Resolutions of Hilbert Modules and Similarity
aa r X i v : . [ m a t h . F A ] S e p RESOLUTIONS OF HILBERT MODULES AND SIMILARITY
RONALD G. DOUGLAS, CIPRIAN FOIAS, AND JAYDEB SARKAR
Abstract.
Let H m be the Drury-Arveson (DA) module which is the reproducing kernelHilbert space with the kernel function ( z, w ) ∈ B m × B m → (1 − m P i =1 z i ¯ w i ) − . We investigatefor which multipliers θ : B m → L ( E , E ∗ ) with ran M θ closed, the quotient module H θ , givenby · · · −→ H m ⊗ E M θ −→ H m ⊗ E ∗ π θ −→ H θ −→ , is similar to H m ⊗ F for some Hilbert space F . Here M θ is the corresponding multiplicationoperator in L ( H m ⊗ E , H m ⊗ E ∗ ) for Hilbert spaces E and E ∗ and H θ is the quotient module( H m ⊗ E ∗ ) /M θ ( H m ⊗ E ), and π θ is the quotient map. We show that a necessary condition isthe existence of a multiplier ψ in M ( E ∗ , E ) such that θψθ = θ. Moreover, we show that the converse is equivalent to a structure theorem for complementedsubmodules of H m ⊗ E for a Hilbert space E , which is valid for the case of m = 1. Thelatter result generalizes a known theorem on similarity to the unilateral shift, but the abovestatement is new. Further, we show that a finite resolution of DA-modules of arbitrarymultiplicity using partially isometric module maps must be trivial. Finally, we discuss theanalogous questions when the underlying operator m -tuple (or algebra) is not necessarilycommuting (or commutative). In this case the converse to the similarity result is alwaysvalid. Introduction
A well known result in operator theory (see [18] and [19]) states that the contractionoperator given by a canonical model is similar to a unilateral shift of some multiplicity ifand only if its characteristic function has a left inverse. Various approaches to this one-variable result have been given (cf. [21]) but a new one is given in this paper which uses thecommutant lifting theorem (CLT). In particular, the proof does not involve, at least explicitly,the geometry of the dilation space for the contraction.The Drury-Arveson (DA) space H m (see [10], [17], [1]) has been intensively studied by manyresearchers over the past few decades. In particular, the CLT has been extended to this spacewith a few necessary changes. Using the CLT, we extend to the DA space one direction of theone variable result on the similarity of quotient modules of the Hardy space on the unit disk.We show that the converse is equivalent to the assertion that each complemented submodule Mathematics Subject Classification.
Key words and phrases. quotient module, shift operator, similarity, Commutant lifting theorem, resolutionsof Hilbert module.This research was partially supported by a grant from the National Science Foundation. of H m ⊗ E for a Hilbert space E is isomorphic to H m ⊗ E ∗ for some Hilbert space E ∗ . Of coursethis result follows trivially from the Beurling-Lax-Halmos theorem (BLHT) in case m = 1.(Actually, for m = 1 the submodule is isometrically isomorphic to H ⊗ E ∗ = H ( D ) ⊗ E ∗ .)In Section 2, we recall some definitions and results in multivariable operator theory. In thenext section, we consider a characterization of those pure co-spherically contractive Hilbertmodules similar to the DA-module of some multiplicity. Using the representation of submod-ules of the DA-module by inner multipliers [16], we are able to obtain the characterizationin terms of inner multiplier associated with a given quotient Hilbert module and the regularinverse of that multiplier.The quotient modules described above are the simplest case of a resolution by DA-modulesfor which the connecting maps are all partial isometries or inner multipliers (see [12]). Moreprecisely, using the results of Arveson [1] and Muller and Vasilescu [17] and McCulloughand Trent [16], for a given pure co-spherical contractive Hilbert module one can obtain aninner resolution. In [3], Arveson suggested that the inner resolution might not terminate asresolutions do in the algebraic context. In this paper we show that the only isometric innermultiplier, V : H m ⊗ E → H m ⊗ E ∗ for Hilbert spaces E and E ∗ , is the trivial one determinedby an isometric operator V : 1 ⊗ E → ⊗ E ∗ . As a consequence, we show that all finite innerresolutions are trivial in a sense that will be explained in Section 4.In Section 5, we are able to apply essentially the same proofs to the non-commutativecase to obtain an analogous result, except here we need the noncommutative analogue of theBLHT due to Popescu ([24], [23]). More precisely, we show that a quotient of the Fock Hilbertspace, F m ⊗ E , for some Hilbert space E , by the range of a multi-analytic map Θ is similar to F m ⊗ F for some Hilbert space F if and only if Θ has a multi-analytic regular inverse.In a concluding section we indicate that many of these results can be extended to completeNevanlinna-Pick kernel Hilbert spaces and to other Hilbert modules for which the CLT holds. Acknowledgement:
The authors wish to thank the referee for a careful reading of the manu-script and useful remarks which led to an improved paper.2.
Preliminaries
We consider two cases, the first one in which the operators commute, or for which thealgebra is C [ z , . . . , z m ] and hence commutative, and the second in which the operators arenot assumed to commute or the algebra is F [ Z , . . . , Z m ]. We begin with the commutativecase.Let { T , . . . , T m } be a commuting m -tuple of bounded linear operators on a Hilbert space H ; that is, [ T i , T j ] = T i T j − T j T i = 0 for i, j = 1 , . . . , m . A Hilbert module H over thepolynomial algebra C [ z , . . . , z m ] of m commuting variables is defined so that the modulemultiplication C [ z , . . . , z m ] × H → H is defined by p ( z , . . . , z m ) · h = p ( T , . . . , T m ) h, ESOLUTIONS OF HILBERT MODULES AND SIMILARITY 3 where p ( z , . . . , z m ) ∈ C [ z , . . . , z m ] and h ∈ H . We denote by M , . . . , M m the operatorsdefined to be module multiplication by the coordinate functions. More precisely, M i h = z i · h = T i h, ( h ∈ H , i = 1 , . . . , m ) . All submodules in this paper are assumed to be closed in the norm topology.A Hilbert module over C [ z , . . . , z m ] is said to be co-spherically contractive , or define a rowcontraction , if k m X i =1 M i h i k ≤ m X i =1 k h i k , ( h , . . . , h m ∈ H ) , or, equivalently, if m X i =1 M i M ∗ i ≤ I H . Natural examples of co-spherically contractive Hilbert modules over C [ z , . . . , z m ] are theDA-module, the Hardy module and the Bergman module, all defined on the unit ball B m in C m . These are all reproducing kernel Hilbert spaces over B m and, among them, the DA-module plays the key role for the class of co-spherically contractive Hilbert modules over C [ z , . . . , z m ]. In order to be more precise, we briefly recall that a scalar reproducing kernel K on a set X is a function K : X × X → C which satisfies l X i,j =1 ¯ c i c j K ( x i , x j ) > , for x , . . . , x l ∈ X , c , . . . , c l ∈ C with not all c i zero and l ∈ N . The reproducing kernelHilbert space H K , corresponding to the kernel K , is the Hilbert space of functions defined on X with the following reproducing property f ( x ) = h f, K x i , f ∈ H K , where for each x ∈ X , K x : X → C is the vector in H K defined by K x ( w ) = K ( w, x ), w ∈ X . The DA-module H m is the reproducing kernel Hilbert space corresponding to thekernel K : B m × B m → C defined by K ( z, w ) = (1 − m X i =1 z i ¯ w i ) − , ( z, w ) ∈ B m × B m . We identify the Hilbert tensor product H m ⊗ E with the E -valued H m space H m ( E ) or the L ( E )-valued reproducing kernel Hilbert space with the kernel ( z, w ) (1 − m P i =1 z i ¯ w i ) − I E .Consequently, H m ⊗ E = { f ∈ O ( B m , E ) : f ( z ) = X k ∈ N m a k z k , a k ∈ E , k f k := X k ∈ N m k a k k γ k < ∞} , where O ( B m , E ) is the space of E -valued holomorphic functions on B m , k = ( k , . . . , k m ) and γ k = ( k + ··· + k m )! k ! ··· k m ! are the multinomial coefficients. A function ϕ ∈ O ( B m , L ( E , E ∗ )) is said to DOUGLAS, FOIAS, AND SARKAR be a multiplier if ϕf ∈ H m ⊗ E ∗ = H m ( E ∗ ) for all f ∈ H m ⊗ E = H m ( E ). By the closed graphtheorem, such a multiplier ϕ defines a bounded module map M ϕ : H m ⊗ E → H m ⊗ E ∗ , M ϕ f = ϕf, f ∈ H m ⊗ E . Equivalently, we can consider ϕ ∈ O ( B m , L ( E , E ∗ )) for which M ϕ defines a bounded operatorfrom H m ⊗ E to H m ⊗ E ∗ . The set of all such bounded multipliers ϕ ∈ O ( B m , L ( E , E ∗ )) willbe denoted by M ( E , E ∗ ). A multiplier ϕ ∈ M ( E , E ∗ ) is said to be inner if M ϕ is a partialisometry in L ( H m ⊗ E , H m ⊗ E ∗ ).We recall an analogue of the CLT due to Ball-Trent-Vinnikov (Theorem 5.1 in [4]) onDA-modules which will be used to prove some of the main results of this paper. Theorem . (Ball-Trent-Vinnikov) Let N and N ∗ be quotient modules of H m ⊗E and H m ⊗E ∗ for some Hilbert spaces E and E ∗ , respectively. If X : N → N ∗ is a bounded module map, thatis, XP N ( M z i ⊗ I E ) | N = P N ∗ ( M z i ⊗ I E ∗ ) | N ∗ X, for i = 1 , . . . , m , then there exists a multiplier ϕ ∈ M ( E , E ∗ ) such that(i) k X k = k M ϕ k and(ii) P N ∗ M ϕ = X. In the language of Hilbert modules, one has π N ∗ M ϕ = Xπ N , where π N and π N ∗ are thequotient maps. The above statement of the CLT for C [ z , . . . , z m ] is due to Ball-Trent-Vinnikov as indicated.However, Popescu pointed out that the result follows from its noncommutative analogueestablished earlier by him in [22, 23]. A more recent paper on this topic is due to Davidsonand Le ([5]).We now recall the notion of pureness for a co-spherically contractive Hilbert module H over C [ z , . . . , z m ]. Define the completely positive map P H : L ( H ) → L ( H )by P H ( A ) = m X i =1 M i AM ∗ i , ( A ∈ L ( H )) . Now I H ≥ P H ( I H ) ≥ P H ( I H ) ≥ · · · ≥ P l H ( I H ) ≥ · · · ≥ , so that P ∞ = SOT − lim l →∞ P l H ( I H )exists and 0 ≤ P ∞ ≤ I H . The Hilbert module H is said to be pure if P ∞ = 0 . A canonical example of a pure co-spherically contractive Hilbert module over C [ z , . . . , z m ]is the DA-module H m ⊗ F , where F is a Hilbert space. Moreover, quotients of DA-modulescharacterize all pure co-spherically contractive Hilbert modules. ESOLUTIONS OF HILBERT MODULES AND SIMILARITY 5
Theorem . Let H be a pure co-spherically contractive Hilbert module. Then(i)( Arveson [1] , Muller-Vasilescu [17] ) H is isometrically isomorphic with a quotient module ( H m ⊗ E ∗ ) / S , where E is a Hilbert space and S is a submodule of H m ⊗ E ∗ .(ii) ( McCullough-Trent [16] ) If S is a submodule of H m ⊗ E ∗ , then there exists a multiplier θ ∈ M ( E , E ∗ ) for some Hilbert space E such that M θ is inner and S = M θ ( H m ⊗ E ) .Therefore, H is isometrically isomorphic to ( H m ⊗ E ∗ ) / ran M θ for some inner multiplier θ ∈ M ( E , E ∗ ) and Hilbert space E . Note that in the statement of Theorem 2.1, Ball-Trent-Vinnikov [4] made the additionalassumption that the submodules N ⊥ and N ⊥∗ are invariant under the scalar multipliers.However, that this condition is redundant follows from part (ii) of Theorem 2.2 above due toMcCullough-Trent.We now consider some preliminaries for the case of noncommuting operators. Let F + m denote the free semigroup with the m generators g , . . . , g m and let F m be the full Fock spaceof m variables, which is a Hilbert space. More precisely, if we let { e , . . . , e m } be the standardorthonormal basis of C m , then F m = M k ≥ ( C m ) ⊗ k , where ( C m ) ⊗ = C . The creation, or left shift, operators S , . . . , S m on F m are defined by S i f = e i ⊗ f, for all f in F m and i = 1 , . . . , m .Let { T , . . . , T m } be m bounded linear operators on a Hilbert space K which are not nec-essarily commuting. One can make K into a Hilbert module over the algebra of polynomials F [ Z , . . . Z m ], in m noncommuting variables, as follows: F [ Z , . . . Z m ] × K → K , p ( Z , . . . , Z m ) · h p ( T , . . . , T m ) h, h ∈ K . The module K over F [ Z , . . . , Z m ] is said to be co-spherically contractive if the row operatorgiven by module multiplication by the coordinate functions is a contraction.A bounded linear operator Θ ∈ L ( F m ⊗ E , F m ⊗ E ∗ ), for some Hilbert spaces E and E ∗ , issaid to be a multi-analytic operator if it is a module map; that is, ifΘ( S i ⊗ I E ) = ( S i ⊗ I E ∗ )Θ , i = 1 , . . . , m. Given a multi-analytic operator Θ as above, one can define a bounded linear operator θ : E → F m ⊗ E ∗ by θx = Θ(1 ⊗ x ) ( x ∈ E ) . In this correspondence of Θ and θ , each uniquely determines the other. Moreover, one definesthe operator coefficients θ α ∈ L ( E , E ∗ ) of Θ by h θ α t x, y i = h θx, e α ⊗ y i = h Θ(1 ⊗ x ) , e α ⊗ y i ( x ∈ E , y ∈ E ∗ )for each α ∈ F + m , where α t = g i p · · · g i for α = g i · · · g i p . It was proved by Popescu (cf. [24])that Θ = SOT − lim r → − ∞ X l =0 X | α | = l r | α | R α ⊗ θ α , DOUGLAS, FOIAS, AND SARKAR where R i = U ∗ S i U for i = 1 , . . . , m, are the right creation operators on F m , R α = R g i · · · R g ip for α = g i · · · g i p , and U is the unitary operator on F m defined by U e α = e α t for α ∈ F + m .The set of all multi-analytic operators in L ( F m ⊗ E , F m ⊗ E ∗ ) coincides with R ∞ m ¯ ⊗L ( E , E ∗ ),the WOT closed algebra generated by the spatial tensor product of R ∞ m and L ( E , E ∗ ), where R ∞ m = U ∗ F ∞ m U and F ∞ m is the WOT closed algebra generated by the left creation operators, S , . . . , S m , and the identity operator on F m .Notice that the definition of a pure co-spherically contractive Hilbert module can be ex-tended to the noncommutative case; that is, with appropriate change of notation, the conceptof a pure co-spherically contractive Hilbert module K over F [ Z , . . . , Z m ] can be defined ina similar way. Popescu proved that any pure co-spherically contractive Hilbert module over F [ Z , . . . , Z m ] can be realized as a quotient module of F m ⊗ E for some Hilbert space E (see[23] and Theorem 2.10 and references in [24]). Theorem . (Popescu) Given a pure co-spherically contractive Hilbert module K over F [ Z , . . . , Z m ] , there is a multi-analytic operator Θ in L ( F m ⊗ E , F m ⊗ E ∗ ) for some Hilbertspaces E and E ∗ , which is isometric such that K is isometrically isomorphic to the quotientof F m ⊗ E ∗ by the range of Θ . Moreover, the characteristic operator function Θ is a completeunitary invariant for K . Finally we need to extend the analogue of the BLHT to the noncommuting setting whichis due to Popescu ([23], [24]).
Theorem . (Popescu) If S is a closed subspace of F m ⊗ F for some Hilbert space F , thenthe following are equivalent:(i) S is a submodule of F m ⊗ F .(ii) There exists a Hilbert space E and an (isometric) inner multi-analytic operator Φ : F m ⊗ E → F m ⊗ F such that S = Φ( F m ⊗ E ) . Hilbert modules over C [ z , . . . , z m ]Let θ ∈ M ( E , E ∗ ) be a multiplier for Hilbert spaces E and E ∗ such that M θ has closed rangeand let H θ be the quotient module defined by the sequence · · · −→ H m ⊗ E M θ −→ H m ⊗ E ∗ π θ −→ H θ −→ , where π θ is the quotient map of H m ⊗ E ∗ onto the quotient of H m ⊗ E ∗ by the range of M θ .There are several possible relationships between these objects: Statement . The sequence splits or π θ is right invertible; that is, there exists a module map σ θ : H θ → H m ⊗ E ∗ such that π θ σ θ = I H θ . Statement . The multiplication operator M θ has a left inverse. Equivalently, there existsa multiplier ψ ∈ M ( E ∗ , E ) satisfying ψ ( z ) θ ( z ) = I E for z ∈ B m . ESOLUTIONS OF HILBERT MODULES AND SIMILARITY 7
Statement . The multiplier θ has a regular inverse. Equivalently, there exists a multiplier ψ ∈ M ( E ∗ , E ) satisfying θ ( z ) ψ ( z ) θ ( z ) = θ ( z ) , for z ∈ B m . Note that in case ker M θ = { } , Statements 2 and 3 are equivalent. Statement . The range of M θ is complemented in H m ⊗ E ∗ or there exists a submodule S of H m ⊗ E ∗ such that ran M θ . + S = H m ⊗ E ∗ . Statement . The quotient Hilbert module H θ is similar to H m ⊗ F for some Hilbert space F . Statement . Suppose H m ⊗ E ∗ is a skew direct sum S . + S , where E ∗ is a Hilbert spaceand S and S are submodules such that S is isomorphic to H m ⊗ E for some Hilbert space E . Then S is isomorphic to H m ⊗ F for some Hilbert space F . Note that Statements 4 and 6 imply Statement 5 and would be the converse to Corollary3.5.
Statement . If S is a complemented submodule of H m ⊗ E ∗ for some Hilbert space E ∗ , then S is isomorphic to H m ⊗ F for some Hilbert space F ? One can reformulate Statement 7 in the following equivalent form.
Statement . Every complemented submodule S of H m ⊗ E , for some Hilbert space E , is therange of M ψ for a multiplier ψ ∈ M ( E , F ) with ker M ψ = { } for some Hilbert space F . Note that one could view an affirmation of this statement as a weak form of the BLHT forDA-modules.Statement 7 raises an important issue for Hilbert modules: are the complemented submod-ules
S 6 = { } of R ⊗ C n always isomorphic to R ⊗ C k for some 0 < k ≤ n . This is certainly nottrue for a general Hilbert module R . However, what if R belongs to the class of “locally-free”Hilbert modules of multiplicity one which is the case for the DA-module H m ? For m = 1, anaffirmation follows trivially from the BLHT. A less obvious argument shows that the resultholds for more general “locally-free” Hilbert modules over the unit disk such as the Bergmanmodule. (Although the language is different, this result was proved by J. S. Fang, C. L. Jiang,X. Z. Guo, K. Ti and H. He. The study of the relationship between the eight statements inthe one-variable case is close to the theme of the book by C. L. Jiang and F. Wang [15], wheredetails can be found.) Further, one can establish an affirmation to Statement 6 if one assumesthat the multiplier θ ∈ M ( E , E ∗ ) is holomorphic on a neighborhood of the closure of B m , atleast if E and E ∗ are finite dimensional. However, what happens in general for “locally-free”Hilbert modules over B m , such as the DA-module, is not clear at this point.Moreover, a necessary condition for S and S with H m ⊗ C n = S . + S to be isomorphicto H m ⊗ C k and H m ⊗ C n − k , respectively, is the existence of a generating set { f , . . . , f n } for H m ⊗ C n with { f , . . . , f k } in S and { f k +1 , . . . , f n } in S . Note one can view each DOUGLAS, FOIAS, AND SARKAR f i ∈ O ( B m , C n ) for i = 1 , . . . , n . If one assumes in addition that the vectors { f i } are in M ( C n , C m ), then that is also sufficient.Note that given a complemented submodule S of H m ⊗ E , that is, for some submodule ˜ S one has S . + ˜ S = H m ⊗ E , there are many choices ˜˜ S so that the skew direct sum S . + ˜˜ S is isomorphic to H m ⊗ E ∗ for some Hilbert space E ∗ . (Here we allow a different space E ∗ .)It is not clear, but seems unlikely that there exists a canonical choice of E ∗ and ˜˜ S , in somesense, or what the “simplest” choice might be. Such ideas are related to the K -theory groupintroduced in [15].In commutative algebra one shows that a short exact sequence of modules0 −→ A ϕ −→ B ϕ −→ C −→ , splits, or ϕ has a right inverse, if and only if ϕ has a left inverse. Using the closed graphtheorem, we can extend this result to Hilbert modules. Moreover, with the CLT we canextend the result to the case where ϕ has a kernel. We begin with the simpler result. Theorem . Let θ ∈ M ( E , E ∗ ) be a multiplier for Hilbert spaces E and E ∗ such that ran M θ is closed. Then ran M θ is complemented in H m ⊗ E ∗ if and only if there exists a module map σ θ : H θ → H m ⊗ E ∗ which is a right inverse for π θ . Proof. If H m ⊗ E ∗ = ran M θ · + S for a (closed) submodule S , then Y = π θ | S is one-to-oneand onto. Hence Y − : ( H m ⊗ E ∗ ) / ran M θ → S is bounded by the closed graph theorem and σ θ = i Y − is a right inverse for π θ , where i : S → H m ⊗ E ∗ is the inclusion map.Conversely, if there exists a right inverse σ θ : H θ → H m ⊗E ∗ for π θ , then σ θ π θ is an idempotenton H m ⊗ E ∗ such that S = ran σ θ π θ is a complementary submodule for the closed submoduleran M θ in H m ⊗ E ∗ .For a multiplier θ ∈ M ( E , E ∗ ) one could consider the quotient of H m ⊗ E ∗ by the closure ofran M θ . Examples in the case m = 1 show that the existence of a right inverse for π θ does notimply that ran M θ is closed. Theorem 3.1 shows that the Statements 1 and 4 are equivalentbut the preceding comment implies the necessity of the assumption that ran M θ is closed.However, we do have the following folklore result which helps clarify matters. Remark . If θ ∈ M ( E , E ∗ ) for Hilbert spaces E and E ∗ and ran M θ is complemented in H m ⊗ E ∗ , then ran M θ is closed. As one knows, by considering the m = 1 case, there is more than one multiplier θ ∈M ( E , E ∗ ) for Hilbert spaces E and E ∗ with the same range and thus yielding the same quotient.While things are even more complicated for m >
1, the following result using the CLTintroduces some order.
Theorem . Let θ ∈ M ( E , E ∗ ) be an inner multiplier for Hilbert spaces E and E ∗ and ϕ ∈ M ( F , E ∗ ) for some Hilbert space F . Then there exists a multiplier ψ ∈ M ( F , E ) suchthat ϕ = θψ if and only if ran M ϕ ⊆ ran M θ . ESOLUTIONS OF HILBERT MODULES AND SIMILARITY 9
Proof. If ψ ∈ M ( F , E ) such that ϕ = θψ , then M ϕ = M θ M ψ and henceran M ϕ = ran M θ M ψ ⊆ ran M θ . Suppose ran M ϕ ⊆ ran M θ and θ ∈ M ( E , E ∗ ) is an inner multiplier. This implies thatran M θ is closed. Consider the module mapˆ M θ : ( H m ⊗ E ) / ker M θ −→ ran M θ defined by ˆ M θ γ θ = M θ , which is invertible since ran M θ is closed. Let γ θ : H m ⊗ E −→ ( H m ⊗ E ) / ker M θ be thequotient module map. Set ˆ X = ˆ M θ − . Then ˆ X : ran M θ → ( H m ⊗ E ) / ker M θ is boundedby the closed graph theorem and so is ˆ XM ϕ : H m ⊗ F → ( H m ⊗ E ) / ker M θ . Appealing tothe CLT yields a multiplier ψ ∈ M ( F , E ) so that γ θ M ψ = ˆ XM ϕ , and hence M θ M ψ = ( ˆ M θ γ θ ) M ψ = ˆ M θ ( ˆ XM ϕ ) = M ϕ , or ϕ = θψ which completes the proof.Note that the result holds for a multiplier θ ∈ M ( E , E ∗ ) for Hilbert spaces E and E ∗ so longas ran M θ is closed since that is all that the proof uses.We now consider our principal result on multipliers and regular inverses. Theorem . Let θ ∈ M ( E , E ∗ ) be a multiplier for Hilbert spaces E and E ∗ . Then thereexists ψ ∈ M ( E ∗ , E ) such that M θ M ψ M θ = M θ if and only if ran M θ is complemented in H m ⊗ E ∗ , or H m ⊗ E ∗ = ran M θ · + S , for some submodule S of H m ⊗ E ∗ . Proof. If H m ⊗ E ∗ = ran M θ · + S for some (closed) submodule S , then ran M θ is closed byRemark 3.2. Consider the module mapˆ M θ : ( H m ⊗ E ) / ker M θ −→ ( H m ⊗ E ∗ ) / S , defined by ˆ M θ γ θ = π S M θ , where γ θ : H m ⊗ E → ( H m ⊗ E ) / ker M θ and π S : H m ⊗ E ∗ → ( H m ⊗ E ∗ ) / S are quotient maps.This map is one-to-one and onto and thus has a bounded inverse ˆ X = ˆ M θ − : ( H m ⊗ E ∗ ) / S → ( H m ⊗ E ) / ker M θ by the closed graph theorem. Since ˆ X satisfies the hypotheses of the CLT,there exists ψ ∈ M ( E ∗ , E ) such that γ θ M ψ = ˆ Xπ S . Further, ˆ M θ γ θ = π S M θ yields π S M θ M ψ = ˆ M θ γ θ M ψ = ˆ M θ ˆ Xπ S = π S , and therefore π S ( M θ M ψ M θ − M θ ) = 0 . Since π S is one-to-one on ran M θ , it follows that M θ M ψ M θ = M θ .Now suppose there exists ψ ∈ M ( E ∗ , E ) such that M θ M ψ M θ = M θ . This implies that( M θ M ψ ) = M θ M ψ , and hence M θ M ψ is an idempotent. From the equality M θ M ψ M θ = M θ we obtain both thatran M θ M ψ contains ran M θ and that ran M θ M ψ is contained in ran M θ . Therefore,ran M θ M ψ = ran M θ , and S = ran ( I − M θ M ψ ) , is a complementary submodule of ran M θ in H m ⊗ E ∗ .Note that Theorem 3.4 implies that Statements 3 and 4 are equivalent. Corollary . Assume θ ∈ M ( E , E ∗ ) for Hilbert spaces E and E ∗ such that ran M θ is closedand H θ is defined by H m ⊗ E M θ −→ H m ⊗ E ∗ −→ H θ −→ . If H θ is similar to H m ⊗ F for some Hilbert space F , then the sequence splits. Proof.
First, assume that there exists an invertible module map X : H m ⊗ F → H θ , andlet ϕ ∈ M ( F , E ∗ ) be defined by the CLT so that π θ M ϕ = X , where π θ : H m ⊗ E ∗ → ( H m ⊗ E ∗ ) / ran M θ is the quotient map. Since X is invertible we have H m ⊗ E ∗ = ran M ϕ . + ran M θ . Thus ran M θ is complemented and hence it follows from the previous corollary that thesequence splits.Corollary 3.5 shows that Statement 6 implies Statement 1. If Statement 6 is valid, then theconverse to Corollary 3.5 holds. Moreover, we see that Statement 8 implies that H θ is similarto H m ⊗ F for some Hilbert space F or that Statement 4 is valid. Finally, the followingweaker converse to Corollary 3.5 always holds. Corollary . Let θ ∈ M ( E , E ∗ ) for Hilbert spaces E and E ∗ , and set H θ = ( H m ⊗E ∗ ) / clos [ ran M θ ] . Then the following statements are equivalent:(i) there exists ψ ∈ M ( E ∗ , E ) such that ψ ( x ) θ ( z ) = I E for z ∈ B m , and(ii) ran M θ is closed, ker M θ = { } and H θ is similar to a complemented submodule S of H m ⊗ E ∗ . Proof.
If (i) holds, then ran M θ is closed and ker M θ = { } . Further, M θ M ψ is an idempotenton H m ⊗ E ∗ such that ran M θ M ψ = ran M θ and H θ is isomorphic to S = ran ( I − M θ M ψ ) ⊆ H m ⊗ E ∗ and H m ⊗ E ∗ = ran M θ . + S so S is complemented. ESOLUTIONS OF HILBERT MODULES AND SIMILARITY 11
Now assume that (ii) holds and there exists an isomorphism X : H θ → S ⊆ H m ⊗ E ∗ , where S is a complemented submodule of H m ⊗ E ∗ . Then Y = Xπ θ : H m ⊗ E ∗ → H m ⊗ E ∗ is amodule map and hence there exists a multiplier ω ∈ M ( E ∗ , E ∗ ) so that Y = M ω . Since X isinvertible, ran M ω = S , which is complemented by assumption, and hence by Theorem 3.4there exists ψ ∈ M ( E ∗ , E ∗ ) such that M ω = M ω M ψ M ω or M ω ( I − M ψ M ω ) = 0. Therefore,ran ( I − M ψ M ω ) ⊆ ker M ω = ker Y = ker π θ = ran M θ . Applying Theorem 3.3, we obtain ϕ ∈ M ( E ∗ , E ) so that I − M ψ M ω = M θ M ϕ . Thus using M ω M θ = 0 we see that M θ M ϕ M θ = ( I − M ψ M ω ) M θ = M θ , or M θ = M θ M ϕ M θ . Since ker M θ = { } , we have M ϕ M θ = I H m ⊗E , which completes the proof.Combining Theorem 3.4 and Corollary 3.5 yields our main result in the commutative settingfor similarity. Corollary . Given θ ∈ M ( E , E ∗ ) for Hilbert spaces E and E ∗ such that ran M θ is closed,consider the quotient Hilbert module H θ defined as above. If H θ is similar to H m ⊗ F forsome Hilbert space F , then there exists a multiplier ψ ∈ M ( E ∗ , E ) satisfying θ ( z ) ψ ( z ) θ ( z ) = θ ( z ) , for z ∈ B m . In conclusion, Corollary 3.7 shows that Statement 5 implies Statement 3.4.
Resolutions of Hilbert modules over C [ z , . . . , z m ]Consideration of resolutions such as those in the preceding section and the ones givenin Theorem 2.2 raises the question of what kind of resolutions exist for pure co-sphericallycontractive Hilbert modules over C [ z , . . . , z m ]. In particular, Theorem 2.2 yields a uniqueresolution of an arbitrary pure co-spherically contractive Hilbert module M over C [ z , . . . , z m ]in terms of DA-modules and inner multipliers. More specifically, consider DA-modules { H m ⊗E k } for Hilbert spaces {E k } and inner multipliers ϕ k ∈ M ( E k , E k − ), or partially isometricmodule maps { M ϕ k } for k ≥
1; set X k = M ϕ k : H m ⊗ E k → H m ⊗ E k − , k ≥ X = π M : H m ⊗ E → M , which is exact. That is, ran X k +1 = ker X k for k ≥
1. Here k = 0 , , . . . , N , with thepossibility of N = + ∞ . A basic question is whether such a resolution can have finite lengthor, equivalently, whether we can take E N = { } for some finite N . That will be the case if and only if some X k is an isometry or, equivalently, if ker X k = { } . Unfortunately, thefollowing result shows that this is not possible when m >
1, unless M is a DA-module andthe resolution is a trivial one. Theorem . For m > , if V : H m ⊗ E → H m ⊗ E ∗ is an isometric module map for Hilbertspaces E and E ∗ , then there exists an isometry V : E → E ∗ such that V ( z k ⊗ x ) = z k ⊗ V x, for k ∈ N m , x ∈ E ∗ . Moreover, ran V is a reducing submodule of H m ⊗ E ∗ of the form H m ⊗ ( ran V ) . Proof.
For x ∈ E , k x k = 1, we have V (1 ⊗ x ) = f ( z ) = X k ∈ N m a k z k , for { a k } ⊆ E . Then V ( z ⊗ x ) = V M z (1 ⊗ x ) = M z V (1 ⊗ x ) = M z f = z f, and k z f k = k z V (1 ⊗ x ) k = k z ⊗ x k = 1 = k f k . Therefore, we have X k ∈ N m k a k k E ∗ k z k k = X k ∈ N m k a k k E ∗ k z k + e k , where k + e = ( k + 1 , . . . , k m ) , or X k ∈ N m k a k k E ∗ {k z k + e k − k z k k } = 0 . If k = ( k , . . . k m ), then k z k + e k = ( k + 1)! · · · k m !( k + · · · + k m + 1)! = k ! · · · k m !( k + · · · + k m )! k + 1 k + · · · + k m + 1 < k ! · · · k m !( k + · · · + k m )! = k z k k , unless k = k = . . . = k m = 0. Since, a k = 0 implies k z k + e k = k z k k we have k = · · · = k m = 0. Repeating this argument using i = 2 , . . . , m , we see that a k = 0 unless k = (0 , . . . , f ( z ) = 1 ⊗ y for some y ∈ E ∗ . Set V x = y to complete the first part of theproof.Finally, since ran V = H m ⊗ (ran V ), we see that ran V is a reducing submodule, whichcompletes the proof.Note that this result generalizes Corollary 3.3 of [9] and is related to an earlier result ofGuo, Hu and Xu [14].The theorem implies that all resolutions by DA-modules with partially isometric maps aretrivial in a sense we will make precise. We start with a definition. ESOLUTIONS OF HILBERT MODULES AND SIMILARITY 13
Definition . An inner resolution of length N , for N = 1 , , , . . . , ∞ , for a pure co-spherical contractive Hilbert module M is given by a collection of Hilbert spaces {E k } Nk =0 ,inner multipliers ϕ k ∈ M ( E k , E k − ) for k = 1 , . . . , N with X k = M ϕ k and a co-isometricmodule map X : H m ⊗ E → M so thatran X k = ker X k − , for k = 0 , , . . . , N . To be more precise, for N < ∞ one has the finite resolution −→ H m ⊗ E N X N −→ H m ⊗ E N − −→ · · · −→ H m ⊗ E X −→ H m ⊗ E X −→ M −→ , and for N = ∞ , the infinite resolution · · · −→ H m ⊗ E N X N −→ H m ⊗ E N − −→ · · · −→ H m ⊗ E X −→ H m ⊗ E X −→ M −→ . Theorem . If the pure, co-spherically contractive Hilbert module M possesses a finiteinner resolution, then M is isometrically isomorphic to H m ⊗ F for some Hilbert space F . Proof.
Applying the previous theorem to M ϕ N , we decompose E N − = E N − ⊕ E N − sothat ˜ M ψ N − = M N − | H m ⊗E N − ∈ L ( H m ⊗ E N − , H m ⊗ E N − ) is an isometry onto ran M N − .Hence, we can apply the theorem to ˜ M N − . Therefore, using induction we obtain the desiredconclusion.The following statement proceeds directly from the theorem. Corollary . If θ ∈ M ( E , E ∗ ) is an inner multiplier for the Hilbert spaces E and E ∗ withker M θ = { } , then the quotient module H θ = ( H m ⊗ E ∗ ) / ran M θ is isometrically isomorphicto H m ⊗ F for a Hilbert space F . Moreover, F can be identified with ( ran V ) ⊥ , where V isthe isometry from E to E ∗ given in Theorem 4.1. Note that in the preceding corollary, one has dim E ∗ = dim E + dim F .A resolution of M can always be made longer in a trivial way. Suppose we have theresolution 0 −→ H m ⊗ E N X N −→ H m ⊗ E N − −→ · · · −→ H m ⊗ E X −→ M −→ . If E N +1 is a nontrivial Hilbert space, then define X N +1 as the inclusion map of H m ⊗ E N +1 ⊆ H m ⊗ ( E N ⊕ E N +1 ). Further, set ˜ X N equal to X N on H m ⊗ E N ⊆ H m ⊗ ( E N +1 ⊕ E N ) and equalto 0 on H m ⊗ E N +1 ⊆ H m ⊗ ( E N ⊕ E N +1 ). Extending ˜ X N to all of H m ⊗ E N +1 linearly, weobtain a longer resolution essentially equivalent to the original one0 −→ H m ⊗ E N +1 X N +1 −→ H m ⊗ ( E N +1 ⊕ E N ) ˜ X N −→ · · · −→ M −→ . Moreover, the new resolution will be inner if the original one is.The proof of the preceding theorem shows that any finite inner resolution by DA-modulesis equivalent to a series of such trivial extensions of the resolution0 −→ H m ⊗ E X −→ H m ⊗ E −→ , for some Hilbert space E and X = I H m ⊗E . We will refer to such a resolution as a trivial innerresolution . We use that terminology to summarize this supplement to the theorem in thefollowing statement. Corollary . All finite inner resolutions for a pure co-spherically contractive Hilbertmodule M are trivial inner resolutions. What happens when we relax the conditions on the module maps { X k } so that ran X k =ker X k − for all k but do not require them to be partial isometries? In this case, non-trivialfinite resolutions do exist, completely analogous to what happens for the case of the Hardyor Bergman modules over C [ z , . . . , z m ] for m >
1. We describe a simple example.Consider the module C (0 , over C [ z , z ] defined so that p ( z , z ) · λ = p (0 , λ, where p ∈ C [ z , z ] and λ ∈ C , and the resolution: 0 −→ H X −→ H ⊕ H X −→ H X −→ C (0 , −→ , where X f = f (0 ,
0) for f ∈ H , X ( f ⊕ f ) = M z f + M z f for f ⊕ f ∈ H ⊕ H ,and X f = M z f ⊕ ( − M z f ) for f ∈ H . One can show that this sequence, which is closelyrelated to the Koszul complex, is exact and non-trivial; in particular, it does not split astrivial resolutions do.Another question one can ask is the relationship between the inner resolution for a pureco-spherically contractive Hilbert module and more general, not necessarily inner , resolutionsby DA-modules. In particular, is there any relation between the minimal length of a notnecessarily inner resolution and the inner resolution. Theorem 3.3 provides some informationon this matter.A parallel notion of resolution for Hilbert modules was studied by Arveson [3], which isdifferent from the one considered in this paper. For Arveson, the key issue is the behaviorof the resolution at ∈ B m or the localization of the sequence of connecting maps at . Hismain goal, which he accomplishes and is quite non trivial, is to extend an analogue of theHilbert’s syzygy theorem. In particular, he exhibits a resolution of Hilbert modules in hisclass which ends in finitely many steps. The resolutions considered in ([7], [6]) and this paperare related to dilation theory although the requirement that the connecting maps are partialisometries is sometimes relaxed.5. Hilbert modules over F [ Z , . . . , Z m ]Although we use the following lemma only in the non-commutative case, it also holds inthe commutative case as indicated. Lemma . If H is a co-spherically contractive Hilbert module over C [ z , . . . , z m ] or F [ Z , . . . , Z m ] ,respectively, which is similar to H m ⊗ F , or F m ⊗ F , respectively, for some Hilbert space F ,then H is pure. Proof.
We use the notation for the commutative case but the proof in both cases is thesame. Let X : H → H m ⊗ F be an invertible module map. Then M i = X − M z i X for all i = 1 , . . . , m . Since { P l H ( I H ) } ∞ l =0 is a decreasing sequence of positive operators, it suffices toshow that WOT − lim l →∞ P l H ( I H ) = 0 . ESOLUTIONS OF HILBERT MODULES AND SIMILARITY 15
To see that this is the case, let f be a vector in H and set f = X ∗− f . Then h X | k | = l M k M ∗ k f , f i = h X | k | = l X − M k z XX ∗ M ∗ k z X ∗− f , f i = h X | k | = l M k z XX ∗ M ∗ k z f, f i≤ k X k X | k | = l h M kz M ∗ kz f, f i . Letting l → ∞ in the last expression, we conclude that the required limit is zero, whichcompletes the proof.Actually, the proof shows that two similar co-spherically contractive Hilbert modules over C [ z , . . . , z m ], or two similar contractive Hilbert modules over F [ Z , . . . , Z m ], are either bothpure or both not pure. Theorem . Let H be a pure co-spherically contractive Hilbert module over F [ Z , . . . , Z m ] .Then H is similar to F m ⊗F for some Hilbert space F if and only if the characteristic operator Θ of H in L ( F m ⊗ E , F m ⊗ E ∗ ) , for some Hilbert spaces E and E ∗ , is left invertible; that is, ifand only if there exists a multi-analytic operator Ψ : F m ⊗ E ∗ → F m ⊗ E such that ΨΘ = I F m ⊗E . Proof.
First, using Theorem 2.3 we realize the pure contractive Hilbert module H as thequotient module given by its characteristic function Θ, which is an isometric multi-analyticmap. That is, H ∼ = H Θ = ( F m ⊗ E ∗ ) / Θ( F m ⊗ E ) . Now given a module map X : F m ⊗ F → H Θ , we appeal to the noncommutative analogue ofthe CLT (see Theorem 6.1 in [23] or Theorem 5.1 in [24]) to obtain a multi-analytic operatorΦ : F m ⊗ F → F m ⊗ E ∗ such that P H Θ Φ = X. Consider, the bounded module map Z : ( F m ⊗ F ) ⊕ ( F m ⊗ E ) → F m ⊗ E ∗ defined by Z ( f ⊕ g ) = Φ f + Θ g, for all f ⊕ g ∈ ( F m ⊗ F ) ⊕ ( F m ⊗ E ). Then Z is invertible if and only if X is invertible. Thisfollows by noting that X is invertible if and only if the range of Z , which is the span of H Θ and ran Θ is F m ⊗ E ∗ , and X is one-to-one if and only if Z is.To prove the necessity part of the theorem, assume that X is invertible or, equivalently,that Z is invertible. Consequently, we can define a module idempotent Q on F m ⊗ E ∗ suchthat Q Θ = Θand ran Q = ran Θ . Then the bounded module map ˆ Q : F m ⊗ E ∗ → F m ⊗ E defined byˆ Q (Φ f + Θ g ) = g, Φ f + Θ g ∈ ( F m ⊗ E ∗ ) satisfies Q = Θ ˆ Q. Since ˆ Q is a module map, there exists a multi-analytic operator Ψ : F m ⊗ E ∗ → F m ⊗ E suchthat ˆ Q = Ψ . Hence Θ = Q Θ = Θ ˆ Q Θ = ΘΨΘ . Since Θ is an isometry, the necessity part follows; that is, Θ has a left inverse.To prove the sufficiency part, let Ψ : F m ⊗ E ∗ → F m ⊗ E be a multi-analytic operator suchthat ΨΘ = I F m ⊗E . Then Q = ΘΨ is an idempotent on F m ⊗ E ∗ and any f in F m ⊗ E ∗ can be expressed as f = ( f − ΘΨ f ) + ΘΨ f, where f − ΘΨ f is in ker Ψ and ΘΨ f is in ran Θ. Thus,ran Q = ran Θ , and ker Ψ = ran ( I − Q ) . Since ker Ψ is a submodule of F m ⊗ E ∗ , by Theorem 2.4, the noncommutative version of theBLHT, there exists an inner multi-analytic operator Φ : F m ⊗ F → F m ⊗ E ∗ for some Hilbertspace F such that ker Ψ = ran ( I − Q ) = Φ( F m ⊗ F ) , Consequently, F m ⊗ E ∗ = ran Φ . + ran Θ . Then one can define the invertible module map Z as in the necessity part. Setting X = P H Θ Φdefines the required similarity between H Θ and F m ⊗ F , which completes the proof.As mentioned in the introduction, specializing the preceding proof to the (commutative) m = 1 case yields a new proof of the old result on the similarity of contraction operators tounilateral shifts.The main difference in the above proof and that of Corollary 3.7 for the commutative caseis that here we can assume that Θ has no kernel and one of the complemented submodule isisomorphic to a DA-module.In the proof of Theorem 5.2, we did not use the fact that the characteristic function isan isometry but the fact that ker Θ = { } and ran Θ is closed. Hence we can state a moregeneral result in terms of a module resolution. Theorem . Let E and E ∗ be Hilbert spaces and Θ : F m ⊗ E → F m ⊗ E ∗ be a multi-analyticoperator such that ker Θ = { } and ran Θ is closed. Then the quotient space H Θ , given by ( F m ⊗ E ∗ ) / ran Θ is similar to F m ⊗ F for some Hilbert space F if and only if ΘΨΘ = Θ forsome multi-analytic operator
Ψ : F m ⊗ E ∗ → F m ⊗ E . ESOLUTIONS OF HILBERT MODULES AND SIMILARITY 17 Concluding remarks
Observe that if H is a Hilbert module over C [ z , . . . , z m ] (or A (Ω), where Ω is a boundedconnected open subset of C m ), Corollary 3.7 remains true under the assumption that theanalogue of the CLT holds for the class of Hilbert modules under consideration. In particular,Corollary 3.7 can be generalized to any reproducing kernel Hilbert module where the kernelis given by a complete Nevanlinna-Pick kernel. References [1] W. B. Arveson,
Subalgebras of C ∗ -algebras. III. Multivariable operator theory , Acta Math. 181 (1998),no. 2, 159–228. MR 2000e:47013.[2] W. B. Arveson, The curvature invariant of a Hilbert module over C [ z , · · · , z d ], J. Reine Angew. Math.522 (2000), 173–236. MR 1758582.[3] W. B. Arveson, The free cover of a row contraction , Doc. Math. 9 (2004), 137–161 MR 2054985.[4] J. A. Ball, T. Trent and V. Vinnikov,
Interpolation and commutant lifting for multipliers on reproducingkernel Hilbert spaces , Operator theory and analysis (Amsterdam, 1997), 89–138, Oper. Theory Adv.Appl., 122, Birkh¨auser, Basel, 2001. MR 2002f:47028.[5] K. R. Davidson and T. Le,
Commutant Lifting for Commuting Row Contractions ,http://arxiv.org/abs/0906.4526[6] R. G. Douglas and G. Misra,
On quasi-free Hilbert modules , New York J. Math. 11 (2005), 547–561;MR 2007b:46044.[7] R. G. Douglas and G. Misra,
Quasi-free resolutions of Hilbert modules , Integral Equations OperatorTheory 47 (2003), no. 4, 435–456; MR 2004i:46109.[8] R. G. Douglas and V. I. Paulsen,
Hilbert Modules over Function Algebras , Research Notes in MathematicsSeries, 47, Longman, Harlow, 1989. MR 91g:46084.[9] R. G. Douglas and J. Sarkar,
On unitarily equivalent submodules , Indiana Univ. Math. J. 57 (2008), no.6, 2729–2743. MR 2482998[10] S. W. Drury,
A generalization of von Neumann’s inequality to the complex ball , Proc. Amer. Math. Soc.,68 (1978), 300304, MR 80c:47010.[11] C. Foias and A. Frazho,
The commutant lifting approach to interpolation problems , Operator Theory:Advances and Applications, 44. Birkh¨auser Verlag, Basel, 1990. MR 92k:47033.[12] D. C. V. Greene,
Free resolutions in multivariable operator theory , J. Funct. Anal. 200 (2003), no. 2,429–450. MR 2004c:47014[13] D. C. V. Greene, S. Richter and C. Sundberg,
The structure of inner multipliers on spaces with completeNevanlinna-Pick kernels , J. Funct. Anal. 194 (2002), 311–331. MR 2003h:46038[14] K. Guo, J. Hu and X. Xu,
Toeplitz algebras, subnormal tuples and rigidity on reproducing C [ z , . . . , z d ] -modules , J. Funct. Anal. 210 (2004), 214–247. MR 2005a:47007.[15] C. Jiang and Z. Wang, Structure of Hilbert space operators , World Scientific Publishing Co. Pvt. Ltd.,Hackensack, NJ, 2006. MR 2008j:47001.[16] S. McCullough and T. T. Trent,
Invariant subspaces and Nevanlinna-Pick kernels , J. Funct. Anal. 178(2000), no. 1, 226–249. MR 2002b:47006.[17] V. Muller and F.-H. Vasilescu,
Standard models for some commuting multioperators , Proc. Amer. Math.Soc. 117 (1993), 979–989. MR 93e:47016[18] B. Sz.-Nagy and C. Foias,
Sur les contractions de l’espace de Hilbert. X. Contractions similaires `a destransformations unitaires , Acta Sci. Math. (Szeged) 26 (1965), 79–91. MR 34 1856.[19] B. Sz.-Nagy and C. Foias,
On contractions similar to isometries and Toeplitz operators , Ann. Acad. Sci.Fenn. Ser. A I Math. 2 (1976), 553–564. MR 58 30376. [20] B. Sz.-Nagy and C. Foias,
Harmonic Analysis of Operators on Hilbert Space , North Holland, Amsterdam,1970. MR 43 947.[21] N. Nikolski,
Operators, functions, and systems: an easy reading. Vol. 2: Model operators and sys-tems , Mathematical Surveys and Monographs, 93. American Mathematical Society, Providence, RI.MR 2003i:47001b.[22] G. Popescu,
Isometric dilations for infinite sequences of noncommuting operators , Trans. Amer. Math.Soc. 316 (1989), no. 2, 523–536. MR 90c:47006.[23] G. Popescu,
Characteristic functions for infinite sequences of noncommuting operators , J. Operator The-ory 22 (1989), no. 1, 51–71. MR 91m:47012.[24] G. Popescu,
Operator theory on noncommutative varieties , Indiana Univ. Math. J. 55 (2006), no. 2,389–442. MR 2007m:47008.
Texas A & M University, College Station, Texas 77843, USA
E-mail address : [email protected] E-mail address : [email protected] (Present Address of Jaydeb Sarkar) Department of Mathematics, The University of Texas atSan Antonio, San Antonio, TX 78249, USA
E-mail address ::