A Delorme-Guichardet Theorem for quantum groups - Hausdorff ...

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 for discussions revolving around the notion ofproperty (T). The work presented in the paper was initiated during the authorsstay at the Hausdorff Research Institute for 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 quantum groupsWe choose here the approach to compact quantum groups developed by Woronowiczin [Wor87a], [Wor87b] and [Wor98]. Thus, a compact quantum 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 quantum group possesses a unique Haar state; i.e. a stateh: C(G) → C such that(id⊗h)∆a = h(a)1 = (h⊗id)∆(a) for 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 quantum 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))Wfor every a ∈ C(G). One notes that the right hand side of this formula makessense for 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 algebraicquantum 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 groups,such as direct sums, tensor products, intertwiners and irreducibility, have naturalcounterparts in the corepresentation theory for compact quantum groups. Inparticular the following important theorem holds true.Theorem 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 thequantum Peter-Weyl theorem [Wor98] ensures that Irred(G) is a countable set.We label its elements by an auxiliary countable set I and choose for 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 } for 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 quantum 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, Hausdorff 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 therefore 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 quantum 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 quantum 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 therefore nothing but the group von Neumann algebraL(Γ). Each element in Γ is a one-dimensional corepresentation for this quantumgroup and the Hopf ∗-algebra therefore 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 quantumgroup”,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 for quantum groups as studied in[BMT01].Any compact quantum group G has a dual quantum 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 quantum 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 KYEDfor 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: for 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 for 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 compactquantum groups 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 quantum groupsetting by Fima in [Fim08] and the definition is as follows.Definition 1.5 (Fima). Let Ĝbe a discrete quantum 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 for 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‖ for all α ∈ E and all η ∈ H α ,and V is said to have almost-invariant vectors if it has an (E,δ)-invariantvector for each finite, non-empty E ⊆ Irred(G) and each δ > 0.(iii) The discrete quantum 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.Theorem 1.7 (Fima). If Ĝ is a discrete quantum group with property (T) thenthe following holds.(i) The quantum 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 quantum group is finitely generated; i.e. the corepresentationcategory Corep(G) of the compact dual is a finitely generated tensorcategory.(iii) The quantum group allows Kazhdan pairs; i.e. for 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 quantum 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 reformulate property (T) for discrete quantum groups 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 for 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‖,for 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 for 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 thatfor 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 quantum 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 for V then it is also (E,δ)-invariant for π V .(iii) If ξ ∈ H is (E,δ)-invariant for π V then it is (E,K E δ)-invariant for V.Thus, V has almost-invariant vectors if and only if π V has almost invariantvectors. In particular, the discrete quantum 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 for 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 for 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 for 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 basisfor 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 for V. Then for 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 vectorfor 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 for 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 for 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 for V. If we put K E = max{n α | α ∈ E} andξ ∈ H is an (E,K −1Eδ)-invariant vector for π then, by Proposition 2.3 (iii), ξ is an(E,δ)-invariant vector for V. Hence V allows a non-zero invariant vector whichis then also invariant for π by Proposition 2.3 (i).□Recall from Theorem 1.7 that a discrete property (T) quantum group is automaticallyfinitely generated. Consider now any finitely generated, discrete quantumgroup Ĝ and let E ⊆ Irred(G) be a finite generating set for 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.Theorem 2.5. The discrete quantum group Ĝ has property (T) if and only ifzero is not in the spectrum of π(x) for 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 for Ĝ and therefore the aboveconvergence forces 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 quantum 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 ′) for 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 uniformly 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 for 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.Theorem 3.1. A discrete quantum group Ĝ has property (T) if and only if anynet of states on C(G u ) converging pointwise to the counit ε converges in theuniform norm.Here the uniform 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 uniform convergence.The fact that property (T) implies the convergence property was proved independentlyby Fima (private communication) in the dual picture. For the proofof Theorem 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 for 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 for (pairs of)groups 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 for 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) forevery v ∈ U. But since the elements in U span C(G u ) linearly the vector η isinvariant.□Proof of Theorem 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 forward 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)for 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 for 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 ‖ −→λ0for 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 Delorme-Guichardet Theorem for groups, stated in the introduction, expressesproperty (T) in terms of vanishing of the first cohomology of the groupin question. In order to prove a quantum version of this result we first introducethe relevant notion of cohomology for our purposes.Definition 4.1. Let Ĝ be a a discrete quantum group and let π: Pol(G) →B(H) be a ∗-representation. A 1-cocycle for the representation π is a linear mapc: Pol(G) → H satisfyingc(ab) = π(a)c(b)+c(a)ε(b),for 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) for 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)〉 for 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 willtherefore 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 before 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 for any v ∈ U we have that α(v)ξ 0 is the Chebyshev center for α(v)Xand that ξ 0 ε(v) is the Chebyshev center for 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) for any v ∈ U. Thus c(v) = π(v)(−ξ 0 )−(−ξ 0 )ε(v) forany 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 ) for everyγ ∈ Γ and if ψ, for any finite subset {γ 1 ,...,γ n } ⊆ Γ, furthermore satisfiesn∑i=1ᾱ i α j ψ(γ −1i γ j ) ≤ 0 for all α 1 ,...,α n ∈ C withn∑α i = 0.Generalizing this to quantum groups 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 for all x ∈ ker(ε). Moreover, ψ is called normalized if ψ(1) = 0and hermitian if ψ(x ∗ ) = ψ(x) for all x ∈ Pol(G).The conditionally negative, normalized and hermitian functionals are also calledinfinitesimal generators because of the following quantum group version ofSchönberg’s Theorem whose proof can be found in [Sch90].Theorem 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 for every t ≥ 0.Here positivity of the map ϕ t simply means that ϕ t (x ∗ x) ≥ 0 for every x ∈Pol(G). Note that [BMT01] Theorem 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 for 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. for all s,t ≥ 0 we have ϕ t ⋆ ϕ s = ϕ t+s , ϕ 0 = ε and for every x ∈ Pol(G)we have ε(x) = lim t→0 ϕ t (x). Such 1-parameter semigroups 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 uniform, 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 ; for quantum groups we have thefollowing analogues result.Theorem 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)〉 for 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) for ∆x. We shall use this notation without further elaboration in thefollowing and refer the reader to [KS97] for a detailed treatment. Theorem 4.6is due to R. Vergnioux and the author would like to express his gratitude toVergnioux for communicating it and for allowing its appearance in the presentpaper. Since the result is not published elsewhere we include Vergnioux’s proof,but before 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〉andforT ∈ 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 therefore 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; for instance [KS97].For the proof of Theorem 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 theformula m(id⊗S)∆x = ε(x)1. The equation (5) follows from (4) and the formula∆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 Theorem 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 formula (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 formula (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 therefore 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 Theorem 4.6.5. The Delorme-Guichardet TheoremUsing the material gathered in the previous sections we are now able to proveour main result.Theorem 5.1. For a discrete quantum 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 therefore 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 Theorem 3.1 the convergence has tobe uniform. Applying [SL09] Proposition 2.3, this implies that the infinitesimalgenerator ψ is bounded. That Ĝ is Kac follows from Theorem 1.7.Next we prove (ii)⇒(iii). Assume therefore 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 thereforegives 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 for this representation. Furthermore, itis easy to check thatPol(G) ∋ a δ↦−→ (c(a),c op (a)) ∈ H ⊕H opisareal(π⊕π op )-cocyclewhichisboundedifandonlyifcisbounded. Itthereforesuffices to treat the case where the cocycle c is real. In this case, Theorem 4.6provides us with a conditionally negative functional ψ: Pol(G) → C such thatψ(x ∗ x) = −2‖c(x)‖ 2 for all x ∈ ker(ε). By assumption the functional ψ isbounded with respect to ‖·‖ u so for 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 therefore 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 for groups in [BdlHV08]. Assume therefore the vanishing of allthe first cohomology groups 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, for 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 21for 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 therefore c is innerand B 1 (Pol(G),H) closed. Conversely, assume that B 1 (Pol(G),H) is closed andtherefore 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‖for all ξ ∈ H, and therefore 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 for a discretequantum group Ĝ with property (T). The corresponding statement for groupswas 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, for instance, easily be deducedfrom [TP09] Theorem 2.2. In this section we show how their argument passesthrough to quantum groups. Before doing so, we briefly remind the reader ofthe necessary definitions concerning L 2 -Betti numbers for groups and quantumgroups.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 discretequantum group Ĝ of Kac type it is therefore 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 therefore 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 dimensionsofbothrightandleftmodulesforL ∞ (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]. Thereforeβ (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] Theorem 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 therefore arrive at the formulaβ (2)1 (Ĝ) = dim L ∞ (G) op H1 (Pol(G),M(G)).Since Ĝ has property (T), Theorem 5.1 implies that H1 (Pol(G),L 2 (G)) vanishesand we are therefore 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] Theorem 2.4, it is enough to see that for 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 quantum 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 for 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 therefore 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)wecanthereforefindanaffiliatedoperatorξ ∈ M(G) such that L(c(a)) = λ(a)ξ −ξε(a) for every a ∈ Pol(G). Choose anincreasing sequence of projections p n ∈ L ∞ (G) such that ξp n ∈ L ∞ (G) for 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 therefore closedintheFrechet topologyonZ 1 (Pol(G),L 2 (G)) introduced andstudied inthe proofof Theorem 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 quantum groups 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-amenabilityofcompactquantumgroups. 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 quantumgroups. C. R. Math. Acad. Sci. Paris, 347(5-6):271–276, 2009.[CJ85] Alain Connes and Vaughan Jones. Property T for von Neumann algebras. Bull.London Math. Soc., 17(1):57–62, 1985.

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26 DAVID KYED[Tho08a] Andreas Thom. L 2 -cohomology for von Neumann algebras. Geom. Funct. Anal.,18(1):251–270, 2008.[Tho08b] Andreas Thom. L 2 -invariants and rank metric. In C ∗ -algebras and elliptic theoryII, Trends Math., pages 267–280. Birkhäuser, Basel, 2008.[Tom06] Reiji Tomatsu. Amenable discrete quantum groups. J. Math. Soc. Japan, 58(4):949–964, 2006.[TP09] Andreas Thom and Jesse Peterson. Group cocycles and the ring of affiliated operators.preprint, 2009. arXiv:0708.4327.[VD96] Alfons Van Daele. Discrete quantum groups. J. Algebra, 180(2):431–444, 1996.[Ver09] Roland Vergnioux. Paths in quantum Cayley trees and L 2 -cohomology. preprint,2009. arXiv:0905.1732.[Wor87a] Stanis̷law L. Woronowicz. Compact matrix pseudogroups. Comm. Math. Phys.,111(4):613–665, 1987.[Wor87b] Stanis̷law L. Woronowicz. Twisted SU(2) group. An example of a noncommutativedifferential calculus. Publ. Res. Inst. Math. Sci., 23(1):117–181, 1987.[Wor98]Stanis̷law L. Woronowicz. Compact quantum groups. In Symétries quantiques (LesHouches, 1995), pages 845–884. North-Holland, Amsterdam, 1998.David Kyed, Mathematisches Institut, Georg-August-Universität Göttingen,Bunsenstraße 3-5, D-37073 Göttingen, Germany.E-mail address: kyed@uni-math.gwdg.deURL: www.uni-math.gwdg.de/kyed