A Delorme-Guichardet Theorem for quantum groups - Hausdorff ...
A Delorme-Guichardet Theorem for quantum groups - Hausdorff ...
A Delorme-Guichardet Theorem for quantum groups - Hausdorff ...
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A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 3Notation. Throughout the paper, the symbol ⊙ will be used to denote algebraictensor products while the symbol ¯⊗ will be used to denote tensor products inthe category of Hilbert spaces or in the category of von Neumann algebras. Alltensor products between C ∗ -algebras are assumed minimal/spatial and these willbe denoted by the symbol ⊗. All Hilbert spaces are assumed to be complexand their inner products are assumed linear in the first variable. Furthermore,∗-representations of unital algebras on Hilbert spaces will always be assumed tobe unit-preserving.Acknowledgements. The author wishes to thank Pierre Fima, Ryszard Nest, JessePeterson and Roland Vergnioux <strong>for</strong> discussions revolving around the notion ofproperty (T). The work presented in the paper was initiated during the authorsstay at the <strong>Hausdorff</strong> Research Institute <strong>for</strong> Mathematics and it is a pleasure tothank the organizers, Wolfgang Lück and Nicolas Monod, as well as the participantsof the trimester program on Rigidity.1. Preliminaries on <strong>quantum</strong> <strong>groups</strong>We choose here the approach to compact <strong>quantum</strong> <strong>groups</strong> developed by Woronowiczin [Wor87a], [Wor87b] and [Wor98]. Thus, a compact <strong>quantum</strong> group Gconsists of a (not necessarily commutative) separable, unital C ∗ -algebra C(G)together with a unital, coassociative ∗-homomorphism ∆: C(G) → C(G)⊗C(G)satisfying a certain density condition. The map ∆ is referred to as the comultiplication.Such a <strong>quantum</strong> group possesses a unique Haar state; i.e. a stateh: C(G) → C such that(id⊗h)∆a = h(a)1 = (h⊗id)∆(a) <strong>for</strong> every a ∈ C(G).The GNS-construction applied to the Haar state yields a separable Hilbert spaceL 2 (G) together with a representation λ: C(G) → B(L 2 (G)) and a linear mapΛ: C(G) → L 2 (G) with dense image. In general, the Haar state need not befaithful and hence λ might have a kernel. We denote by C(G red ) the imageλ(C(G)). This C ∗ -algebra inherits a <strong>quantum</strong> group structure from G and thecomultiplication ∆ red on C(G red ) is implemented by the so-called multiplicativeunitary W ∈ B(L 2 (G)¯⊗L 2 (G)) given byW ∗ (Λ(a)⊗Λ(b)) = Λ⊗Λ(∆(b)(a⊗1)).The statement that W implements ∆ red means that ∆ red (λ(a)) = W ∗ (1⊗λ(a))W<strong>for</strong> every a ∈ C(G). One notes that the right hand side of this <strong>for</strong>mula makessense <strong>for</strong> any a ∈ B(L 2 (G)) and it turns out that the enveloping von Neumannalgebra L ∞ (G) = C(G red ) ′′ is turned into a compact von Neumann algebraic<strong>quantum</strong> group (see [KV03]) when endowed with this map as comultiplication.A unitary corepresentation of G on a Hilbert space H is of a unitary elementu ∈ M(K(H) ⊗ C(G)) such that (id⊗∆)u = u (12) u (13) . Here K(H) denotes
4 DAVID KYEDthe compact operators on H, M(K(H) ⊗ C(G)) is the multiplier algebra ofK(H) ⊗ C(G) and the subscripts (12) and (13) is the standard leg-numberingconvention. Representation theoretic notions from the theory of compact <strong>groups</strong>,such as direct sums, tensor products, intertwiners and irreducibility, have naturalcounterparts in the corepresentation theory <strong>for</strong> compact <strong>quantum</strong> <strong>groups</strong>. Inparticular the following important theorem holds true.<strong>Theorem</strong> 1.1 (Woronowicz). Any irreducible unitary corepresentation of G isfinite dimensional and an arbitrary unitary corepresentation decomposes as adirect sum irreducible ones.We denote by Irred(G) the set of equivalence classes of irreducible, unitarycorepresentations of G. The separability assumption on C(G) together with the<strong>quantum</strong> Peter-Weyl theorem [Wor98] ensures that Irred(G) is a countable set.We label its elements by an auxiliary countable set I and choose <strong>for</strong> each α ∈ Ia Hilbert space H α and a concrete representative u α ∈ B(H α )⊗C(G). Abusingnotation slightly, we shall often identify the index α with the corresponding classof u α . Fix an orthonormal basis {e 1 ,...,e nα } <strong>for</strong> H α and consider the correspondingfunctionals ω ij : B(H α ) → C given by ω ij (T) = 〈Te i |e j 〉. The matrixcoefficients of u α , relative to the chosen basis, are then defined asu α ij = (ω ij ⊗id)u α ∈ C(G).It turns out that these matrix coefficients are linearly independent and that theirlinear span constitutes a dense ∗-subalgebra Pol(G) of C(G). Furthermore, thecomultiplication descends to a comultiplication ∆: Pol(G) → Pol(G) ⊙ Pol(G)and with this comultiplication Pol(G) becomes a Hopf ∗-algebra; i.e. there existsan antipode S: Pol(G) → Pol(G) as well as a counit ε: Pol(G) → C satisfyingthe standard Hopf ∗-algebra relations [KS97]. The fact that Pol(G) is spannedby matrix coefficients arising from finite dimensional, unitary corepresentationsof G also ensures that the relation‖a‖ u = sup{‖π(x)‖ | π: Pol(G) → B(H) a ∗-representation}defines a C ∗ -norm ‖ · ‖ u on Pol(G). The C ∗ -completion of Pol(G) with respectto this norm is called the universal C ∗ -algebra associated with G and is denotedC(G u ). By definition of ‖·‖ u , the comultiplication extends to a comultiplication∆ u : C(G u ) → C(G u ) ⊗ C(G u ) turning C(G u ) into a compact <strong>quantum</strong> group.Note that the ∗-representations of C(G u ) are in one-to-one correspondence withthe ∗-representations of Pol(G) via restriction/extension.Example 1.2. Thefundamentalexample, onwhichthegeneraldefinitionismodeled,is obtained by considering a compact, second countable, <strong>Hausdorff</strong> topologicalgroup G and its commutative C ∗ -algebra C(G) of continuous, complex valuedfunctions. In this case the comultiplication is the Gelfand dual of the multiplicationmap G ×G → G and the Haar state is given by integration against theunique Haar probability measure µ on G. The GNS-space there<strong>for</strong>e identifies
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 5with L 2 (G,µ) and the representation λ with the action of C(G) on L 2 (G,µ)by pointwise multiplication. Similarly, the von Neumann algebra identifies withL ∞ (G,µ) and the Hopf ∗-algebra Pol(G) is the subalgebra of C(G) generated bymatrix coefficients arising from irreducible, unitary representations of G. Theantipode in Pol(G) is the Gelfand dual of the inversion map and the counit isgiven by evaluation at the identity element in G.In the previous example there is no real difference between the reduced anduniversal version of the compact <strong>quantum</strong> group. The next example, however,will illustrate this difference more clearly.Example 1.3. Consider a countable, discrete group Γ. Denote by Cred ∗ (Γ) itsreduced group C ∗ -algebra acting on l 2 (Γ) via the left regular representation anddefine a comultiplication on group elements by ∆ red γ = γ⊗γ. This turns Cred ∗ (Γ)into a compact <strong>quantum</strong> group whose Haar state is given by the natural traceon Cred ∗ (Γ). Hence the GNS-space and GNS-representation can be identified,respectively, with l 2 (Γ) and the left regular representation, and the envelopingvon Neumann algebra is there<strong>for</strong>e nothing but the group von Neumann algebraL(Γ). Each element in Γ is a one-dimensional corepresentation <strong>for</strong> this <strong>quantum</strong>group and the Hopf ∗-algebra there<strong>for</strong>e identifies with the complex group algebraCΓ. Thus, the universal C ∗ -algebra is, by definition, equal to the maximal groupC ∗ -algebra Cu(Γ).∗Remark 1.4. The three C ∗ -algebras C(G),C(G red ) and C(G u ), together withtheir comultiplications, can be thought of as ”different pictures of the same <strong>quantum</strong>group”,eachhaving itsadvantagesanddisadvantages. Forinstance, whereasthe the Haar state is always faithful on C(G red ) this is in general not the caseon C(G u ) and, conversely, the counit is always well defined on all of C(G u ) butnot necessarily on C(G red ). The latter difference is the fundamental observationleading to the notion of (co)-amenability <strong>for</strong> <strong>quantum</strong> <strong>groups</strong> as studied in[BMT01].Any compact <strong>quantum</strong> group G has a dual <strong>quantum</strong> group Ĝ of so-calleddiscrete type. As in the compact case, Ĝ comes with both a C ∗ -algebra c 0 (Ĝ)∞and a von Neumann algebra l (Ĝ) defined, respectively, asc 0c 0 (Ĝ) = ⊕l ∞B(H α ) and l ∞ (Ĝ) = ∏B(H α ).α∈IInthediscretepicturewewillprimarilybeworkingwiththevonNeumannalgebral ∞ (Ĝ). This algebra is endowed with a natural comultiplication ˆ∆: l ∞ (Ĝ) →l ∞ ∞(Ĝ)¯⊗l (Ĝ) arising from the <strong>quantum</strong> group structure on G. Since c0(Ĝ) is adirect sum of finite dimensional C ∗ -algebras we have isomorphismsl ∞l ∞ (Ĝ)¯⊗B(H) ≃ M(c 0 (Ĝ)⊗B(H)) ≃ ∏B(H α )⊗B(H)α∈Iα∈I
6 DAVID KYED<strong>for</strong> any Hilbert space H. For an element T ∈ l ∞ (Ĝ)¯⊗B(H) we will denote by(T α ) α∈I the corresponding element in ∏ l ∞α∈I B(Hα ) ⊗ B(H) and in the sequelwe will freely identify T and (T α ) α∈I . By a unitary corepresentation of Ĝ on aHilbert space H we shall mean a unitary operator V ∈ l ∞ (Ĝ)¯⊗B(H) satisfying(ˆ∆⊗id)V = V (13) V (23) .Consider the unitary V u = (u α ) α∈I ∈ ∏ l ∞α∈I B(H α) ⊗ C(G u ). This unitary encodesthe duality between G and Ĝ in the following sense: <strong>for</strong> every unitarycorepresentation V ∈ l ∞ (Ĝ)¯⊗B(H) of Ĝ there exists a unique ∗-representationπ V : C(G u ) → B(H) such that(id⊗π V )u α = V α <strong>for</strong> each α ∈ I.Conversely, every ∗-representation π: C(G u ) → B(H) defines a unitary corepresentationof Ĝ on H by the above relation. For detailed introductions to compact<strong>quantum</strong> <strong>groups</strong> and their discrete duals we refer the reader to the survey articles[MVD98] and [KT99] as well as the original articles [VD96] and [Wor98].The notion of property (T) was recently introduced in the <strong>quantum</strong> <strong>groups</strong>etting by Fima in [Fim08] and the definition is as follows.Definition 1.5 (Fima). Let Ĝbe a discrete <strong>quantum</strong> group andconsideraunitarycorepresentation V = (V α ) α∈I ∈ l ∞ (Ĝ)¯⊗B(H) of Ĝ on a Hilbert space H.(i) A vector ξ ∈ H is said to be invariant if V α (η ⊗ξ) = η ⊗ξ <strong>for</strong> all α ∈ Iand η ∈ H α .(ii) For a finite, non-empty subset E ⊆ Irred(G) and a δ > 0 a vector ξ ∈ His called (E,δ)-invariant if‖V α (η ⊗ξ)−η ⊗ξ‖ < δ‖η‖‖ξ‖ <strong>for</strong> all α ∈ E and all η ∈ H α ,and V is said to have almost-invariant vectors if it has an (E,δ)-invariantvector <strong>for</strong> each finite, non-empty E ⊆ Irred(G) and each δ > 0.(iii) The discrete <strong>quantum</strong> group Ĝ is said to have property (T) if any unitarycorepresentation of Ĝ with almost invariant vectors has a non-zeroinvariant vector.Remark 1.6. For notational smoothness we will adopt the convention that,unless explicitly specified otherwise, subsets E of Irred(G) are always both finiteand non-empty.The main results in [Fim08] are summarized in the following theorem.<strong>Theorem</strong> 1.7 (Fima). If Ĝ is a discrete <strong>quantum</strong> group with property (T) thenthe following holds.(i) The <strong>quantum</strong> group is automatically a Kac-algebra; i.e. the Haar stateh: C(G) → C is a trace.
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 7(ii) The discrete <strong>quantum</strong> group is finitely generated; i.e. the corepresentationcategory Corep(G) of the compact dual is a finitely generated tensorcategory.(iii) The <strong>quantum</strong> group allows Kazhdan pairs; i.e. <strong>for</strong> every finite subsetE ⊆ Irred(G) generating the corepresentation category and containingthe trivial corepresentation there exists a δ > 0 such that whenever V isa unitary corepresentation of Ĝ having an (E,δ)-invariant vector, then Vhas a non-trivial invariant vector.Moreover, if G is an infinite, compact <strong>quantum</strong> group such that L ∞ (G) is a factorthen Ĝ has property (T) iff L∞ (G) is a type II 1 -factor with property (T) in thesense of Connes-Jones [CJ85].In the following section we show how property (T) can be expressed using∗-representations of the Hopf ∗-algebra Pol(G).2. Property (T) from the dual point of viewIn this section we re<strong>for</strong>mulate property (T) <strong>for</strong> discrete <strong>quantum</strong> <strong>groups</strong> interms of their compact duals and use this description to give a spectral characterizationof property (T). In the compact setting it is natural to consider thefollowing notions of invariance and almost invariance.Definition 2.1. Let π: Pol(G) → B(H) be a ∗-representation. A vector ξ ∈ H issaid to be invariant if π(a)ξ = ε(a)ξ <strong>for</strong> all a ∈ Pol(G). If a non-zero invariantvector exists, π is said to contain the counit. For a subset E ⊆ Irred(G) andδ > 0 a vector ξ ∈ H is said to be (E,δ)-invariant if‖π(u α ij )ξ −ε(uα ij )ξ‖ < δ‖ξ‖,<strong>for</strong> all α ∈ E and i,j ∈ {1,...,n α }. The representation π is said to havealmost invariant vectors if it allows a non-zero (E,δ)-invariant vector <strong>for</strong> everyE ⊆ Irred(G) and every δ > 0.Remark 2.2. Since the set {u α ij | α ∈ I,1 ≤ i,j ≤ n α} spans Pol(G) linearly it isnot difficult to see that a representation π: Pol(G) → B(H) has almost invariantvectors iff there exists a sequence of unit vectors ξ n ∈ H such that<strong>for</strong> every a ∈ Pol(G).lim ‖π(a)ξ n −ε(a)ξ n ‖ = 0n→∞The following proposition contains the translation of property (T) from thediscrete to the compact picture.Proposition 2.3. Let Ĝ be a discrete <strong>quantum</strong> group and consider a unitarycorepresentation V ∈ l ∞ (Ĝ)¯⊗B(H) as well as the the corresponding ∗-representationπ V : Pol(G) → B(H). Let furthermore E ⊆ Irred(G) and δ > 0 be givenand define K E = max{n α | α ∈ E}. Then the following holds.
8 DAVID KYED(i) A vector ξ ∈ H is V-invariant if and only if it is π V -invariant.(ii) If ξ ∈ H is (E,δ)-invariant <strong>for</strong> V then it is also (E,δ)-invariant <strong>for</strong> π V .(iii) If ξ ∈ H is (E,δ)-invariant <strong>for</strong> π V then it is (E,K E δ)-invariant <strong>for</strong> V.Thus, V has almost-invariant vectors if and only if π V has almost invariantvectors. In particular, the discrete <strong>quantum</strong> group Ĝ has property (T) iff any ∗-representation π: Pol(G) → B(H) with almost invariant vectors has a non-zeroinvariant vector.Proof. Consider the fixed basis {e 1 ,...,e nα } of H α and the corresponding functionalse ′ i: H α → C and ω ij : B(H α ) → C given, respectively, bye ′ i (x) = 〈x|e i〉 and ω ij (T) = 〈Te i |e j 〉.A vector ξ ∈ H is V-invariant exactly when V α (e j ⊗ξ) = e j ⊗ξ <strong>for</strong> any α ∈ Iand j ∈ {1,...,n α }. This in turn holds iff(e ′ i ⊗id)V α (e j ⊗ξ) = e ′ i (e j)ξ <strong>for</strong> all α ∈ I and i,j ∈ {1,...,n α }.Keeping in mind that ε(u α ij ) = δ ij, the above equation translates intoπ V (u α ij)ξ = ε(u α ij)ξ <strong>for</strong> all α ∈ I and i,j ∈ {1,...,n α },which is equivalent to ξ being π V -invariant since the u α ij ’s constitute a linear basis<strong>for</strong> Pol(G). This proves (i). To prove (ii), fix E ⊆ Irred(G) and δ > 0 and assumethat ξ ∈ H is an (E,δ)-invariant unit vector <strong>for</strong> V. Then <strong>for</strong> each α ∈ E andi,j ∈ {1,...,n α } we have‖π V (u α ij )ξ−ε(uα ij )ξ‖ = ‖(e′ i ⊗id)[V α (e j ⊗ξ)−e j ⊗ξ]‖ ≤ ‖V α (e j ⊗ξ)−e j ⊗ξ‖ < δ,as desired. To prove (iii), assume that ξ ∈ H is an (E,δ)-invariant unit vector<strong>for</strong> π V . For j ∈ {1,...,n α } we now get∑n α‖V α (e j ⊗ξ)−e j ⊗ξ‖ 2 = ‖(e ′ i ⊗id)[V α (e j ⊗ξ)−e j ⊗ξ]‖ 2i=1n α∑= ‖π V (u α ij)ξ −ε(u α ij)ξ‖ 2 < n α δ 2 .Hence <strong>for</strong> η = ∑ n αi=1 η ie i ∈ H α we get by Hölder’s inequalityi=1‖V α (η⊗ξ)−η ⊗ξ‖ ≤n α∑i=1< ‖η‖ 1√nα δ≤ ‖η‖ 2 n α δ,which shows that ξ is (E,K E δ)-invariant <strong>for</strong> V.|η i |‖V α (e i ⊗ξ)−e i ⊗ξ‖Similarly, the existence of Kazhdan pairs also translates to the dual picture.□
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 9Corollary 2.4. Let Ĝ have property (T) and let E ⊆ Irred(G) be a finitesubset containing the trivial corepresentation 1 which generates the corepresentationcategory of G. Then there exists δ > 0 such that any representationπ: Pol(G) → B(H) having an (E,δ)-invariant vector has a non-zero invariantvector.Proof. Assume that Ĝ has property (T) and let π: Pol(G) → B(H) be given.Denote by V the corresponding corepresentation of Ĝ on H and choose δ > 0such that (E,δ) is a Kazhdan pair <strong>for</strong> V. If we put K E = max{n α | α ∈ E} andξ ∈ H is an (E,K −1Eδ)-invariant vector <strong>for</strong> π then, by Proposition 2.3 (iii), ξ is an(E,δ)-invariant vector <strong>for</strong> V. Hence V allows a non-zero invariant vector whichis then also invariant <strong>for</strong> π by Proposition 2.3 (i).□Recall from <strong>Theorem</strong> 1.7 that a discrete property (T) <strong>quantum</strong> group is automaticallyfinitely generated. Consider now any finitely generated, discrete <strong>quantum</strong>group Ĝ and let E ⊆ Irred(G) be a finite generating set <strong>for</strong> Corep(G)containing the trivial corepresentation. For each α ∈ E and i,j ∈ {1,...,n α }define x α ij = uα ij −ε(uα ij )1 and put x = ∑ α∈E,i,j xα∗ ij xα ij . Property (T) can then beread of the element x by means of the following result.<strong>Theorem</strong> 2.5. The discrete <strong>quantum</strong> group Ĝ has property (T) if and only ifzero is not in the spectrum of π(x) <strong>for</strong> any representation π: Pol(G) → B(H)not containing the counit.Proof. Assume that Ĝ has property (T) and let π: Pol(G) → B(H) be a representationsuch that π(x) is not bounded away from zero. Then there exists asequence (ξ k ) k∈N in the unit ball of H such that π(x)ξ k → 0. Hence0 = limk〈π(x)ξ k |ξ k 〉∑= limk→∞∑= limk→∞n α∑α∈E i,j=1∑n αα∈E i,j=1〈π(x α ij) ∗ π(x α ij)ξ k |ξ k 〉‖π(u α ij )ξ k −ε(u α ij )ξ k‖ 2 .For suitable δ > 0 the pair (E,δ) is a Kazhdan pair <strong>for</strong> Ĝ and there<strong>for</strong>e the aboveconvergence <strong>for</strong>ces π to have a non-trivial invariant vector; hence π contains ε.Conversely, assume that Ĝ does not have property (T). Then there exists arepresentation π: Pol(G) → B(H) with almost invariant vectors, but withoutnon-zero invariant vectors. In particular we may find a sequence of unit vectors(ξ k ) k∈N in H such thatlim∑n α∑k→∞α∈E i,j=1‖π(u α ij )ξ k −ε(u α ij )ξ k‖ 2 = 0.
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 11Corollary 2.8. A discrete <strong>quantum</strong> group Ĝ has property (T) if and only if thefollowing holds: For any δ > 0 there exist E 0 ⊆ Irred(G) and δ 0 > 0 such that anyrepresentation π: Pol(G) → B(H) with an (E 0 ,δ 0 )-invariant unit vector ξ ∈ Hhas an invariant vector η ∈ H such that ‖ξ −η‖ < δ.Proof. Assume Ĝ has property (T) and let δ > 0 be given. Assume without lossof generality that δ ≤ 1 and choose a Kazhdan pair (E 0 ,δ 0 ′) <strong>for</strong> Ĝ. Put δ 0 = δ 0 ′ δ.Then (E 0 ,δ 0 ) is also a Kazhdan pair and from (the proof of) Proposition 2.7 weget that the pair (E 0 ,δ 0 ) satisfies the claim.□3. Property (T) in terms of states on the universal C ∗ -algebraAs already mentioned in the introduction, a discrete group Γ has property(T) exactly when every sequence of normalized, positive definite functions onΓ converging pointwise to 1 actually converges uni<strong>for</strong>mly to 1. Recall that afunction ϕ: Γ → C is called normalized if ϕ(e) = 1 and positive definite ifn∑ᾱ i α j ϕ(γi −1 γ j ) ≥ 0 <strong>for</strong> all n ∈ N,γ 1 ,...,γ n ∈ Γ and α 1 ,...,α n ∈ C.i=1Hence, there is a one-to-onecorrespondence between normalized, positive definitefunctions on Γ and states on the universal group C ∗ -algebra Cu ∗ (Γ). Having thiscorrespondence in mind, the following theorem generalizes the classical result.<strong>Theorem</strong> 3.1. A discrete <strong>quantum</strong> group Ĝ has property (T) if and only if anynet of states on C(G u ) converging pointwise to the counit ε converges in theuni<strong>for</strong>m norm.Here the uni<strong>for</strong>m norm is the norm on the state space of C(G u ) given by‖ϕ‖ = sup{|ϕ(a)| | ‖a‖ u ≤ 1},and convergence in this norm will often be referred to as uni<strong>for</strong>m convergence.The fact that property (T) implies the convergence property was proved independentlyby Fima (private communication) in the dual picture. For the proofof <strong>Theorem</strong> 3.1 we need the following lemma.Lemma 3.2. Let δ > 0 and π: C(G u ) → B(H) be a ∗-representation. If ξ ∈ His a unit vector such that ‖π(v)ξ −ε(v)ξ‖ ≤ δ <strong>for</strong> every unitary v ∈ C(G u ) thenthere exists an invariant vector η ∈ H such that ‖ξ −η‖ ≤ δ.The proof of Lemma 3.2 is inspired by the corresponding proof <strong>for</strong> (pairs of)<strong>groups</strong> which can be found in [Jol05]. For the proof, and throughout the rest ofthe paper, we denote the unitary group of C(G u ) by U.Proof of Lemma 3.2. Denote by C the closed, convex hull of the setΩ = {π(v)ξε(v ∗ ) | v ∈ U}.
12 DAVID KYEDFor any element η = ∑ nk=1 t kπ(v k )ξε(vk ∗ ) in the convex hull of Ω we haven∑‖ξ −η‖ = ‖ t k (π(v k )ξε(vk)−ξ)‖∗≤=k=1n∑t k ‖π(v k )ξε(vk)−ξ‖∗k=1n∑t k ‖π(v k )ξ −ε(v k )ξ‖ ≤ δ,k=1and hence ‖ξ −η‖ ≤ δ <strong>for</strong> any η ∈ C. Now let η ∈ C be the unique element ofminimal norm [KR83, 2.2.1]. For every v ∈ U we haveπ(v)Ω = π(v){π(u)ξε(u ∗ ) | u ∈ U}= π(v){π(v ∗ u)ξε(u ∗ v) | u ∈ U}= Ωε(v),and hence π(v)C = Cε(v). Since π(v)η is the element of minimal norm in π(v)Cand ηε(v) is the ditto element in Cε(v) we conclude that π(v)η = ηε(v) <strong>for</strong>every v ∈ U. But since the elements in U span C(G u ) linearly the vector η isinvariant.□Proof of <strong>Theorem</strong> 3.1. Assume first that Ĝ has property (T) and consider anynet (ϕ λ ) λ∈Λ of states on C(G u ) converging pointwise to ε. Denote by (H λ ,π λ ,ξ λ )the GNS-triple associated with ϕ λ . A straight <strong>for</strong>ward calculation reveals that|ϕ λ (a)−ε(a)| 2 = ‖π λ (a)ξ λ −ε(a)ξ λ ‖ 2 −(ϕ λ (a ∗ a)−ϕ λ (a ∗ )ϕ λ (a)) (1)<strong>for</strong> any a ∈ C(G u ). Note also, that the Cauchy-Schwartz inequality implies thatϕ λ (a ∗ a) − ϕ λ (a ∗ )ϕ λ (a) ≥ 0 and that this quantity converges to zero. Hencelim λ ‖π λ (a)ξ λ −ε(a)ξ λ ‖ = 0 <strong>for</strong> every a ∈ C(G u ). Let δ > 0 be given. Since Ĝhas property (T), Corollary 2.8 allows us to find a Kazhdan pair (E 0 ,δ 0 ) suchthat any representation with an (E 0 ,δ 0 )-invariant unit vector ξ has an invariantvector η such that ‖ξ −η‖ ≤ δ . We now claim that the representation2π = ⊕ λ∈Λ π λ : Pol(G) → B(⊕ l2λ∈ΛH λ )is of this type. To see this, denote by ˜ξ λ the image of ξ λ under the naturalembedding of H λ into H = ⊕ µ∈Λ H µ and note that‖π(a)˜ξ λ −ε(a)˜ξ λ ‖ = ‖π λ (a)ξ λ −ε(a)ξ λ ‖ −→λ0<strong>for</strong> any a ∈ C(G u ). In particular we get an λ 0 ∈ Λ such that∀λ ≥ λ 0 ∀α ∈ E 0 ∀i,j ∈ {1,...,n α } : ‖π(u α ij )˜ξ λ −ε(u α ij )˜ξ λ ‖ < δ 0 ,
14 DAVID KYED4. Cocycles and conditionally negative functionsThe <strong>Delorme</strong>-<strong>Guichardet</strong> <strong>Theorem</strong> <strong>for</strong> <strong>groups</strong>, stated in the introduction, expressesproperty (T) in terms of vanishing of the first cohomology of the groupin question. In order to prove a <strong>quantum</strong> version of this result we first introducethe relevant notion of cohomology <strong>for</strong> our purposes.Definition 4.1. Let Ĝ be a a discrete <strong>quantum</strong> group and let π: Pol(G) →B(H) be a ∗-representation. A 1-cocycle <strong>for</strong> the representation π is a linear mapc: Pol(G) → H satisfyingc(ab) = π(a)c(b)+c(a)ε(b),<strong>for</strong> all a,b ∈ Pol(G). A 1-cocycle c is called inner if there exists ξ ∈ H such thatc(a) = π(a)ξ − ξε(a) <strong>for</strong> all a ∈ Pol(G). The set of cocycles Z 1 (Pol(G),H) isnaturally a complex vector space in which the set of inner cocycles B 1 (Pol(G),H)constitutes a subspace, and the first cohomology H 1 (Pol(G),H) of Pol(G) withcoefficients in H is then defined as the space of cocycles modulo the inner ones.Finally, a cocycle c is called real if〈c(S(y ∗ ))|c((Sx) ∗ )〉 = 〈c(x)|c(y)〉 <strong>for</strong> all x,y ∈ Pol(G).Remark 4.2. Note that a cocycle c: Pol(G) → H is nothing but a derivationinto H where H is considered a Pol(G)-bimodule with left action given by πand right action given by the counit ε. This is the reason why we from time totime write the scalar action via ε on the right. Using the standard descriptionof the first Hochschild cohomology in terms of derivations [Lod98], we see thatH 1 (Pol(G),H) is exactly the first Hochschild cohomology of Pol(G) with coefficientsin the bimodule π H ε . Throughout the paper, we shall only make use ofthe first Hochschild cohomology group and in the sequel the term cocycle willthere<strong>for</strong>e be used to mean 1-cocycle.The following lemma gives an alternative description of the space of innercocycles and is a modified version of a result in [Pet09].Lemma 4.3. If π: Pol(G) → B(H) is a representation and c: Pol(G) → H is acocycle then c is inner if and only if it is bounded with respect to the norm ‖·‖ uon C(G u ).Proof. First note that bothπ andεextend to C(G u ) by definition of theuniversalnorm. It is clear that an inner cocycle is bounded so assume, conversely, that cextends to C(G u ). We denote the extensions of π,ε and c by the same symbolsand defineX = {c(u)ε(u ∗ ) | u ∈ U},where U as be<strong>for</strong>e denotes the unitary group of C(G u ). Since X is a boundedset in the Hilbert space H there is a unique Chebyshev center [BdlHV08, 2.2.7];i.e. there exists a unique ξ 0 ∈ H minimizing the functionH ∋ ξ ↦→ sup{‖x−ξ‖ | x ∈ X} ∈ R.
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 15Consider now the affine isometric action of U on H given by α(v)(ξ) = π(v)ξ +c(v). Then <strong>for</strong> any v ∈ U we have that α(v)ξ 0 is the Chebyshev center <strong>for</strong> α(v)Xand that ξ 0 ε(v) is the Chebyshev center <strong>for</strong> Xε(v). On the other handα(v)X = α(v){c(u)ε(u ∗ ) | u ∈ U}= α(v){c(v ∗ u)ε(u ∗ v) | u ∈ U}= {π(v)c(v ∗ u)ε(u ∗ v)+c(v) | u ∈ U}= {π(v)[π(v ∗ )c(u)+c(v ∗ )ε(u)]ε(u ∗ v)+c(v) | u ∈ U}= {c(u)ε(u ∗ )ε(v)+π(v)c(v ∗ )ε(v)+c(v) | u ∈ U}= {c(u)ε(u ∗ )ε(v)−c(v)ε(v ∗ )ε(v)+c(v) | u ∈ U}= {c(u)ε(u ∗ )ε(v) | u ∈ U}= Xε(v),and hence α(v)ξ 0 = ξ 0 ε(v) <strong>for</strong> any v ∈ U. Thus c(v) = π(v)(−ξ 0 )−(−ξ 0 )ε(v) <strong>for</strong>any v ∈ U and since the elements in U span C(G u ) linearly we conclude that c isinner.□ThenotionofcocyclesonadiscretegroupΓisintimatelylinked(seee.g.section2.10 in [BdlHV08]) to the notion of conditionally negative functions. Recall, thata function ψ: Γ → R is called conditionally negative if ψ(γ) = ψ(γ −1 ) <strong>for</strong> everyγ ∈ Γ and if ψ, <strong>for</strong> any finite subset {γ 1 ,...,γ n } ⊆ Γ, furthermore satisfiesn∑i=1ᾱ i α j ψ(γ −1i γ j ) ≤ 0 <strong>for</strong> all α 1 ,...,α n ∈ C withn∑α i = 0.Generalizing this to <strong>quantum</strong> <strong>groups</strong> we arrive at the following definition.Definition 4.4. A functional ψ: Pol(G) → C is said to be conditionally negativeif ψ(x ∗ x) ≤ 0 <strong>for</strong> all x ∈ ker(ε). Moreover, ψ is called normalized if ψ(1) = 0and hermitian if ψ(x ∗ ) = ψ(x) <strong>for</strong> all x ∈ Pol(G).The conditionally negative, normalized and hermitian functionals are also calledinfinitesimal generators because of the following <strong>quantum</strong> group version ofSchönberg’s <strong>Theorem</strong> whose proof can be found in [Sch90].<strong>Theorem</strong> 4.5 (Schürmann). A functional ψ: Pol(G) → C is conditionally negative,normalized and hermitian if and only if ϕ t = exp(−tψ): Pol(G) → C is apositive and unital functional <strong>for</strong> every t ≥ 0.Here positivity of the map ϕ t simply means that ϕ t (x ∗ x) ≥ 0 <strong>for</strong> every x ∈Pol(G). Note that [BMT01] <strong>Theorem</strong> 3.3 states that such functionals automaticallyextend to states on C(G u ). Perhaps the definition of the ϕ t ’s require a bitof explanation. For two functionals µ,ω: Pol(G) → C their convolution producti=1
16 DAVID KYEDω ⋆µ is defined as (ω ⊗µ)∆. For a single functional ψ, the co-semisimplicity ofPol(G) makes the series∞∑ (−t) kψ ⋆k (x)k!k=0convergent <strong>for</strong> each x ∈ Pol(G) and its sum is denoted exp(−tψ(x)). For aninfinitesimal generator ψ, the family ϕ t defined above is actually a 1-parameterconvolution semigroup of states on C(G u ) converging pointwise to the counit;i.e. <strong>for</strong> all s,t ≥ 0 we have ϕ t ⋆ ϕ s = ϕ t+s , ϕ 0 = ε and <strong>for</strong> every x ∈ Pol(G)we have ε(x) = lim t→0 ϕ t (x). Such 1-parameter semi<strong>groups</strong> of states on C ∗ -bialgebras have been studied by Lindsay and Skalski in [SL09] where it is alsoproved that iflimt→0‖ϕ t −ε‖ = 0,i.e. if the convergence is uni<strong>for</strong>m, then the infinitesimal generator ψ is boundedwith respect to the norm ‖·‖ u .In the classical case, a group cocycle c: Γ → π H gives rise to an infinitesimalgenerator ψ: Γ → C by setting ψ(γ) = ‖c(γ)‖ 2 ; <strong>for</strong> <strong>quantum</strong> <strong>groups</strong> we have thefollowing analogues result.<strong>Theorem</strong> 4.6 (Vergnioux). Let π: Pol(G) → B(H) be a ∗-representation andc: Pol(G) → H a cocycle. Then ψ: Pol(G) → C defined byis linear and satisfiesψ(x) = 〈c(x (1) )|c(Sx ∗ (2 )〉ψ(x ∗ y) = −〈c((Sx) ∗ )|c(S(y ∗ ))〉−〈c(y)|c(x)〉 <strong>for</strong> all x,y ∈ ker(ε). (2)If furthermore c is a real cocycle then ψ is an infinitesimal generator; i.e. ψ isconditionally negative, normalized and hermitian.In the definition of ψ we made use of the so-called Sweedler notation, writingx (1) ⊗x (2) <strong>for</strong> ∆x. We shall use this notation without further elaboration in thefollowing and refer the reader to [KS97] <strong>for</strong> a detailed treatment. <strong>Theorem</strong> 4.6is due to R. Vergnioux and the author would like to express his gratitude toVergnioux <strong>for</strong> communicating it and <strong>for</strong> allowing its appearance in the presentpaper. Since the result is not published elsewhere we include Vergnioux’s proof,but be<strong>for</strong>e doing so a bit of notation is needed.Notation 4.7. The dual of the Hilbert space H is denoted H op and the innerproduct will considered asalinear map〈·|·〉: H⊙H op → C. Forξ ∈ H we denotebyξ op ∈ H op thedualelement〈·|ξ〉and<strong>for</strong>T ∈ B(H)wedenotebyT op ∈ B(H op )the operator T op ξ op = (Tξ) op . The symbol m will denote both the multiplicationmap Pol(G)⊙Pol(G) → Pol(G) as well as the action π(Pol(G))⊙H → H. TheantipodeinPol(G)isdenoted byS andasusual we denotethecounit byε. Recallthat in the general (i.e. non-Kac) case S 2 ≠ id, but the relation S(S(x ∗ ) ∗ ) = x
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 17holds always. We will often consider the ∗-operation as a self-map of Pol(G) andmay there<strong>for</strong>e write ∗(a) instead of a ∗ ; the above relation involving the antipodemay then be written as S ∗ S ∗ = id. Likewise, we will consider ξ ↦→ ξ op as amap op: H → H op an write op(ξ) in stead of ξ op whenever convenient. By σ wewill denote the flip-map on Pol(G) ⊙Pol(G) as well as the flip-map on H ⊙H.Similarly, σ (13) will denote the map on a three-fold tensor product which flips thefirst and the third leg and leaves the middle leg untouched. In the following weshall furthermoremake useoftheabundance ofrelationsvalidinaHopf ∗-algebrawithout further reference. These may be found in any standard book on Hopf∗-algebras; <strong>for</strong> instance [KS97].For the proof of <strong>Theorem</strong> 4.6 we will need a small lemma concerning the interplaybetween the cocycle and the antipode.Lemma 4.8 (Vergnioux). For x ∈ Pol(G) with ε(x) = 0 we haveProof. The equation m(S ⊗id)∆x = ε(x)1 givesc(Sx) = −π(Sx (1) )c(x (2) ); (3)c(x) = −π(x (1) )c(Sx (2) ); (4)c((Sx) ∗ ) = −π((Sx (2) ) ∗ )c(x ∗ (1)). (5)0 = c(m(S ⊗id)∆x)= c((Sx (1) )x (2) )= π(Sx (1) )c(x (2) )+c(Sx (1) )ε(x (2) )= π(Sx (1) )c(x (2) )+c(S((id⊗ε)∆x))= π(Sx (1) )c(x (2) )+c(Sx),proving equation (3). In the same manner, the equation (4) follows from the<strong>for</strong>mula m(id⊗S)∆x = ε(x)1. The equation (5) follows from (4) and the <strong>for</strong>mula∆S = (S ⊗S)σ∆:c((Sx) ∗ ) = −π(((Sx) ∗ ) (1) )c(S(((Sx) ∗ ) (2) ))= π((Sx (2) ) ∗ )c(S((Sx (1) ) ∗ ))= π((Sx (2) ) ∗ )c(x ∗ (1) ).This concludes the proof of the lemma.□Proof of <strong>Theorem</strong> 4.6. We may write ψ = 〈·|·〉c⊗(op c S ∗)∆ which shows thatψ is well defined and linear. It is first proved that ψ satisfies equation (2). Using
18 DAVID KYEDthe cocycle condition we getψ(x ∗ y) = 〈·|·〉c⊗(op c S ∗)∆(x ∗ y)= 〈c(x ∗ (1)y (1) )|c((Sx (2) )S(y(2)))〉∗= 〈π(x ∗ (1) )c(y (1))|π(Sx (2) )c(S(y(2) ∗ ))〉+〈π(x∗ (1) )c(y (1))|c(Sx (2) )ε(S(y(2) ∗ ))〉+〈c(x ∗ (1) )ε(y (1))|π(Sx (2) )c(S(y(2) ∗ ))〉+〈c(x∗ (1) )ε(y (1))|c(Sx (2) )ε(S(y(2) ∗ ))〉.We now treat the four terms one by one. Since ε(x) = 0, the first term vanishes:〈π(x ∗ (1))c(y (1) )|π(Sx (2) )c(S(y ∗ (2)))〉 = 〈c(y (1) )|π(m(id⊗S)∆x)c(S(y ∗ (2)))〉= 〈c(y (1) )|π(ε(x)1)c(S(y ∗ (2) ))〉= 0.Using the <strong>for</strong>mula (3) and the fact that εS = ε, the second term becomes〈π(x ∗ (1) )c(y (1))|c(Sx (2) )ε(S(y ∗ (2) ))〉 = 〈ε(y (2))c(y (1) )|π(x (1) )c(Sx (2) )〉= −〈c((id⊗ε)∆y)|c(x)〉= −〈c(y)|c(x)〉.Similarly, using the <strong>for</strong>mula (5) the third term becomes〈c(x ∗ (1))ε(y (1) )|π(Sx (2) )c(S(y(2)))〉 ∗ = 〈π((Sx (2) ) ∗ )c(x ∗ (1))|ε(y(1))c(S(y ∗ (2)))〉∗= −〈c((Sx) ∗ )|c((ε⊗id)(S ⊗S)∆(y ∗ ))〉= −〈c((Sx) ∗ )|c((ε⊗id)σ∆(S(y ∗ )))〉= −〈c((Sx) ∗ )|c(S(y ∗ ))〉.We are there<strong>for</strong>e done if we can show that the fourth term vanishes. This followsfrom the assumption ε(y) = 0 and the following calculation:〈c(x ∗ (1) )ε(y (1))|c(Sx (2) )ε(S(y ∗ (2) )))〉 = ε(y (1))ε(y (2) )〈c(x ∗ (1) )|c(Sx (2))〉= ε((id⊗ε)∆y)〈c(x ∗ (1))|c(Sx (2) )〉= 0.Hence ψ satisfies (2). Assume now that c is a real cocycle. The equation (2)then gives that ψ(x ∗ x) = −2‖c(x)‖ 2 whenever x ∈ ker(ε) which shows that ψ isconditionally negative. Since c(1) = 0 it is clear that ψ is normalized. That ψ is
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 19hermitian is seen by the following calculation.ψ(x ∗ ) = 〈·|·〉(c⊗(op c S ∗))x ∗ (1) ⊗x∗ (2)= 〈c(x ∗ (1) )|c(Sx (2))〉= 〈c(Sx (2) )|c(x ∗ (1))〉= −〈π(S(x (2)(1) ))c(x (2)(2) )|c(x ∗ (1))〉 (by (3))= −〈c(x (2)(2) )|π(S(x (2)(1) ) ∗ )c(x ∗ (1) )〉= −〈·|·〉c(x (2)(2) )⊗(π((Sx (2)(1) ) ∗ ) op c(x ∗ (1) )op= −〈·|·〉c⊗[m((op π ∗ S)⊗(op c ∗))](x (2)(2) ⊗x (2)(1) ⊗x (1) )= −〈·|·〉c⊗[m((op π ∗ S)⊗(op c ∗))]σ (13) (x (1) ⊗x (2)(1) ⊗x (2)(2) )= −〈·|·〉c⊗[m((op π ∗ S)⊗(op c ∗))]σ (13) (x (1)(1) ⊗x (1)(2) ⊗x (2) )= −〈c(x (2) )|π((S(x (1)(2) )) ∗ )c(x (1)∗(1) )〉= 〈c(x (2) )|c((Sx (1) ) ∗ )〉 (by (5))= 〈c(S(S(x ∗ (2)) ∗ ))|c((Sx (1) ) ∗ )〉= 〈c(x (1) )|c(S(x ∗ (2)))〉 (c real)= ψ(x).This concludes the proof of <strong>Theorem</strong> 4.6.5. The <strong>Delorme</strong>-<strong>Guichardet</strong> <strong>Theorem</strong>Using the material gathered in the previous sections we are now able to proveour main result.<strong>Theorem</strong> 5.1. For a discrete <strong>quantum</strong> group Ĝ the following are equivalent.(i) Ĝ has property (T).(ii) Ĝ is Kac and every normalized, hermitian, conditionally negative functionalψ: Pol(G) → C is bounded with respect to ‖·‖ u .(ii) For every ∗-representation π: Pol(G) → B(H) the first cohomology groupH 1 (Pol(G),H) vanishes.Proof. We first prove (i)⇒(ii). Assume there<strong>for</strong>e that Ĝ has property (T) andlet ψ: Pol(G) → C be normalized, conditionally negative and hermitian. Byexponentiation (see section 4) we get a 1-parameter family of states on C(G u )converging pointwise to the counit and by <strong>Theorem</strong> 3.1 the convergence has tobe uni<strong>for</strong>m. Applying [SL09] Proposition 2.3, this implies that the infinitesimalgenerator ψ is bounded. That Ĝ is Kac follows from <strong>Theorem</strong> 1.7.Next we prove (ii)⇒(iii). Assume there<strong>for</strong>e that Ĝ is Kac and that everyinfinitesimal generator is bounded, and let a representation π: Pol(G) → B(H)□
20 DAVID KYEDas well as a cocycle c: Pol(G) → H be given. By Lemma 4.3 it suffices to showthat c is bounded with respect to the norm ‖·‖ u . Since Ĝ is assumed Kac theantipode S: Pol(G) → Pol(G) is ∗-preserving. The ∗-representation π there<strong>for</strong>egives rise to a dual ∗-representation π op : Pol(G) → B(H op ) on the dual Hilbertspace H op given by π op (a)ξ op = (π(Sa ∗ )ξ) op and the map c op : Pol(G) → H opgiven by c op (a) = (c(Sa ∗ )) op is a cocycle <strong>for</strong> this representation. Furthermore, itis easy to check thatPol(G) ∋ a δ↦−→ (c(a),c op (a)) ∈ H ⊕H opisareal(π⊕π op )-cocyclewhichisboundedifandonlyifcisbounded. Itthere<strong>for</strong>esuffices to treat the case where the cocycle c is real. In this case, <strong>Theorem</strong> 4.6provides us with a conditionally negative functional ψ: Pol(G) → C such thatψ(x ∗ x) = −2‖c(x)‖ 2 <strong>for</strong> all x ∈ ker(ε). By assumption the functional ψ isbounded with respect to ‖·‖ u so <strong>for</strong> x ∈ ker(ε) we get‖x‖ 2 u‖ψ‖ ≥ |ψ(x ∗ x)| = 2‖c(x)‖ 2 ,and hence the restriction c 0 of c to ker(ε) extends boundedly to a map ˜c 0 onthe closure J of ker(ε) inside C(G u ). The ideal J is exactly the kernel of theextension of the counit ε: C(G u ) → C and there<strong>for</strong>e the mapis bounded and extends c.C(G u ) ∋ a ↦→ ˜c 0 (a−ε(a)1) ∈ HThe proof of the implication (iii)⇒(i) is a minor modification of the correspondingproof <strong>for</strong> <strong>groups</strong> in [BdlHV08]. Assume there<strong>for</strong>e the vanishing of allthe first cohomology <strong>groups</strong> and fix a ∗-representation π: Pol(G) → B(H) on aHilbert space H without non-zero invariant vectors. We then need to show thatπ can not have invariant vectors either. Define, <strong>for</strong> a finite subset E ⊆ Irred(G),a seminorm on Z 1 (Pol(G),H) by‖c‖ E = sup{‖c(u α ij)‖ | α ∈ E,1 ≤ i,j ≤ n α }.Since the matrix coefficients span Pol(G) linearly, it is a routine to check thatZ 1 (Pol(G),H) becomes a Frechet space when endowed with the topology arisingfrom this family of seminorms. The claim now is that π does not have almostinvariant vectors iff B 1 (Pol(G),H) is closed in Z 1 (Pol(G),H). Clearly theclaim implies the desired result since the vanishing of H 1 (Pol(G),H) in particularimplies that B 1 (Pol(G),H) is closed. To prove the claim, consider the mapΦ: H → B 1 (Pol(G),H) mapping a vector ξ to the corresponding inner cocycle.Then Φ is linear, continuous and surjective and since π is assumed to have nonon-zero fixed vectors it follows that Φ is also injective. Assume first that π doesnot have almost invariant vectors either. Then there exists E 0 ⊆ Irred(G) andδ 0 > 0 such that‖Φ(ξ)‖ E0 = sup{‖π(u α ij )ξ −ε(uα ij )ξ‖ | α ∈ E 0,1 ≤ i,j ≤ n α } ≥ δ 0 ‖ξ‖ (6)
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 21<strong>for</strong> all ξ ∈ H. Let c be in the closure of B 1 (Pol(G),H) and choose a sequence(ξ n ) n∈N such that Φ(ξ n ) converges to c in the Frechet topology. Then, in particular,(Φ(ξ n )) n∈N is a Cauchy sequence with respect to the seminorm ‖·‖ E0 and by(6) this implies that (ξ n ) n∈N is a Cauchy sequence in H; hence it has a limit pointξ ∈ H. By continuity of Φ we conclude that c = Φ(ξ) and there<strong>for</strong>e c is innerand B 1 (Pol(G),H) closed. Conversely, assume that B 1 (Pol(G),H) is closed andthere<strong>for</strong>e a sub-Frechet space in Z 1 (Pol(G),H). Then the open mapping theorem[Rud73, 2.12] implies that Φ: H → B 1 (Pol(G),H) is bi-continuous and thusbounded away from zero in at least one of the seminorms ‖ · ‖ E0 . Hence thereexists a δ 0 > 0 such thatsup{‖π(u α ij )ξ −ε(uα ij )ξ‖ | α ∈ E 0,1 ≤ i,j ≤ n α } = ‖Φ(ξ)‖ E0 ≥ δ 0 ‖ξ‖<strong>for</strong> all ξ ∈ H, and there<strong>for</strong>e π can not have almost invariant vectors.6. An application to L 2 -invariantsInthis section we prove the vanishing of the first L 2 -Betti number <strong>for</strong> a discrete<strong>quantum</strong> group Ĝ with property (T). The corresponding statement <strong>for</strong> <strong>groups</strong>was known to Gromov (see [Gro93]), but the first detailed proof was given byBekka and Valette in [BV97]. The modern algebraic approach to L 2 -invariantsdeveloped by Lück [Lüc02] (see also [Tho08a], [TP09], [Tho08b], [Sau05],[Far96])makes this statement easier to prove and it can, <strong>for</strong> instance, easily be deducedfrom [TP09] <strong>Theorem</strong> 2.2. In this section we show how their argument passesthrough to <strong>quantum</strong> <strong>groups</strong>. Be<strong>for</strong>e doing so, we briefly remind the reader ofthe necessary definitions concerning L 2 -Betti numbers <strong>for</strong> <strong>groups</strong> and <strong>quantum</strong><strong>groups</strong>.For a discrete group Γ, its L 2 -Betti numbers can be described/defined inpurely algebraic terms (see [Lüc02]) as β p (2) (Γ) = dim L(Γ) Tor CΓp (L(Γ),C) wheredim L(Γ) (−) us Lück’s extended Murray-von Neumann dimension. For a discrete<strong>quantum</strong> group Ĝ of Kac type it is there<strong>for</strong>e natural to define its L2 -Betti numbersasβ p (2) (Ĝ) = dim L ∞ (G)Tor Pol(G)p (L ∞ (G),C),where dim L ∞ (G)(−) is the Murray-von Neumann dimension arising from the tracialHaar state. These L 2 -Betti numbers have been studied in [CHT09], [Ver09],[Kye08b] and [Kye08a] 1 , and the aim of the present section is to prove the following.Corollary 6.1. IfĜ has property (T) then β(2) 1 (Ĝ) = 0.Note that if Ĝ has property (T) then G is automatically of Kac type, and itsHaar state h: L ∞ (G) → C is there<strong>for</strong>e a trace, so that the first L 2 -Betti number1 Note that the notation β (2)p (G) is used in [Kye08b] and [Kye08a].□
22 DAVID KYEDmakes sense. As mentioned above, the proof of Corollary 6.1 follows the lines ofthe corresponding proof in [TP09]. During the proof we will have to consider dimensionsofbothrightandleftmodules<strong>for</strong>L ∞ (G), andtoavoidconfusionwewilllet dim L ∞ (G)(X) denote the dimension of a left module X whereas dim L ∞ (G) op(Y)will denote the dimension of a right module Y.Proof. Denote by M(G) the ∗-algebra of closed, densely defined (potentially unbounded)operators affiliated with L ∞ (G). This is a self-injective and von Neumannregular ring and tensoring L ∞ (G)-modules with M(G) is a flat and dimensionpreserving functor [Rei01]. There<strong>for</strong>eβ (2)1 (Ĝ) = dim L ∞ (G)M(G)⊙ L ∞ (G) Tor Pol(G)1 (L ∞ (G),C)By [Tho08a] Corollary 3.4 we have= dim L ∞ (G)Tor Pol(G)1 (M(G),C).dim L ∞ (G)Tor Pol(G)1 (M(G),C) = dim L ∞ (G) op Hom M(G)(Tor Pol(G)1 (M(G),C),M(G)),and using the self-injectiveness of M(G) (see e.g. [Tho08a] <strong>Theorem</strong> 3.5 and itsproof) we get an isomorphism of right M(G)-modulesHom M(G) (Tor Pol(G)1 (M(G),C),M(G)) ≃ Ext 1 Pol(G)(C,M(G)).By considering the bar-resolution of the trivial Pol(G)-module C, one sees thatExt 1 Pol(G)(C,M(G)) may also be computed as the first Hochschild cohomologyH 1 (Pol(G),M(G)) where M(G) carries the natural left action of Pol(G) andright action given by the counit. We there<strong>for</strong>e arrive at the <strong>for</strong>mulaβ (2)1 (Ĝ) = dim L ∞ (G) op H1 (Pol(G),M(G)).Since Ĝ has property (T), <strong>Theorem</strong> 5.1 implies that H1 (Pol(G),L 2 (G)) vanishesand we are there<strong>for</strong>e done if we can prove thatdim L ∞ (G) op H1 (Pol(G),L 2 (G)) = dim L ∞ (G) op H1 (Pol(G),M(G)).To see this, we consider the following diagram of right L ∞ (G)-modules:0 B 1 (Pol(G),L ∞ (G)) Z 1 (Pol(G),L ∞ (G))0 B 1 (Pol(G),L 2 (G)) Z 1 (Pol(G),L 2 (G)) H 1 (Pol(G),L ∞ (G)) H 1 (Pol(G),L 2 (G))000 B 1 (Pol(G),M(G)) Z 1 (Pol(G),M(G)) H 1 (Pol(G),M(G)) 0Therowsinthisdiagramsareexact bydefinitionandthetwo firstcolumns clearlyconsist of inclusions. We now prove thatdim L ∞ (G) op B1 (Pol(G),L ∞ (G)) = dim L ∞ (G) op B1 (Pol(G),M(G)); (7)dim L ∞ (G) op Z1 (Pol(G),L ∞ (G)) = dim L ∞ (G) op Z1 (Pol(G),M(G)), (8)
A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS 23and the result then follows from additivity [Lüc02] of the dimension functiondim L ∞ (G) op(−). To prove the equality (7), notice that the first column identifieswith the inclusions L ∞ (G) ⊆ L 2 (G) ⊆ M(G) so it suffices to see thatdim L ∞ (G) op M(G)/L∞ (G) = 0.By [Sau05] <strong>Theorem</strong> 2.4, it is enough to see that <strong>for</strong> every ξ ∈ M(G) and everyδ > 0thereexistsaprojectionp ∈ L ∞ (G)suchthath(p) ≥ 1−δ andξp ∈ L ∞ (G).But this follows from the fact all the spectral projections of the absolute value ofξ are in L ∞ (G). To prove the equality (8), consider a cocycle c: Pol(G) → M(G)and a δ > 0. Again by [Sau05], we have to find a projection p ∈ L ∞ (G) suchthat h(p) ≥ 1−δ and c(−)p ∈ Z 1 (Pol(G),L ∞ (G)). For this, consider again theset of matrix coefficients{u α ij | α ∈ I,1 ≤ i,j ≤ n α }.Since C(G) is assumed separable, this set is at most countable so we may choosea sequence of numbers δ α ij > 0 such that∑∑n αα∈I i,j=1δ α ij ≤ δ.For each u α ij we have c(uα ij ) ∈ M(G) and hence we can find a projection pα ij ∈L ∞ (G) such that c(u α ij)p α ij ∈ L ∞ (G) and such that h(p α ij) ≥ 1−δ α ij. Let p be theinfimum of all these projections. We then haveh(1−p) = h(1− ∧ α,i,jp α ij )) = h(∨ α,i,jp α ij ) ≤ ∑ α,i,jh(p α ij ) ≤ δ.Since the set {u α ij | α ∈ I,1 ≤ i,j ≤ n α } spans Pol(G) linearly we also have thatc(−)p is a cocycle with values in L ∞ (G).□Turning things around, vanishing of the first L 2 -Betti number may be turnedinto an honest vanishing of cohomology result.Corollary 6.2. If Ĝ is non-amenable and of Kac type with β(2) 1 (Ĝ) = 0 thenH 1 (Pol(G),L 2 (G)) vanishes.Recall that a discrete <strong>quantum</strong> group Ĝ is said to be amenable if the counitε: Pol(G) → C extends to a bounded character on C(G red ). A detailed study ofthis notion may be found in [BMT01] and [Tom06]. The proof Corollary 6.2 isagain a modification of the corresponding proof in [TP09].Proof. Since β (2)1 (Ĝ) = 0 we have2 thatdim L ∞ (G)Tor Pol(G)1 (M(G),C) = 0,2 See e.g. the beginning of the proof of Corollary 6.1.
24 DAVID KYEDand by [Tho08a] Corollary 3.3 this implies vanishing of the dual M(G)-moduleHom M(G) (Tor Pol(G)1 (M(G),C),M(G)).As in the proof of Corollary 6.1 we have an isomorphism of right M(G)-modulesHom M(G) (Tor Pol(G)1 (M(G),C),M(G)) ≃ H 1 (Pol(G),M(G)),and hence every cocycle with values in M(G) is inner. Denote by J ∈ B(L 2 (G))the modular conjugation arising from the tracial state h and recall that <strong>for</strong> eachx ∈ L 2 (G) the operator L(x) 0 : Λa ↦→ Jλ(a) ∗ Jx is pre-closed and its closure L(x)is affiliated with L ∞ (G). We there<strong>for</strong>e have L 2 (G) embedded into M(G) via themap L: x ↦→ L(x) which is easily seen to bee an embedding of Pol(G)-bimodules.Foragivencocyclec: Pol(G) → L 2 (G)wecanthere<strong>for</strong>efindanaffiliatedoperatorξ ∈ M(G) such that L(c(a)) = λ(a)ξ −ξε(a) <strong>for</strong> every a ∈ Pol(G). Choose anincreasing sequence of projections p n ∈ L ∞ (G) such that ξp n ∈ L ∞ (G) <strong>for</strong> everyn ∈ N and such that (p n ) n∈N converges in the strong operator topology to 1. Foreach n ∈ N and a ∈ Pol(G) we now haveL(Jp n J(c(a))) = L(c(a))p n = λ(a)(ξp n )−(ξp n )ε(a),and evaluating the operators on Λ(1) we getJp n J(c(a)) = λ(a)Λ(ξp n )−Λ(ξp n )ε(a).But also (Jp n J) n∈N converges in the strong operator topology to 1 and hencec(a) = lim λ(a)Λ(ξp n )−Λ(ξp n )ε(a),n→∞which proves that c is the pointwise limit of a sequence inner cocycles with valuesin L 2 (G). But since Ĝ is non-amenable, the left regular representation λ can nothave almost invariant vectors and the space B 1 (Pol(G),L 2 (G)) is there<strong>for</strong>e closedintheFrechet topologyonZ 1 (Pol(G),L 2 (G)) introduced andstudied inthe proofof <strong>Theorem</strong> 5.1. This topology is exactly the topology of pointwise convergenceand we conclude that c ∈ Z 1 (Pol(G),L 2 (G)) is inner.□References[Ban99] Teodor Banica. Representations of compact <strong>quantum</strong> <strong>groups</strong> and subfactors. J.Reine Angew. Math., 509:167–198, 1999.[BdlHV08] Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s property (T), volume11 of New Mathematical Monographs. Cambridge University Press, Cambridge,2008.[BMT01] ErikBédos,GerardJ.Murphy, andLarsTuset.Co-amenabilityofcompact<strong>quantum</strong><strong>groups</strong>. J. Geom. Phys., 40(2):130–153, 2001.[BV97] Mohammed E. B. Bekka and Alain Valette. Group cohomology, harmonic functionsand the first L 2 -Betti number. Potential Anal., 6(4):313–326, 1997.[CHT09] Benoît Collins, Johannes Härtel, and Andreas Thom. Homology of free <strong>quantum</strong><strong>groups</strong>. C. R. Math. Acad. Sci. Paris, 347(5-6):271–276, 2009.[CJ85] Alain Connes and Vaughan Jones. Property T <strong>for</strong> von Neumann algebras. Bull.London Math. Soc., 17(1):57–62, 1985.
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