) = Z ∂ Σ u ( ζ ) dω ( ζ )for every function u that is harmonic on Σ and continuous on Σ. We denote by H (Σ) theclosure in L ( ω ) of the space A (Σ) := Hol(Σ) ∩ C (Σ). In case that Σ is not connected we let H (Σ) be the direct sum of the H spaces of the connected components.The multiplier algebra of H (Σ) is H ∞ (Σ), the bounded analytic functions on Σ. Notethat the norm in H (Σ) depends on the choice of base-point a , but the norm in H ∞ (Σ) doesnot, as it is the supremum of the modulus on Σ; for more details see [3].We say that a proper holomorphic map G from a finite Riemann surface Σ into a boundedopen set U ⊆ C d is a holomap if there is a finite subset Λ of Σ with the property that G is on-singular and injective on Σ \ Λ. We say that G is transversal at the boundary if h DG ( ζ ) , G ( ζ ) i 6 = 0 for all ζ ∈ ∂ Σ . The first result on this problem [4] showed that if G : D → W is a biholomorphic unramified C -map that is transversal at the boundary, then there is an isomorphism of multiplieralgebras from M D = H ∞ ( D ) to M W (the assumptions appearing in [4] are slightly weaker— they only required C and did not ask for the map to be unramified — but it seems thatone needs a little more; see [5, p. 1132]). This was extended to planar domains in [5, Section2.3.6], and to finite Riemann surfaces in [23]. Later, it was proved that a holomorphic C embedding of a finite Riemann surface is automatically transversal at the boundary [10,Theorem 3.3]. Combining this automatic transversaility result with [23, Theorem 4.2] weobtain: Theorem 5.17 ([4, 5, 10, 23]) . Let Σ be a finite Riemann surface and W a variety in B d .Let G : Σ → B d be a holomap that maps Σ onto W , is C up to ∂ Σ , and is one-to-one on ∂ Σ . Then the map α : h h ◦ G for h ∈ F W is an isomorphism from F W onto H G (Σ) := H (Σ) ∩ { h ◦ G : h ∈ Hol( W ) } . Consequently,the map f f ◦ G implements an isomorphism of M W onto H ∞ G (Σ) := H ∞ (Σ) ∩ { h ◦ G : h ∈ Hol( W ) } . The main idea of the proof goes back to [4]. One first shows that α , given by the formula h h ◦ G , is a well defined bounded and invertible map from F W onto H G (Σ), by computing α ∗ and αα ∗ , and showing that αα ∗ is an injective Fredholm operator. The key trick is tobreak up αα ∗ as the sum of a Toeplitz operator and a Hilbert-Schmidt operator (see [23,Theorem 4.2] for details). Being positive and Fredholm, injectivity implies invertibility, andthe first claim in the theorem follows. A straightforward computation then shows that theasserted isomorphism between M W and H ∞ G (Σ) is the similarity induced by α . Corollary 5.18.
Let Σ be a finite Riemann surface, and let V and W be varieties in B d such that W = G (Σ) , where G : Σ → B d is a holomap which is C on Σ and is one-to-oneon ∂ Σ . Let F : W → V be a biholomorphism that extends to be C and one-to-one on W .Then the map ϕ : M V → M W , given by ϕ ( f ) = f ◦ F for f ∈ M V , is an isomorphism. As an application of the above results, we give the following theorem on extension ofbounded holomorphic maps from a one dimensional subvariety of the ball to the entireball (under rather general assumptions). Such an extension theorem is difficult to proveusing complex-analytic techniques, and it is pleasing to obtain it from operator theoreticconsiderations.
Corollary 5.19 ([4] and [23], Corollary 4.12) . Let W be as in Theorem 5.17. Then M W = H ∞ ( W ) , and the norms are equivalent. Consequently, every h ∈ H ∞ ( W ) extends to amultiplier in M d , and in particular to a bounded holomorphic function on B d . Moreover,there exists a constant C such that for all h ∈ H ∞ ( W ) , there is an ˜ h ∈ M d such that ˜ h (cid:12)(cid:12) W = h and k ˜ h k ∞ ≤ k ˜ h k M d ≤ C k h k ∞ . .4. A class of counter-examples.
In the last two subsections we saw classes of varieties,for which (well behaved) biholomorphism of the varieties implies isomorphism of the mul-tiplier algebras. We now turn to exhibiting a class of examples that show that, in general,biholomorphism of the varieties does not imply that the multiplier algebras are isomorphic.In particular, these examples show that biholomorphic varieties need not be multiplier bi-holomorphic.
Proposition 5.20.
Suppose that G : D → B d is a proper injective holomorphic map whichextends to a differentiable map on D ∪ {− , } such that the extension, also denoted by G ,satisfies G (1) = G ( − . If V = G ( D ) is a variety, then G −
6∈ M V . In particular, theembedding M V → M D = H ∞ , f f ◦ G is not surjective. One way to prove this proposition is to observe that such a map G can not be bi-Lipschitzwith respect to the pseudohyperbolic metric, and then invoke Corollary 3.5 (see [10, Remark6.3] for details). For an alternative proof, we refer the reader to [10, Theorem 5.1]. Example 5.21.
Fix r ∈ (0 , b ( z ) = z − r − rz . Note that b (1) = 1 and b ( −
1) = −
1. Define G ( z ) = 1 √ (cid:0) z , b ( z ) (cid:1) . It is not hard to verify that this map is a biholomorphism satisfying the hypotheses ofProposition 5.20. Therefore, M V ( H ∞ ( V ), and G − is not a multiplier. By Corollary 6.4below we obtain that M V is not isomorphic to M D = H ∞ .6. Embedded discs in B ∞ Some general observations.
In this section we will examine multiplier algebras M V where V = G ( D ) ⊆ B d is a biholomorphic image of a disc via a biholomorphism G : D → B d .The case that interests us most is d = ∞ . Theorem 6.1 ([10], Theorem 2.5) . Let V and W be two varieties in B d , biholomorphic toa disc via the maps G V and G W , respectively. Furthermore, assume that(a) for every λ ∈ V , the fiber π − { λ } is the singleton { ρ λ } , and(b) π ( M ( M V )) ∩ B d = V .If ϕ : M V → M W is an algebra isomorphism, then F = F ϕ | W is a multiplier biholomorphism F : W → V , such that ϕ ( f ) = f ◦ F for all f ∈ M V . Here F = F ϕ is the function provided by Proposition 3.1. By saying that F is a multiplierbiholomorphism we mean that (i) F = ( F , F , . . . ) where every F i ∈ M W , i.e., is a multiplier,and (ii) F is holomorphic on W , in the sense that for every λ ∈ W there is a ball B ǫ ( λ )and a holomorphic function ˜ F : B ǫ ( λ ) → C d such that F (cid:12)(cid:12) B ǫ ( λ ) ∩ W = ˜ F (cid:12)(cid:12) B ǫ ( λ ) ∩ W . We requireslightly different terminology (compared to Section 3) because we are dealing with d = ∞ ,and we are not making any complete boundedness assumptions (see Remark 3.2). For moredetails about holomorphic maps in this setting of discs embedded in B ∞ see [10, Section 2]. roof. We assume that d = ∞ . There are two issues here: we need to prove that F isa biholomorphism, and that F ( W ) = V in the isomorphic case. For the first issue, let α = ( α i ) ∞ i =1 ∈ ℓ . Then h F ◦ G W ( z ) , α i = ∞ X i =1 α i h i ( z ) , where h i ( z ) := F i ◦ G W ( z ). As characters are completely contractive, we have ∞ X i =1 | h i ( z ) | = k F ( G W ( z )) k = k ρ G W ( z ) ( ϕ ( Z | W )) k ≤ k Z | W k = 1 . Thus, P ∞ i =1 α i h i converges uniformly on W since by the Cauchy-Schwartz inequality, ∞ X n = N | α n h n ( z ) | ≤ ∞ X n = N | α n | ! N →∞ −−−→ . Therefore, h F ◦ G W ( · ) , α i is holomorphic for all α , and it follows that F is holomorphic (see[10, Section 2]).We now show that the injectivity of ϕ implies that F is not constant, and that this implies F ( W ) ⊆ B ∞ . Suppose that F is the constant function λ ( λ ∈ B d ). Then for every i we have ϕ ( λ i − Z i | V ) = λ i − F i = 0. By the injectivity of ϕ , Z i | V = λ i , which is impossible as V isnot a singleton. Thus, F is not constant. If µ = F ( λ ) lies in ∂ B ∞ for some λ ∈ W , then h F ◦ G W ( · ) , µ i is a holomorphic function into D , which is equal to 1 at λ . The maximummodulus principle would then imply that this function is constant, so this cannot happen.In view of the previous paragraph, F ( W ) ⊆ B ∞ . Since for every λ ∈ W , ϕ ∗ ( ρ λ ) ∈ π − { F ( λ ) } ⊆ π − B ∞ , by the assumptions (a) and (b), we conclude that F maps W into V ,and therefore (by Corollary 3.3) that ϕ ( f ) = f ◦ F . In particular, ϕ is weak- ∗ continuous, andso (as ϕ is an isomorphism) ϕ − is weak- ∗ continuous too. Thus, both ϕ ∗ and ( ϕ − ) ∗ mappoint evaluations to point evaluations. We conclude that F is a biholomorphism, mapping W onto V . (cid:3) Remark . We do not know when precisely conditions (a) and (b) in the above theoremhold. We do not have an example in which they fail. We do know that if a variety V in B ∞ is the intersection of zero sets of a family of polynomials (or more generally, elements in M ∞ that are norm limits of polynomials) then (b) holds (see [10, Proposition 2.8]).By a familiar result [21, p. 143] the automorphisms of H ∞ are the maps C θ ( h ) := h ◦ θ forsome M¨obius map θ (i.e. θ ( z ) = λ (cid:0) z − a − az (cid:1) for a ∈ D , and λ ∈ ∂ D ). If G is a biholomorphicmap of the disc onto a variety V in B d , then one can transfer the M¨obius maps to conformalautomorphisms of V by sending θ to G ◦ θ ◦ G − . Since this can be reversed, these areprecisely the conformal automorphisms of V . We say that M V is automorphism invariant if composition with all these conformal maps yields automorphisms of M V . Proposition 6.3.
Let V and W be two varieties in B d , biholomorphic to a disc via the maps G V and G W , respectively. Assume that V satisfies the conditions (a) and (b) of Theorem6.1. Let ϕ : M V → M W be an algebra isomorphism. Then there is a M¨obius map θ suchthat the diagram V M W H ∞ H ∞ ϕC G V C G W C θ commutes. The proof follows by Theorem 6.1 and the above discussion. We omit the details.Suppose that the automorphism θ can be chosen to be the identity, or equivalently, that C F , where F = G V ◦ G − W , is an isomorphism of M V onto M W . Then we will say that M V and M W are isomorphic via the natural map . Corollary 6.4.
Let V and W be two varieties in B d , biholomorphic to a disc via the maps G V and G W , respectively. Assume that V satisfies the conditions (a) and (b) of Theorem6.1. If M V or M W is automorphism invariant, then M V and M W are isomorphic if andonly if they are isomorphic via the natural map C F , where F = G V ◦ G − W . In particular, if M V is isomorphic to H ∞ , then C G V implements the isomorphism. A special class of embeddings.
We now consider a class of embedded discs in B ∞ .The principal goal is to exhibit a large class of multiplier biholomorphic discs in B ∞ forwhich we may classify the obtained multiplier algebras. Though this goal is not obtainedfully, we are able to tell when one of these multiplier algebras is isomorphic to H ∞ := H ∞ ( D ).Moreover, we obtain an uncountable family of embeddings of the disc into B ∞ such that allobtained multiplier algebras are mutually non-isomorphic, while the one dimensional varietiesassociated with them are all multiplier biholomorphic to each other, via a biholomorphismthat extends continuously and one-to-one up to the boundary.Let ( b n ) ∞ n =1 be an ℓ -sequence of norm 1 and b = 0. Define G : D → B ∞ by G ( z ) = ( b z, b z , b z , . . . ) for z ∈ D . Then G : D → G ( D ) ⊆ B ∞ is a biholomorphism with inverse b − Z | G ( D ) and these mapsare multipliers. Moreover, G ( D ) is a variety because the conditions on the sequence ( b n )(namely, that it has norm 1 and that b = 0) imply that V := V ( { b n z n − b n z n : n ≥ } ) = G ( D ) . It is easy to see that any two varieties arising this way are multiplier biholomorphic.
Remark . One may also consider embeddings similar to the above but with the differencethat P | b n | <
1, and the results obtained are in some sense analogous to what we describehere, but also contain some surprises. Since the varieties involved are technically differentfrom those on which we concentrate in this survey, we do not elaborate; the reader is referredto [10, Section 8].Define a kernel on D by k G ( z, w ) = 11 − h G ( z ) , G ( w ) i for z, w ∈ D , nd let H G be the Hilbert function space on D with reproducing kernel k G . Then we candefine a linear map U : F V → H G by U h = h ◦ G . Since h k G ( z ) , k G ( w ) i = 11 − h G ( z ) , G ( w ) i = h ( k G ) z , ( k G ) w i for all z, w ∈ D , it follows that U k G ( z ) = ( k G ) z extends to a unitary map of F V onto H G . Hence compositionwith G determines a unitarily implemented completely isometric isomorphism C G : M V → Mult( H G ). Therefore, we can work with multiplier algebras of Hilbert function spaces onthe disc rather than the algebras M V itself.Now write k G ( z, w ) = 11 − P ∞ n =1 | b n | ( zw ) n =: ∞ X n =0 a n ( zw ) n for a suitable sequence ( a n ) ∞ n =0 . A direct computation shows that the sequence ( a n ) satisfiesthe recursion a = 1 and a n = n X k =1 | b k | a n − k for n ≥ . Moreover, 0 < a n ≤ n ∈ N .Due to the special form of the kernel k G , we may compute the multiplier norm of monomialsin H G . Lemma 6.6 ([10], Lemma 7.2) . For every n ∈ N , it holds that k z n k H G ) = k z n k H G = 1 a n . We now compare between two varieties embedded discs V and W as above. We let ( b V n ) ∞ n =1 and ( b Wn ) ∞ n =1 be two ℓ -sequence of norm 1 and b V = 0 = b W , and define G V , G W : D → B ∞ by G V ( z ) = ( b V z, b V z , b V z , . . . ) and G W ( z ) = ( b W z, b W z , b W z , . . . ) . As before, we consider also the sequences ( a Vn ) ∞ n =0 and ( a Wn ) ∞ n =0 which satisfy k G V ( z, w ) = ∞ X n =0 a Vn ( zw ) n and k G W ( z, w ) = ∞ X n =0 a Wn ( zw ) n . Theorem 6.7 ([10], Proposition 7.5) . The algebras M V and M W are isomorphic via thenatural map of composition with G V ◦ G − W if and only if the sequences ( a Vn ) and ( a Wn ) arecomparable, i.e., if and only if there is some c > such that c − | a Vn | ≤ | a Wn | ≤ c | a Vn | for all n . Furthermore, if π − { λ } = { ρ λ } for every λ ∈ W and M W is automorphism invariant,then M V and M W are isomorphic if and only if they are isomorphic via the natural map.Proof. If ( a Vn ) and ( a Wn ) are comparable, then by Lemma 6.6 the norms in H G V and H G W ofthe orthogonal base { z n : n ∈ N } are comparable. Thus, the identity map is an invertiblebounded operator between H G V and H G W . Therefore, Mult( H G V ) = Mult( H G W ), so that M V and M W are isomorphic via the natural map.Conversely, if M V and M W are isomorphic via the natural map then Mult( H G V ) =Mult( H G W ). Therefore the identity map is an isomorphism between these two semisimple anach algebras, so the isomorphism is topological. By Lemma 6.6, the sequences ( a Vn ) and( a Wn ) are comparable.If if π − { λ } = { ρ λ } for every λ ∈ W and M W is automorphism invariant, then byCorollary 6.4, this is equivalent to M V being isomorphic to M W via any isomorphism. (cid:3) Corollary 7.4 of [10] states that if M W is automorphism invariant and sup n ≥ ( a Wn /a Wn − ) < ∞ , then π − { λ } = { ρ λ } for every λ ∈ W . This gives rise to examples in which the secondpart of Theorem 6.7 is meaningful. For example, the following corollary follows by the aboveby setting ( b W , b W , b W , . . . ) = (1 , , , . . . ), and noting that a Wn = 1 for all n ∈ N . Corollary 6.8. M V is isomorphic to H ∞ if and only if the sequence ( a Vn ) is bounded below. In terms of the sequence ( b n ) the result reads as follows. Corollary 6.9.
Let V = G ( D ) where G ( z ) = ( b z, b z , b z , . . . ) , where k ( b n ) k ℓ = 1 and b = 0 . Then M V is isomorphic to H ∞ if and only if ∞ X n =1 n | b n | < ∞ . Proof.
By the Erd˝os-Feller-Pollard theorem (see [17, Chapter XIII, Section 11]) we knowthat lim n →∞ a n = 1 P ∞ n =1 n | b n | , where 1 / ∞ = 0. Hence, ( a n ) is bounded below if and only is the series converges. (cid:3) Example 6.10 ([10], Example 7.9) . For every s ∈ [ − , H s with kernel k s ( z, w ) = ∞ X n =0 ( n + 1) s ( zw ) n for z, w ∈ D . It is shown in [10] that these kernels arise from embeddings as above, and also that theseembeddings satisfy all the conditions of Theorem 6.7. We have that a sn = ( n +1) s in this case,and obviously the sequences (cid:0) ( n + 1) s (cid:1) ∞ n =0 and (cid:0) ( n + 1) s ′ (cid:1) ∞ n =0 are not comparable for s = s ′ .Thus the family of algebras Mult( H s ) is an uncountable family of multiplier algebras of thetype we consider which are pairwise non-isomorphic. Note that all these algebras live onvarieties that are multiplier biholomorphic via a biholomorphism that extends continuouslyto the boundary. 7. Open problems
Though we have accumulated a body of satisfactory results, and although we have arich array of examples and counter examples, the isomorphism problem for irreducible Pickalgebras is far from being solved. We close this survey by reviewing some open problems. .1. Finite unions of irreducible varieties.
Theorem 5.5 implies that in the case where V and W are finite unions of irreducible varieties in B d (for d < ∞ ), we have that if M V and M W are isomorphic then V and W are multiplier biholomorphic. It is not known whether theconverse holds. We did see an example of multiplier biholomorphic varieties which are infinite unions of irreducible varieties but with non-isomorphic multiplier algebras; see Example 5.7.We also saw an example (Example 5.21) of biholomorphic irreducible varieties, with non-isomorphic multiplier algebras; this, however, was not a multiplier biholomorphism. Andso the question, whether a multiplier biholomorphism of varieties which are a finite unionof irreducible ones implies that the multiplier algebras are isomorphic, remains unsolved for d < ∞ (for d = ∞ the answer is no , see Example 6.10).7.2. Maximal ideal spaces of multiplier algebras.
As we remarked in the introduction,in the case d = ∞ there are multiplier algebras M V for which there are points in π − B ∞ ⊆ M ( M V ) which are not point evaluations; similarly, there are also multiplier algebras M V with characters in fibers over points in B ∞ \ V [10, Example 2.4]. Nevertheless, when werestrict attention to “sufficiently nice” varieties, it might be the case that the characters overthe varieties do behave appropriately, in the sense that for every λ ∈ V the fiber π − { λ } isthe singleton { ρ λ } , and π ( M ( M V )) ∩ B ∞ = V . In particular, it will be interesting to obtainsuch a result for the family of discs embedded in B ∞ by G ( z ) = ( b z, b z , . . . ) as in Section6.2. This will amount to obtaining a better understanding of the maximal ideal space of thealgebras Mult( H G ).7.3. The correct equivalence relation.
Theorem 5.5 says (under some assumptions) thatif M V and M W are isomorphic then V and W are multiplier biholomorphic. We have seen acouple of counter examples showing that the converse is not true, but to clarify the nature ofthe obstruction let us point out the following: multiplier biholomorphism is not an equivalencerelation , while, on the other hand, isomorphism is an equivalence relation; see [10, Remark6.7]. This leads to the problem: describe the equivalence relation ∼ = on varieties given by V ∼ = W iff M V is isomorphic to M W in complex geometric terms.7.4. Structure theory.
The central problem dealt with up to now was the isomorphismproblem: when are M V and M W isomorphic (or isometrically isomorphic)? For isometricisomorphisms the problem is completely resolved: the structure of the Banach algebra M V is completely determined by the conformal structure of V . As for algebraic isomorphisms,we know that the biholomorphic structure of V is an invariant of the algebra M V . Thisopens the door for a profusion of delicate questions on how to read the (operator) algebraicinformation from the variety, and vice versa. For example, how is the dimension of V reflectedin M V ? If V is a finite Riemann surface with m handles and n boundary components, whatin the algebraic structure of M V reflects the m handles and the n boundary components?What about algebraic-geometric invariants, such as number of irreducible components ordegree?7.5. Embedding dimension.
A particular question in the flavour of the above broadquestion, is this: given an irreducible complete Pick algebra A , what is the minimal d ∈{ , , . . . , ∞} such that A is isomorphic to M V , with V ⊆ B d ? This question is interesting and the answer is unknown — even for the case of the multiplier algebra of the wellstudied Dirichlet space D (see [6]).7.6. Other algebras. Norm closed algebras of multipliers.
The isomorphism problemmakes sense on many natural algebras, for examples, one may wonder whether, given twovarieties
V, W ⊆ B d , is it true that the algebra H ∞ ( V ) is (isometrically) isomorphic to H ∞ ( W ) precisely when V is biholomorphic to W ? Answering this question will require anunderstanding of the maximal ideal spaces of the bounded analytic functions of a variety.Another natural class of algebras is given by the norm closures of the polynomials in M V , A V = C [ z ] k·k M V . (These algebras are sometimes referred to as the continuous multipliers on F V , but thisterminology is misleading since in general A V ( C ( V ) ∩ M V ; see [29, Section 5.2]). In fact,the isomorphism problem was studied in [15] first for the algebras A V . It was later realizedthat the norm closed algebras present some delicate difficulties; see [16, Section 7]. In fact,subtleties arise already in the case d = 1; see [16, Section 8].7.7. Approximation and Nullstellensatz.
One of the problems in studying the isomor-phism problem for the norm closed algebras A V is the following (see [16, Section 7] for anexplanation of how these issues relate). Denote by A d the norm closed algebra generatedby the polynomials in M d . Let V ⊆ B d be a variety, and assume that d < ∞ , and that V is determined by polynomials. Consider the following ideals K V = { p ∈ C [ z ] : p (cid:12)(cid:12) V = 0 } , I V = { f ∈ A d : f (cid:12)(cid:12) V = 0 } , and J V = { f ∈ M d : f (cid:12)(cid:12) V = 0 } . A natural question is whether I V is the norm closure of K V , and whether J V is the wot -closure of I V . In other words, weknow that every f ∈ I V is the norm limit of polynomials, but does the fact that f vanisheson V imply that it can be approximated in norm using only polynomials from K V ? Like-wise, is every function in J V the limit of a bounded and pointwise convergent sequence ofpolynomials in K V (or functions in I V )?It is very natrual to conjecture that the answer is yes , and this was indeed proved forhomogeneous ideals; see [16, Corollary 6.13] (see also [27, Corollary 2.1.31] for the wot case). As may be expected, this approximation result is closely related to an analytic Null-stellensatz: √I = I ( V ( I )) (here I is some norm closed ideal in A d , V ( I ) is the the zerolocus of the ideal I , I ( V ( I )) is the ideal of all functions in A d vanishing on V ( I ), and √I is an appropriately defined radical; see [16, Theorem 6.12] and [27, 2.1.30]). However, weunderstand very little about these issues in the non-homogeneous case. References [1] J. Agler and J.E. McCarthy. Complete Nevanlinna–Pick Kernels.
J. Funct. Anal. , 175:11–124, 2000.[2] J. Agler and J.E. McCarthy.
Pick interpolation and Hilbert function spaces , Graduate Studies in Math-ematics, 44, Providence, RI: American Mathematical Society, 2002.[3] P.R. Ahern and D. Sarason. The H p spaces of a class of function algebras. Acta Math.
Commun. Pure Appl. Anal. , 2(2): 139–145, 2003.[5] N. Arcozzi, R. Rochberg and E.T. Sawyer. Carleson measures for the Drury-Arveson Hardy space andother Besov-Sobolev spaces on complex balls.
Adv. Math. , 218(4): 1107–1180, 2008.
6] N. Arcozzi, R. Rochberg, E.T. Sawyer, and B. Wick. The Dirichlet space: a survey.
New York J. Math ,17:45–86, 2011.[7] L. Bers. On rings of analytic functions.
Bull. Amer. Math. Soc. , 54:311–315, 1948.[8] F.F. Bonsall and J. Duncan.
Complete normed algebras . Ergebnisse der Mathematik und ihrer Gren-zgebiete, Band 80. Springer-Verlag, New York-Heidelberg, 1973.[9] H.G. Dales. Automatic continuity: a survey.
Bull. London Math. Soc. , 10(2):192–183, 1978.[10] K.R. Davidson, M. Hartz and O.M. Shalit. Multipliers of embedded discs. arXiv:1307.3204 [math.OA] ,2014. Also in
Complex Anal. Oper. Theory , to appear.[11] K.R. Davidson, M. Hartz and O.M. Shalit. Erratum to: Multpliers of embedded discs.
Complex Anal.Oper. Theory , to appear.[12] K.R. Davidson and D.R. Pitts. The algebraic structure of non-commutative analytic Toeplitz algebras.
Math. Ann. , 311(2):275–303, 1998.[13] K.R. Davidson and D.R. Pitts. Nevanlinna-Pick interpolation for non-commutative analytic Toeplitzalgebras.
Integral Equations Operator Theory , 31:321–337, 1998.[14] K.R. Davidson and D.R. Pitts. Invariant subspaces and hyper-reflexivity for free semigroup algebras.
Proc. Lond. Math. Soc. , 78:401–430, 1999.[15] K.R. Davidson, C. Ramsey and O.M. Shalit. The isomorphism problem for some universal operatoralgebras.
Adv. Math. , 228(1):167–218, 2011.[16] K.R. Davidson, C. Ramsey and O.M. Shalit. Operator algebras for analytic varieties. arXiv:1201.4072[math.OA] , 2014. Also in
Trans. Amer. Math. Soc. , 367:1121–1150, 2015.[17] W. Feller.
An introduction to probability theory and its applications , Vol. I, 3rd ed., Jhon Wiley & sonsInc., 1968.[18] I.M. Gelfand, D.A. Raikov and G.E. Shilov. Commutative normed rings.
Uspehi Matem. Nauk (N. S.)
Introduction to holomorphic functions of several variables
Vol. II, Local theory. Monterey,CA, 1990.[20] M. Hartz. Topological isomorphisms for some universal operator algebras.
J. Funct. Anal. , 263(11):3564–3587, 2012.[21] K. Hoffman.
Banach spaces of analytic functions . Prentice-Hall Series in Modern Analysis Prentice-Hall,Inc., Englewood Cliffs, N. J. 1962.[22] M. Kennedy and D. Yang. A non-self-adjoint Lebesgue decomposition.
Anal. PDE , 7(2):497–512, 2014.[23] M. Kerr, J.E. McCarthy and O.M. Shalit. On the isomorphism question for complete pick multiplieralgebras.
Integral Equations Operator Theory , 76(1):39–53, 2013.[24] V.I. Paulsen.
Completely Bounded Maps and Operator Algebras . Cambridge University Press, Cam-bridge, 2002.[25] G. Pick. Uber die Beschrankungen analytischer Funktionen, welche durch vorgegebene Funktionswertebewirkt werden.
Math Ann. , 77(1):7–23, 1915.[26] G. Popescu. Operator theory on noncommutative varieties.
Indiana Univ. Math. J. , 5(2):389–442, 2006.[27] C. Ramsey.
Maximal ideal space techniques in non-selfadjoint operator algebras . PhD. thesis, Universityof Waterloo, 2013. http://hdl.handle.net/10012/7464.[28] W. Rudin.
Function theory in the unit ball of C n . Springer-Verlag, 1980.[29] O.M. Shalit. Operator theory and function theory in Drury-Arveson space and its quotients . to appearin
Handbook of Operator Theory , editor Daniel Alpay, Springer.[30] O.M. Shalit and B. Solel. Subproduct Systems.
Doc. Math. , 14:801–868, 2009.
Department of Mathematics, Technion - Israel Institute of Technology, Haifa 3200003,Israel
E-mail address : [email protected] Department of Mathematics, Technion - Israel Institute of Technology, Haifa 3200003,Israel
E-mail address : [email protected]@tx.technion.ac.il