Proc. Natl. Acad. Sci. USA Vol. 96, pp. 11096–11099, September 1999 Mathematics
The curvature of a Hilbert module over [z1;:::;zd] (curvature invariant / Gauss–Bonnet–Chern formula / multivariable operator theory) William Arveson
Department of Mathematics, University of California, Berkeley, CA 94720 ABSTRACT A notion of curvature is introduced in multi- H2d/M denoting the quotient of contractive Hilbert mod- variable operator theory. The curvature invariant of a Hilbert ules.
module over [z1;:::;zd] is a nonnegative real number which To make use of these extremal properties for a given finite has significant extremal properties, which tends to be an in- rank contractive Hilbert module H, one must know the value teger, and which is hard to compute directly. It is shown that of KH. But direct computation appears to be difficult for for graded Hilbert modules, the curvature agrees with the most of the natural examples, and in the few cases where the Euler characteristic of a certain finitely generated algebraic computations can be explicitly carried out, the curvature turns module over the appropriate polynomial ring. This result is out, to be an integer. Thus we were led to ask if the curvature a higher dimensional operator-theoretic counterpart of the invariant can be expressed in terms of some other invariant Gauss–Bonnet formula which expresses the average Gaussian which is (i) obviously an integer and (ii) easier to calculate. curvature of a compact oriented Riemann surface as the alter- We establish such a formula in Theorem B below, which ap- nating sum of the Betti numbers of the surface, and it solves plies to Hilbert A-modules H which are “graded” in the sense the problem of calculating the curvature of graded Hilbert that their operator d-tuple is circularly symmetric. This for- modules. The proof of that result is based on an asymptotic mula is analogous to the Chern–Fenchel generalization of the formula which expresses the curvature of a Hilbert module in Gauss–Bonnet formula to compact oriented Riemannian 2n- terms that allow its comparison to a corresponding asymp- manifolds (2); it asserts that KH agrees with the Euler char- totic expression for the Euler characteristic. acteristic of a certain finitely generated algebraic module that is associated with it in a natural way. Because the Euler char- acteristic of a finitely generated module over z ;:::;z 1. Introduction 1 d is relatively easy to compute by using conventional algebraic methods, the problem of calculating the curvature can be con- By a Hilbert module over the polynomial algebra A = sidered solved for graded Hilbert modules. z ;:::;z of dimension d, we mean a Hilbert space H 1 d The problem of calculating the curvature of ungraded finite together with a commuting d-tuple of bounded operators rank Hilbert modules remains open. T ;:::;T acting on H. The A-module structure is defined 1 d Our proof of Theorem B amounts to establishing appropri- on H by ate asymptotic formulas for both the curvature and the Euler f · ξ = f T ;:::;T ξ; f A; ξ H: characteristic of arbitrary (i.e., perhaps ungraded) finite rank 1 d contractive Hilbert A-modules (Theorems C and D below). A Hilbert module H is said to be contractive if for every Theorem D is proved by showing that KH is actually the trace of a certain self-adjoint trace class operator (the curva- ξ1;:::;ξd H we have ture operator of H), and an asymptotic formula for the trace 2 2 2 T1ξ1 +···+Tdξd < ξ1 +···+ξd ; [1] of that operator is developed. The problem of determining more precisely the distribution and the rank of a contractive Hilbert module is defined as the of eigenvalues of this curvature operator is an attractive one ∗ dimension of the range of the defect operator 1 =1−T1T1 − about which we as yet know very little. ∗ 1/2 ···−TdTd . The rank of H is written rankH. This note concerns finite rank contractive Hilbert modules. 2. Basic Definitions These are Hilbert space counterparts of finitely generated modules over polynomial rings. We introduce a numerical in- We now describe these results in more detail, beginning variant KH for such Hilbert modules H, called the curva- with the definition of the curvature invariant KH.Let ture invariant of H. KH is a real number satisfying 0 < T1;:::;Td "H be the defining d-tuple of a contrac- KH < rankH. We give almost no proofs here, but full de- tive finite rank Hilbert A-module H. With every point tails can be found in ref. 1, which is available at the author’s d z =z1;:::;zd in complex d-space , we associate the web site (http://math.berkeley.edu/7arveson.) operator The curvature invariant carries key information about the operator theory and structure of Hilbert modules. For exam- T z=z T +···+z T "H: ple, it is sensitive enough to detect exactly when H is a “free 1 1 d d Hilbert module of finite rank" in the sense that H is unitarily Because of the inequality [1], we have equivalent to a finite direct sum of copies of the free Hilbert module H2d (to be described more pecisely below) if, and T z < z= z 2 +···+z 2 1/2;z d; only if, KH=rankH. This is the first extremal property 1 d of KH. and in particular for every point z in the open unit ball B = The second extremal property of KH is related to the d z d xz + 1; we have T z + 1, and hence 1 − T z structure of the invariant subspaces of the rank-one free is invertible. Thus for every z B , we can define a positive Hilbert module H2 in the following sense: a closed submod- d operator Fz which acts on the finite dimensional Hilbert ule M H2d is associated with an “inner sequence” space 1H as follows, iff ∗ −1 −1 KH2d/M=0; Fzξ = 11 − T z 1 − T z 1ξ; ξ 1H:
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To define the curvature invariant, we require the bound- F/M of a finite rank free Hilbert A-module F by a closed
ary values of the function z Bd 7→ trace Fz. These do not submodule M F is a pure Hilbert A-module of rank at exist in a conventional sense because in all significant cases most r = rankF. Conversely, the basic result on the dila- this function is unbounded. However, “renormalized" bound- tion theory of d-contractions (see ref. 3, theorem 8.5 for ex- ary values do exist almost everywhere relative to the natural ample) implies the following characterization of pure Hilbert rotation-invariant probability measure σ defined on the unit A-modules.
sphere ∂Bd, in the following sense. Dilation Theorem. Every pure contractive Hilbert A- Theorem A. For σ-almost every ζ ∂Bd; the limit module H of finite or infinite rank r is unitarily equivalent to a quotient F/M; where F = r · H2 is the free Hilbert module of 2 K0ζ=lim1 − r trace Frζ rank r and M F is a closed submodule. r→1 We conclude that to understand finite rank Hilbert A-
exists and satisfies 0 < K0ζ < rankH. modules, one should focus attention on free Hilbert modules The curvature invariant of a finite rank contractive Hilbert of finite rank, their closed submodules and quotients. In par- module is defined as follows: ticular, how does one determine whether a given Hilbert Z module is free? What are the properties of submodules of free Hilbert modules? In the following discussion, we re- KH= K0ζ dσζ; [2] ∂Bd late these questions to extremal properties of the curvature invariant. These are fundamental results in the subject. and we have 0 < KH < rankH. First Extremal Property. Let H be a pure Hilbert A- module of finite rank r. Then H is unitarily equivalent to a free 3. Extremal Properties of KH Hilbert A-module iff KH=r; and in that case H is equivalent to r · H2. In a recent paper (3), the author developed the theory of To discuss the second extremal property of KH; we a particular Hilbert module over A = z1;:::;zd, called consider closed submodules M H2 of the free Hilbert 2 d 2 H or more simply H when there is no possibility of con- A-module of rank 1. Such a submodule M is itself a con- fusion over the dimension. There is a natural inner product on tractive Hilbert module, it is pure, but it is frequently of 2 A and H is obtained by completing this inner product space. infinite rank (see Theorem E below). In any case, the Dilation 2 The action of polynomials on H extends the natural multi- Theorem readily implies that there is a (finite or infinite) se- plication of A. The associated d-tuple of operators S1;:::;Sd quence of bounded holomorphic functions φ ;φ ;::: on the 2 1 2 acting on H is called the d-shift. The d-shift satisfies [1] but d open unit ball Bd of whose multipliers define bounded does not satisfy the natural counterpart of von Neumann’s in- operators on H2d, which in fact satisfy equality for the unit ball Bd. Indeed, there is no constant K ∗ ∗ satisfying Mφ Mφ + Mφ Mφ +···=PM ; [3] 1 1 2 2 2 f S1;:::;Sd < K sup f z PM denoting the orthogonal projection of H onto the zB d subspace M. Formula [3] makes the following assertion. If we consider the row operator defined by the sequence for every polynomial f A (see ref. 3, Theorem 3.3). It fol- R =M ;M ;::: as an operator from a direct sum of lows from this fact that the d-shift is not a subnormal d-tuple. φ1 φ2 2 2 Nevertheless, finite direct sums of copies of the d-shift serve copies of H into H , then R is a partially isometric homo- morphism of Hilbert A-modules whose range is precisely the as an effective substitute for free modules in the algebraic the- 2 ory of finitely generated modules over polynomial rings. To given invariant subspace M H . discuss this we require the notion of a pure Hilbert A-module. The sequence φn of multipliers appearing in [3] is cer- Let H be a contractive Hilbert A-module with canonical op- tainly not uniquely determined by the invariant subspace M, but if it is carefully chosen, it is unique up to a “unitary rota- erators T1;:::;Td and consider the linear map of operators φ x "H→"H defined by tion matrix.” We also emphasize that it is usually impossible to find a single multiplier φ or even a finite sequence of mul- ∗ ∗ φA=T AT +···+T AT ;A "H: tipliers φ1;:::;φr which satisfies [3]. Indeed, the minimum 1 1 d d length r of such a sequence turns out to be the rank of M φ is completely positive map satisfying φ=φ1 < 1. as a Hilbert A-module, and as we have pointed out above, It follows that the sequence of positive operators φn1, n = the latter is usually infinite. Thus, while such sequences exist n 2 0; 1; 2;::: is decreasing and hence the limit limn→: φ 1 ex- for arbitrary closed submodules M H , they are typically ists relative to the strong operator topology as a positive con- infinite sequences. traction. The Hilbert A-module H is said to be pure if In dimension d = 1 there is a single multiplier φ satisfy- ∗ ing MφMφ = PM , and [3] reduces essentially to the assertion lim φn1=0: of Beurling’s theorem: every invariant subspace of H2 is n→: the range of an isometry which commutes with the unilateral Let r be a positve integer. By a free Hilbert module of rank r shift. In turn, the multiplier φ is an inner function, that is, we mean a direct sum F = r ·H2 = H2 ⊕···⊕H2 of r copies of a bounded analytic function defined on the open unit disk the Hilbert module H2 = H2d, where the module structure whose boundary values satisfy φeiθ = 1 almost everywhere is defined by the d-shift, i.e., dθ on the unit circle. This description of the boundary values of the multiplier φ is easily established by using subnormality: f ·g1;:::;gd=f · g1;:::;f · gr ; the unilateral shift is the restriction of a unitary operator (the f A; g ;:::;g H2: bilateral shift) to an irreducible invariant subspace. 1 d To what extent does such a function-theoretic description This definition has a natural and obvious interpretation when of invariant subspaces hold in higher dimensions? Alexandrov r = :, and : · H2 is called the free Hilbert module of infi- has shown that inner functions exist for the unit ball in di- nite rank. Notice that if we are given any closed submodule mension d > 2 (see ref. 4; also see refs. 5–7), but as [3] and M of a (contractive) Hilbert A-module H, then the quotient the remarks following it imply, one cannot hope to achieve H/M is naturally a Hilbert space and it inherits the struc- [3] with a single multiplier or even with a finite sequence of ture of a contractive Hilbert module from that of H. In gen- multipliers. Thus, we require a notion of innerness that can eral, one has rankH/M < rankH. Every such quotient be applied to finite or infinite sequences. Downloaded by guest on September 25, 2021 11098 Mathematics: Arveson Proc. Natl. Acad. Sci. USA 96 (1999)
Now Eq. [3] implies that the sequence of functions φn define the Euler characteristic of H by has the following property on the open unit ball Xn k+1 2 2 χH= −1 βk: [6] φ1z +φ2z +···< 1;z Bd: k=1
In particular, each φn is a bounded holomorphic function on For algebraic reasons one always has χH > 0. ˜ Bd and thus it has radial limits φζ for σ-almost every point We now describe the principal formula which evaluates the ζ on the sphere ∂Bd (8). The preceding inequality implies that curvature invariant of graded Hilbert modules. By a graded Hilbert space we mean a pair H; 0 where H is a (separable) ˜ 2 ˜ 2 φ1ζ +φ2ζ +···< 1 Hilbert space and 0 is a strongly continuous unitary represen- tation of the circle group =λ xλ=1. 0 is called holds almost everywhere dσζ on ∂Bd. φn is called an inner the gauge group of H. For each n we can define the asso- sequence if equality holds ciated spectral subspace
˜ 2 ˜ 2 n φ1ζ +φ2ζ +···=1 [4] Hn =ξ H x 0λξ = λ ξ; λ ;
for σ-almost every point ζ ∂Bd. and thus H can be regarded as a -graded Hilbert space in We have already pointed out above that in dimension d > the sense that we have an orthogonal decomposition 2; the d-shift is not a subnormal d-tuple. Hence the argument alluded to above for establishing [4] in one dimension breaks H =···⊕H−1 ⊕ H0 ⊕ H1 ⊕···: [7] down in a fundamental way, and the issue of whether or not an invariant subspace can be associated with an inner sequence Conversely, if we are given a -graded Hilbert space [7], then becomes significant. we can construct an associated gauge group simply by defining n Second Extremal Property. Let M be a closed sub- 0λ on the nth summand to be ξ 7→ λ ξ and then extend- module of the rank-one free Hilbert module H2d, and let ing by linearity. Algebraists tend to prefer the description in φ be any sequence of holomorphic functions in the unit terms of -gradings because that formulation works for arbi- n trary scalar fields, but for our purposes it is more convenient ball of Bd whose multipliers satisfy [3]. 2 d to think in terms of gauge groups. Then φn is an inner sequence iff KH /M=0. We will make use of these extremal properties at several A graded Hilbert A-module is a contractive Hilbert module points in the discussion to follow. H, which is also a graded Hilbert space such that the gauge group 0 is related as follows to the operators T1;:::;Td asso- ciated with the module structure 4. Euler Characteristic, Graded Hilbert Modules, Principal Formula −1 0λTk0λ = λTk;λ ;k= 1;:::;d: We have pointed out in the Introduction that the curvature Thus, graded Hilbert modules are precisely those whose un- invariant is not easily calculated directly, nor is it obviously an derlying operator d-tuple is circularly symmetric. One can eas- integer in even the simplest cases. In this section we introduce ily verify that the previous formula is satisfied if and only if another invariant which is an integer by virtue of its definition, the Tk shift the terms Hn of the associated -grading [7] as and which can be readily computed for many examples by follows using the basic tools of commutative algebra (cf. 9, 10). Throughout this section, H will denote a Hilbert module TkHn Hn+1;n ;k= 1;:::;d: over A = z1;:::;zd of finite rank r. We will work not with H itself but with the following linear submanifold of H There are many natural examples of graded Hilbert modules, and in fact there is a “graded” variation of the Dilation The-
MH = spanf · 1ξ x f A; ξ H: orem above that characterizes them in much the same way as finite rank pure Hilbert A-modules are characterized. The definition and basic properties of the Euler characteristic We view the following result as an analogue of the the are independent of any topology associated with the Hilbert Gauss–Bonnet–Chern formula for graded Hilbert A-modules space H, and depend solely on the linear algebra of MH .Ifwe H. It implies that KH is an integer in such cases, and choose any linear basis ζ1;:::;ζr for the r-dimensional space because there are well-developed algebraic tools for com- 1H; then every element of MH is a finite sum of vectors of puting the Euler characteristic of finitely generated modules the form over polynomial rings, it can also be viewed as a solution of the problem of calculating the curvature invariant of graded f1 · ζ1 +···+fr · ζr ; Hilbert modules. Theorem B. For every pure finite rank graded Hilbert A- where f1;:::;fr are polynomials in A. Thus, MH is a finitely module H; generated A-module. By Hilbert’s Syzygy Theorem (refs. 11
and 12, or ref. 9 or 10), MH has finite free resolutions in the KH=χH: category of finitely generated A-modules 5. Something About the Proof: Asymptotics of K(H) and 0 → Fn →···→F2 → F1 → MH → 0; [5] ·(H)
each Fk being a sum of a finite number βk of copies of the Theorem B is proved by establishing asymptotic formulas for rank-one module A. The sequence [5] is an exact sequence both the curvature invariant and Euler characteristic of arbi- of A-modules, and is called a finite free resolution of MH . trary (i.e., perhaps ungraded) finite rank Hilbert A-modules. Free resolutions are not unique, but there do exist free res- We briefly discuss these results in this section and say some- olutions whose length n does not exceed the dimension d of thing about the underlying curvature operator.
the underlying polynomial ring A = z1;:::;zd. Let H be a finite rank Hilbert A-module with canonical The alternating sum of the ranks β1 − β2 + β3 −+··· does operators T1;:::;Td, and let φ be the corresponding com- ∗ ∗ not depend on the particular free resolution of MH , and we pletely positive map on "H, φA=T1AT1 +···+TdATd , Downloaded by guest on September 25, 2021 Mathematics: Aveson Proc. Natl. Acad. Sci. USA 96 (1999) 11099
A "H. Because 1 − φ1=12 is a finite rank opera- Corollary 1. Every closed invariant subspace of H2 which tor, φn1−φn+11=φn12 is finite rank for every n = contains a nonzero polynomial is associated with an inner se- 0; 1; 2;::: and hence so is quence. Xn More generally, one has Corollary 2. 1 − φn+11= φk12: Corollary 2. Let H be a finite rank Hilbert A-module k=0 whose canonical operators T1;:::;Td satisfy an equation f T1;:::;Td=0 for some nonzero polynomial f A. Then We have the following asymptotic formulas, the principal one KH=χH=0. being Theorem D. Returning to the rank-one case, we are led to ask if Theorem C. For every finite rank Hilbert A-module H; the second alternative [11] can occur, and the answer is rank 1 − φn1 yes. Indeed, if M is any closed submodule of H2 which χH=d! lim : (i) contains no nonzero polynomials and (ii) satisfies n→: nd 06=M 6= H2, then from condition (i) one easily shows Theorem D. For every finite rank Hilbert A-module H; that χH2/M=rankH2/M=1. Condition (ii) implies there is a naturally associated self-adjoint trace class operator that the quotient H2/M cannot be a free Hilbert module, d0; and hence from the First Extremal property of the curvature in- trace1 − φn1 variant we have KH2/M + rankH=1. Thus [11] holds KH=trace d0 = d! lim : for all such quotients H = H2/M. Explicit examples are n→: nd described in ref. 1. We will not define the operator d0 here, but we do offer the These remarks also show that Theorem B does not hold for following comment. While KH=trace d0 is always a non- ungraded Hilbert modules. Nevertheless, we still do not know negative real number, d0 is never a positive operator. Indeed, if KH2/M must be 0 in the context of alternative [11].By d0 can be expressed as a difference E −P where E is a projec- making use of the Second Extremal property of the curvature tion whose dimension is rankH and P is a positive operator invariant, this special case of Problem 1 (i.e., the case in which satisfying trace P < rankH. See ref. 1 for more detail. rankH=1) becomes the following more concrete question. n Notice that because 1−φ 1 is a positive operator of norm Problem 2: Is every nonzero closed invariant subspace M at most 1 for every n = 1; 2;:::; we have of H2 associated with an inner sequence?† trace1 − φn1 < rank1 − φn1; Finally, we offer some comments about the rank of submod- ules of the free rank-one Hilbert module H2. We have already and hence Theorems C and D together imply that, for (perhaps pointed out that every closed submodule M of H2 is a (con- ungraded) finite rank Hilbert modules H, we have the general tractive, pure) Hilbert A-module, and that when M 6= H2 the inequality quotient module H2/M is of rank 1. What about the rank of M itself? Because H2 itself is of finite rank, it is easy to see 0 < KH < χH: [8] that any closed submodule M H2 which is of finite codimen- This inequality has significant implications (see Corollaries 1 sion in H2 must also be of finite rank; and the ranks of such and 2 in the following section). finite codimensional submodules can be arbitrarily large. The Finally, we point out that the following upper bound on the following result concerns infinite codimensional submodules Euler number is a straightforward consequence of Theorem C: M which are generated as closed submodules by some set of homogeneous polynomials in A (perhaps of varying degrees). χH < rankH: [9] Theorem E. Let 0 be the natural gauge group acting on the rank-one free Hilbert module H2; and let M be any closed sub- 6. Discussion: Problems and Applications module which is graded in the sense that 0λM M for every λ ; and which is nonzero and of infinite codimension in H2. Theorem B implies that KH is an integer for every pure Then rankM=:. finite rank Hilbert module H which is graded. But we do not Notice that this result stands in rather stark contrast with know if this is so for ungraded modules. the assertion of Hilbert’s basis theorem (9). Indeed, we con- Problem 1: Is KH an integer for every pure finite rank jecture that the answer to the following question is yes. Hilbert A-module H? Problem 3: Must the rank of every closed submodule of We conclude with a discussion of issues and problems re- H2, which is nonzero and of infinite codimension in H2,be lated to Problem 1. Consider the simplest nontrivial case infinite? rankH=1. The Dilation Theorem implies that H can be Additional problems and applications are discussed in ref. 1. realized as a quotient H2/M of the free Hilbert module H2 by a closed submodule M, and the general inequalities [8] †I have received a letter from Stefan Richter and Carl Sundberg, which and [9] imply that implies an affirmative solution of Problem 2. 0 < KH2/M < χH2/M < rankH2/M=1: W.A. is on appointment as a Miller Research Professor in the Miller Institute for Basic Research in Science. This work was supported by Because χH2/M must be either 0 or 1 and we conclude that National Science Foundation Grant DMS-9802474. either 1. Arveson, W. (1999) The Curvature Invariant of a Hilbert Module over z ;:::;z , preprint. KH2/M=χH2/M=0 [10] 1 d 2. Chern, S.-S. (1944) Ann. Math. 45 741–752. or 3. Arveson, W. (1998) Acta Math. 181, 159–228. 4. Alexandrov, A. (1982) Mat. Sbornik 18(2), 1–13 (Russian). 2 2 5. Low, E. (1982) Invent. Math. 67(2), 223–229. 0 < KH /M + χH /M=1: [11] 6. Alexandrov, A. (1984) Funct. Analiz i ego Prilozh 118(2), 147–163 (Rus- sian). Consider the first alternative [10]. Using a general result of 7. Tomaszewski, B. (1999) A Simple Proof of the Existence of Inner Functions Auslander and Buchsbaum (see refs. 9 or 10) it is not hard to on the Unit Ball of Cd , preprint. show that χH2/M=0 iff M contains a nonzero polynomial. 8. Rudin, W. (1980) Function Theory in the Unit Ball of n (Springer, Berlin). On the other hand, recalling the Second Extremal property of 9. Eisenbud, D. (1994) Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics (Springer, Berlin), Vol. 150. KH, we see that when [10] holds, the closed submodule 10. Kaplansky, I. (1970) Commutative Rings (Allyn and Bacon, Boston). 2 M H must be associated with an inner sequence. Thus we 11. Hilbert, D. (1890) Math. Ann. 36, 473–534. may conclude as follows. 12. Hilbert, D. (1893) Math. Ann. 42, 313–373. Downloaded by guest on September 25, 2021