Kernels of Hilbert Module Maps: A Counterexample
aa r X i v : . [ m a t h . OA ] J a n Kernels of Hilbert Module Maps:A Counterexample
Jens Kaad and Michael SkeideJanuary 2021
Abstract
Answering a long standing question, we give an example of a Hilbert module and a nonzerobounded right linear map having a kernel with trivial orthogonal complement. In particular,this kernel is di ff erent from its own double orthogonal complement. * MSC 2020: 46L08. Keywords: Hilbert modules, kernels, right linear bounded maps, orthogonal complement,hereditary subalgebra. Introduction
The complement of a subset S of a pre-Hilbert module E is defined as S ⊥ : = { x ∈ E : h S , x i = { }} . It is well known that, given a subset S of a Hilbert space H , the double complement S ⊥⊥ = ( S ⊥ ) ⊥ can be computed as S ⊥⊥ = span S . Since a bounded linear functional that vanishes on S also vanishes on span S , it vanishes on S ⊥⊥ . In particular, the kernel of a bounded linear functional on a Hilbert space coincides withits double complement. (In some papers this is phrased, saying the kernel is orthogonally com-plete.)The same sort of reasoning shows that the kernel of a bounded B –linear map from a vonNeumann B –module (or W ∗ –module [1] ) E into B coincides with its double complement.But, is it true for Hilbert modules? The question in its purest form is:Given a bounded right linear map Φ from a Hilbert B –module E to B , does Φ ( S ) = { } for a subset S ⊂ E with S ⊥ = { } imply that Φ = a from a Hilbert B –module E to a Hilbert B –module E ′ , do we always have ker a = ( ker a ) ⊥⊥ ?These two questions have the same answer. Moreover, for special cases the answer is yes, forinstance, if the map a is required to be adjointable, or if span S B is an ideal submodule (Bakicand Guljas [BG02]) or a closed ternary ideal (Skeide [Ske18]).Several publications point out pleasant consequences of a positive answer to the above ques-tions. (See Frank [Fra02], Bhat and Skeide [BS15, Footnotes 1-3], and [Ske18, Footnote 7].)But, is it true? The scope of this short note is to illustrate by an explicit counterexample that,unfortunately, the general answer is no: There exist a Hilbert B –module E, a bounded B –linear map Φ : E → B , and aclosed submodule F such thatE , ker Φ ⊃ F and F ⊥ = { } . In particular, ker Φ ⊥⊥ ⊃ F ⊥⊥ = E , ker Φ . [1] A W ∗ –module is a self-dual Hilbert module over a W ∗ –algebra; this definition is probably due to Baillet,Denizeau, and Havet [BDH88]. A von Neumann module is a Hilbert module over a von Neumann algebra that isstrongly closed in a uniquely associated operator space; see Skeide [Ske00]. E is a standard Hilbertmodule over a separable unital commutative C ∗ –algebra and F is a full submodule. [2] Before diving into the example in the next section, let us briefly discuss why, we think, thestatement we disprove in this note is suspicious right from the beginning.Hilbert modules are in particular Banach spaces and, of course, share all the good propertiesof general Banach spaces. But as far as their geometric properties are concerned, frequentlyHilbert modules behave much more like pre-Hilbert spaces than like Hilbert spaces. (Closedsubmodules need not be complemented; bounded right linear maps need not have an adjoint; infact, an isometry has an adjoint if and only if its range is complemented. All these failures, inthe end, go back to the fact that a Hilbert module need not be self-dual; and our example is noexception.) And for pre-Hilbert spaces the answer is no. We give a quick example, which weowe to Orr Shalit. This example both indicates how to find a counterexample and explains whatgoes wrong in existing attempts to find a positive answer to the above questions.
Let H be a pre-Hilbert space with orthonormal Hamel basis (cid:0) e n (cid:1) n ∈ N and let S : = { e n − e n + } . Then S ⊥ is { } in H . But, S ⊥ is not { } in the completion H . Therefore, for everyelement ϕ , H , the functional h ϕ, •i vanishes on S , but not on H .Under the passage from H to H , the set S loses the property to be 0–complemented. So,knowing the answer is yes for Hilbert spaces, does not help proving the same for pre-Hilbertspaces by embedding the latter into the former. Based on the observation that for von Neumannmodules the answer is a ffi rmative, there have been attempts to prove the same for Hilbert mod-ules by embedding the latter into the former. Specifically for the embedding into the bidualit follows from a careful analysis of results by Akemann and Pedersen [AP73] (see [Ped79,Section 3.11] in Pedersen’s book) that this can never work.We end this note by providing an application to general C ∗ –algebra theory concerning rep-resentations of hereditary C ∗ –subalgebras. Let B be a C ∗ –algebra. Recall that for a set S the standard Hilbert B –module over S is definedas B S = n(cid:0) b s (cid:1) s ∈ S : X s ∈ S b ∗ s b s exists as norm limit over the finite subsets of S o . If B is unital, then B S has an orthonormal basis (cid:0) e s (cid:1) s ∈ S defined by e s : = (cid:0) δ s , s ′ (cid:1) s ′ ∈ S . It is wellknown that the formula β ∗ s = Φ ( e s ) [2] This does not mean that the statements in [Fra02, BS15, Ske18] cannot be true. In fact, Shalit and Skeide[SS20, Appendix B] prove, by a di ff erent method, a result that would have followed from a positive answer to theabove questions; this is what made the second named author start to think about them. B ) a one-to-one correspondence between bounded right linear maps Φ : B S → B and families (cid:0) β s (cid:1) s ∈ S ( β s ∈ B ) such that there exists a constant M ≥ (cid:13)(cid:13)(cid:13)(cid:13)X s ∈ S ′ β ∗ s β s (cid:13)(cid:13)(cid:13)(cid:13) ≤ M for all finite subsets S ′ ⊂ S . Indeed, suppose that we are given a family (cid:0) β s (cid:1) s ∈ S for which sucha constant M ≥ B S ′ : = (cid:0) β s χ S ′ ( s ) (cid:1) s ∈ S ∈ B S , from k B S ′ B ∗ S ′ k = kh B S ′ , B S ′ ik we see that the net (cid:0) B S ′ B ∗ S ′ (cid:1) S ′ of rank-one operators is bounded by M . Clearly, we havelim S ′ h B S ′ , e s i = Φ ( e s ). This means, the bounded net (cid:0) h B S ′ , •i (cid:1) S ′ converges strongly to Φ onthe dense subset of finite B –linear combinations of the basis vectors e s , hence, everywhere. SeeManuilov and Troitsky [MT05, Section 2.5] for a di ff erent proof for sequences. By the standardHilbert B –module, we mean B N , frequently also written as H B or B ∞ .For our example we fix B : = C [0 , q ∈ (0 ,
1] choose a sequence of elements ( ψ q , m ) m ∈ N in B such that ∞ X m = | ψ q , m ( t ) | = χ [0 , q ) ( t )pointwise for t ∈ [0 , f q , m : t max(0 , min(1 , m ( q − t ))) and f q , = ψ q , m : = p f q , m − f q , m − will do.) It follows that Ψ q = (cid:0) ψ q , m (cid:1) m ∈ N defines a bounded rightlinear map B N → B . (Since (cid:0) ψ q , m (cid:1) m ∈ N is not an element of B N , the map Ψ q is not adjointable.)Next, we put E : = B ⊕ ( B N ) ∞ , and, with the set A : = { } ∪ ( N × N ), we shall write E as E = B A . The index 0 ∈ A refers to the separate copy of B . The second index m of ( n , m ) ∈ A refers to the m th element of the sequence in B N , while the first index n refers to the number of the copy of B N we are talking about.Let (cid:0) q n (cid:1) n ∈ N be a sequence of elements in (0 , Φ : (cid:0) b a (cid:1) a ∈ A b + ∞ X n = − n Ψ q n (cid:16)(cid:0) b ( n , m ) (cid:1) m ∈ N (cid:17) , we define a nonzero right linear bounded map. In fact, Φ : E → B is surjective.If we define the elements ζ k ,ℓ : = (cid:0) δ a , − k ψ ∗ q k ,ℓ − δ a , ( k ,ℓ ) (cid:1) a ∈ A in E , then Φ ( ζ k ,ℓ ) =
0. So, for S : = { ζ k , l : k , ℓ ∈ N } the closed submodule F : = span S B iscontained in ker Φ .For the punchline, we are done proving Theorem 1.1, if we show that the sequence ( q n ) n ∈ N can be chosen such that S ⊥ = { } . 4 .1 Lemma. Suppose that the set { q n : n ∈ N } is dense [0 , . Then S ⊥ = { } . P roof . Let (cid:0) b a (cid:1) a ∈ A ∈ B A and suppose that (cid:10) ζ n , m , (cid:0) b a (cid:1) a ∈ A (cid:11) = n , m . Then2 − n ψ q n , m b − b n , m = . ( ∗ )That is, each b n , m is determined by b . In particular, if b =
0, then so is b .Suppose that b , b n , m as determined by ( ∗ ). Then by density of the subset { q n : n ∈ N } ⊂ [0 ,
1] and continuity of b there are n ∈ N , ε >
0, and d > | b ( t ) | ≥ d for all t ∈ ( q n − ε, q n + ε ) ∩ [0 , f ( t ) : = ∞ X m = | b n , m ( t ) | = ∞ X m = − n | ψ q n , m ( t ) b ( t ) | = − n | b ( t ) | χ [0 , q n ) ( t )for all t ∈ [0 , χ [0 , q n ) ( t ) has a discontinuity at the point q n ∈ (0 ,
1] and since | b ( t ) | ≥ d on a neighbuorhood of q n , also f is not continuous at q n . It follows that P ∞ m = | b n , m | does notconverge uniformly in [0 , (cid:0) b a (cid:1) a ∈ A < B A . Consequently, if (cid:0) b a (cid:1) a ∈ A ∈ S ⊥ ⊂ B A , then b = (cid:0) b a (cid:1) a ∈ A = The following result about representations of hereditary C ∗ –subalgebras might be of interest forgeneral C ∗ –algebra theory. There is a C ∗ –algebra A with hereditary C ∗ –subalgebra B separating the pointsof A (that is, a B = { } implies a = ) such that there is no faithful representation π of A suchthat π ( B ) acts nondegenerately on the representation space. P roof . In the example from the previous section, the C ∗ –algebra K ( E ) and its hereditary sub-algebra K ( F ) do the job. Indeed, the Hilbert K ( E )–module K ( E ) may be identified with the(internal) tensor product E ⊙ E ∗ , and it contains K ( E , F ) = F ⊙ E ∗ as zero-complemented sub-module. Φ gives rise to a nonzero bounded right-linear map Φ ⊙ id E ∗ : E ⊙ E ∗ → E ∗ that vanisheson F ⊙ E ∗ .Let π : K ( E ) → B ( H ) be a representation. Then E ⊙ E ∗ ⊙ H = span π ( K ( E )) H via theidentification x ′ ⊙ x ∗ ⊙ h π ( x ′ x ∗ ) h . So, suppose π is nondegenerate, so that E ⊙ E ∗ ⊙ H = H .The map Φ induces the bounded map Φ ⊙ id E ∗ ⊙ id H from H = E ⊙ E ∗ ⊙ H to E ∗ ⊙ H . Suppose π is faithful, so that Φ ⊙ id E ∗ ⊙ id H , Φ = Φ ⊙ id E ∗ ⊙ id H vanishes on F ⊙ E ∗ ⊙ H = span π ( F ⊙ E ∗ ) H ⊃ span π ( F ⊙ F ∗ ) H , the representation π restrictedto K ( F ) = F ⊙ F ∗ cannot be nondegenerate (because, otherwise, Φ would be 0).5 cknowledgments: This little note is an output of a talk by the second author at the “Hilbert C ∗ –Modules Online Weekend” from December 5 /
6, 2020, presenting the open problem whichwas then answered by the first author. We are deeply grateful to the organizers of the WorkshopMichael Frank, Vladimir Manuilov, and Evgenij Troitsky for having made this possible. Wealso acknowledge useful discussions with Michael Frank, Boris Guljas, and Orr Shalit.
References [AP73] C.A. Akemann and G.K. Pedersen,
Complications of semicontinuity in C ∗ –algebratheory , Duke Math. J. (1973), 785–795.[BDH88] M. Baillet, Y. Denizeau, and J.-F. Havet, Indice d’une esperance conditionnelle ,Compositio Math. (1988), 199–236.[BG02] D. Bakic and B. Guljas, On a class of module maps of Hilbert C ∗ –modules , Math.Commun. (2002), 177–192.[BS15] B.V.R. Bhat and M. Skeide, Pure semigroups of isometries on Hilbert C ∗ –modules ,J. Funct. Anal. (2015), 1539–1562, electronically Jun 2015. Preprint, arXiv:1408.2631.[Fra02] M. Frank, On Hahn-Banach type theorems for Hilbert C ∗ –modules , Int. J. Math. (2002), 675–693.[MT05] V.M. Manuilov and E.V. Troitsky, Hilbert C ∗ –modules , Translations of MathematicalMonographs, no. 226, American Mathematical Society, 2005.[Ped79] G.K. Pedersen, C ∗ –algebras and their automorphism groups , Academic Press, 1979.[Ske00] M. Skeide, Generalized matrix C ∗ –algebras and representations of Hilbert modules ,Mathematical Proceedings of the Royal Irish Academy (2000), 11–38, (Cott-bus, Reihe Mathematik 1997 / M-13).[Ske18] ,
Ideal submodules versus ternary ideals versus linking ideals , Preprint, ar-Xiv: 1804.05233v4, 2018, (to appear in Algebr. Represent. Theory).[SS20] O.M. Shalit and M. Skeide,
CP-Semigroups and dilations, subproduct systemsand superproduct systems: The multi-parameter case and beyond , Preprint, arXiv:2003.05166v1, 2020.Jens Kaad:
Department of Mathematics and Computer Science, The University of Southern Denmark,Campusvej 55, DK-5230 Odense M, Denmark, E-mail: [email protected]
Homepage: https://portal.findresearcher.sdu.dk/en/persons/kaad
Michael Skeide:
Dipartimento di Economia, Universit`a degli Studi del Molise, Via de Sanctis, 86100Campobasso, Italy, E-mail: [email protected]
Homepage: http://web.unimol.it/skeide/http://web.unimol.it/skeide/