PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 4, April 2010, Pages 1349–1359 S 0002-9939(09)10220-4 Article electronically published on November 25, 2009

OPERATOR BIPROJECTIVITY OF COMPACT QUANTUM GROUPS

MATTHEW DAWS

(Communicated by Marius Junge)

Abstract. Given a (reduced) locally compact quantum group A,wecancon- sider the convolution algebra L1(A) (which can be identified as the predual of the von Neumann algebra form of A). It is conjectured that L1(A) is operator biprojective if and only if A is compact. The “only if” part always holds, and the “if” part holds for Kac algebras. We show that if the splitting morphism associated with L1(A) being biprojective can be chosen to be completely posi- tive, or just contractive, then we already have a Kac algebra. We give another proof of the converse, indicating how modular properties of the Haar state seem to be important.

1. Introduction

A Banach algebra A is biprojective if the multiplication map ∆∗ : A⊗
A → A has a right inverse in the category of A-bimodule maps. This can be thought of as a “finiteness condition”. In particular, the group algebra L1(G) is biprojective if and only if G is compact; see [9, Chapter IV, Theorem 5.13]. When dealing with more non-commutative (or “quantum”) algebras (here we focus on L1(G)∗ = L∞(G) when we suggest that the classical situation is commu- tative), there is a large amount of evidence that operator spaces form the correct category to work in. For example, if we consider the Fourier algebra A(G), then A(G)isoperator biprojective if and only if G is discrete, [23]. When G is abelian, ∼ as A(G) = L1(Gˆ), and Gˆ is compact if and only if G is discrete, this result is in full agreement with what we might expect. By contrast, if we ask when A(G) is bipro- jective, then, if G is discrete and almost abelian (contains a finite-index abelian subgroup), then A(G) is biprojective. Conversely, if A(G) is biprojective, then G is discrete, and either almost abelian, or is non-amenable yet does not contain F2; see [16]. In this paper, we shall continue the study of when the convolution algebra of a (reduced) compact quantum group is operator biprojective. It was shown in [1, Theorem 4.12] that if the convolution algebra of a locally compact quantum group G is operator biprojective, then G is already compact. Conversely, if G is a compact Kac algebra, then G is operator biprojective. We shall show that if the right inverse

Received by the editors May 16, 2009, and, in revised form, July 12, 2009. 2010 Mathematics Subject Classification. Primary 46L89, 46M10; Secondary 22D25, 46L07, 46L65, 47L25, 47L50, 81R15. Key words and phrases. Compact quantum group, biprojective, Kac algebra, modular auto- morphism group.

c 2009 American Mathematical Society Reverts to public domain 28 years from publication 1349

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to ∆∗ can be chosen to be completely contractive, then G must already be a Kac algebra. We make some remarks on the general case. We indicate that the modular theory of the Haar state seems to be important outside of the Kac case, and it seems likely that a better understanding of how to deal with how the coproduct interacts with the modular automorphism group will be necessary to completely characterise when the convolution algebra of G is operator biprojective. We shall follow the notation of [6], and in particular, write ⊗
for the operator space projective tensor product, and write CB(E,F) to denote the space of complete bounded linear maps between the operator spaces E and F . We would like to thank the referee for helpful comments which have improved the exposition and for bringing [4] to our attention.

2. Locally compact quantum groups Locally compact quantum groups [11, 10] are an axiomatic framework which encompasses the L1(G) algebras, the Fourier algebra A(G), and various “quantum” examples, for example, Woronowicz’s compact quantum groups. Kac algebras [7] are an earlier axiomatic framework which fails to encompass many of the “quantum” examples, for example [26]. However, we shall concentrate on the compact case, which is technically easier. We shall follow the presentation of [21], which in turn closely follows Woronowicz’s original papers [24] and [25]. See also readable, non-technical accounts in [10], and the survey [13], although be aware that these sources use different notation. A compact quantum semigroup is a unital C∗-algebra A equipped with a unital ∗-homomorphism ∆ : A → A ⊗min A such that (∆ ⊗ ι)∆ = (ι ⊗ ∆)∆. A com- pact quantum group is a compact quantum semigroup (A, ∆) which satisfies the cancellation laws,namelythat ∆(A)(A ⊗ 1) := lin{∆(a)(b ⊗ 1) : a, b ∈ A}, ∆(A)(1 ⊗ A)

are both dense in A⊗minA.IfG is a compact semigroup, then we may set A = C(G) and ∆(f)(s, t)=f(st) to get a compact quantum semigroup (A, ∆). Then the cancellation laws correspond to G having the cancellation laws: namely that if st = sr for s, t, r ∈ G,thent = r, and similarly with the orders reversed. As sketched in [13], these are equivalent to G being a group. From now on, fix a compact quantum group (A, ∆). These axioms imply that A carries a unique Haar state, that is, a state ϕ ∈ A∗ such that (ϕ ⊗ ι)∆(a)=ϕ(a)1 = (ι ⊗ ϕ)∆(a)(a ∈ A). WecanformtheGNSconstruction(H, Λ) for ϕ. We shall always suppose that (A, ∆) is reduced,thatis,thatϕ is faithful. As such, we shall identify A with a concrete C∗-algebra acting on H.Ifϕ is not faithful, then we may quotient by its kernel N = {a ∈ A : ϕ(a∗a)=0} to obtain a reduced compact quantum group. Note that N is an ideal because ϕ is a KMS weight (see below); see the details in [3, Theorem 2.1]. Let M = A be the von Neumann algebra generated by A.Then∆ex- tends to a normal ∗-homomorphism ∆ : M → M⊗M. Then, by [6, Theo- rem 7.2.4], (M⊗M)∗ = M∗⊗
M∗ and normality of ∆ induces a complete contraction ∆∗ : M∗⊗
M∗ → M∗. That ∆ is coassociative implies that ∆∗ is associative, so M∗ becomes a completely contractive Banach algebra. If we started with a compact

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1 group G,thenM∗ is nothing but L (G), and so we refer to M∗ as the convolu- tion algebra of (A, ∆). For more on (locally) compact quantum groups in the von Neumann algebra setting, see [12]. A finite-dimensional corepresentation of (A, ∆) is an invertible matrix u = (ui,j ) ∈ Mn(A) such that n ∆(uij)= uik ⊗ ukj (1 ≤ i, j ≤ n). k=1 There are suitable notions of intertwiner between corepresentations, and what an irreducible corepresentation is. Every finite-dimensional corepresentation can be written as the direct sum of irreducible corepresentations. Using the Haar state, it can be shown that every finite-dimensional corepresentation is equivalent to a unitary one, that is, where u ∈ Mn(A) is unitary. The general corepresentation theory of (A, ∆) parallels the representation theory of compact groups very closely. { α α nα ∈ A} Let u =(uij)i,j=1 : α be a maximal family of mutually inequivalent, finite-dimensional irreducible unitary corepresentations of (A, ∆). Let α0 ∈ A be such that vα0 = 1, the trivial corepresentation. Let A be the algebra generated { α ∈ A ≤ ≤ } A ∗ { α ∈ by uij : α , 1 i, j nα in A.Then is a Hopf -algebra,and uij : α A, 1 ≤ i, j ≤ nα} forms a basis for A. This means that A is a ∗-algebra, that ∆ restricts to give a ∗-homomorphism ∆ : A→A⊗A(the algebraic tensor product) and there exist maps  : A→C and S : A→A,thecounit and antipode, satisfying the usual properties. Indeed, for α ∈ A and 1 ≤ i, j ≤ nα,wehavethat   nα       α α ⊗ α α α ∗ α α ∆ ui,j = ui,k uk,j,S(ui,j )= uj,i ,ui,j = δij,ϕui,j = δα,α0 . k=1 Furthermore, for each α ∈ A, there exists a unique positive invertible matrix F α ∈ M α α −1 nα with Tr F =Tr(F ) ,andsuchthat   ((F α)−1)   F α ϕ (uβ )∗uα = δ δ ki ,ϕuβ (uα )∗ = δ δ lj . ij kl αβ jl Tr(F α) ij kl αβ ik Tr(F α) The Hopf ∗-algebra A is norm dense in A and is the unique such dense Hopf ∗- algebra; see [3, Appendix A]. These “F -matrices” allow us to define characters on A.Forz ∈ C, define   A→C α → α z fz : ,uij (F ) ij. As F α is positive, the matrix (F α)z makes sense. Then, for w, z ∈ C, define

nα A→A α → α α α ρz,w : ,uij fw(uik)fz(ulj)ukl. k,l=1

Then ρz,w is an automorphism of A with inverse ρ−z,−w,andifz and w are purely imaginary, then ρz,w is a ∗-automorphism of A. In particular, set

σz = ρiz,iz,τz = ρ−iz,iz (z ∈ C).

Then (σt)t∈R is the (restriction) of the modular automorphism group for ϕ,and (τt)t∈R is the (restriction) of the scaling group. For example, we can calculate that ϕ(aσ−i(b)) = ϕ(ba)fora, b ∈A, a relation which we expect, as ϕ is KMS for σ.

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See [20] for more details on the modular theory of weights on general von Neu- mann algebras, and [11], [24] and [25] for the specific theory as applied to compact quantum groups (and for details of the scaling group). Proposition 2.1. There exists a maximal family of mutually inequivalent, finite- { α α nα dimensional irreducible unitary corepresentations of (A, ∆),say v =(vij)i,j=1 : α α ∈ A}, with the property that the associated F-matrices are all diagonal, say F α nα α α −1 has diagonal entries (λi )i=1, so that i λi = i(λi ) = Trα,say. { α α nα ∈ A} Proof. Start with some maximal family u =(uij)i,j=1 : α as before. As each α α ∈ M α nα F is positive it can be diagonalised by some unitary matrix Q nα .Let( λi )i=1 α α α α −1 α −1 be the eigenvalues of F ,sothatTr(F )= i λi =Tr((F ) )= i(λi ) . α ∗ α α α nα Then (Q ) F Q is the diagonal matrix with entries (λi )i=1.Set   nα α α ∗ α α α α α ∈ A ≤ ≤ vij = (Q ) u Q ij = QkiuklQlj (α , 1 i, j nα). k,l=1 It is now routine to check that vα is a unitary corepresentation matrix, and that { α } the properties above still hold for the family vij . For example, we see that     ∗   β ∗ α β β β α α α ϕ (vij) vkl = ϕ QriursQsj QtkutpQpl r,s t,p    β β α α β ∗ α = QriQsjQtkQplϕ (urs) utp r,s,t,p 1  = δ Qβ Qβ Qα Qα ((F α)−1) αβ Tr(F α) ri sj tk sl tr r,s,t 1    1 1 = δ (Qα)∗ Qα (Qα)∗(F α)−1Qα = δ δ δ . αβ Tr js sl ki αβ jl ki Tr λα α s α i Similar calculations show that   α β α ∗ λj ϕ vij(vkl) = δαβδikδjl Trα and also α α z α α w α z α
fz(vij)=δij(λi ) ,ρz,w(vij)=(λi ) (λj ) vij . Henceforth, the family of corepresentations shall always be given by this propo- { α ∈ A ≤ ≤ } sition; notice in particular that vij : α , 1 i, j nα forms a linear basis for A.

3. Biprojectivity Let (A, ∆) be a reduced compact quantum group, with associated Haar state ϕ, GNS construction (H, Λ), von Neumann algebra M and convolution algebra M∗. We shall study when M∗ is operator biprojective, that is, whether there is a completely bounded right inverse to ∆∗ : M∗⊗
M∗ → M∗ which is also an M∗- bimodule homomorphism. Henceforth, we shall term such a map θ∗ a splitting morphism. See [1, 2] for further details on the operator space case, and see [9, Chapter IV] or [17, Section 4.3] for the classical Banach space setting.

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Lemma 3.1. M∗ is biprojective if and only if there exists a normal completely bounded map θ : M⊗M → M with θ∆=id, ∆θ =(θ ⊗ id)(id ⊗∆) = (id ⊗θ)(∆ ⊗ id).

Proof. Suppose that such a θ exists; then as θ is normal, there exists θ∗ : M∗ → M∗⊗
M∗ with ∆∗θ∗ =id.Then,forω, τ ∈ M∗ and x ∈ M,

x, θ∗(ω ∗ τ) = θ(x), ∆∗(ω ⊗ τ) = (θ ⊗ id)(id ⊗∆)(x),ω⊗ τ

= (id ⊗∆)(x),θ∗(ω) ⊗ τ = x, θ∗(ω) ∗ τ .

Here we write ∗ for both the product in M∗, and the bimodule action of M∗ on M∗⊗
M∗. Similarly, θ∗(ω ∗ τ)=ω ∗ θ∗(τ), so we see that θ∗ is an M∗-bimodule homomorphism. The converse is simply a case of reversing the argument.

In the following section, we shall carefully study the structure of normal com- pletely bounded maps M⊗M → M.Fromnowon,fixsuchamapθ : M⊗M → M { α nα ∈ A} and let (vij)i,j=1 : α be as in Proposition 2.1. Proposition 3.2. We have that θ∆=idand ∆θ =(θ ⊗ id)(id ⊗∆) = (id ⊗θ)(∆ ⊗ { α ∈ M ∈ A} ∈ A id) if and only if there exists a family X nα : α such that, for α, β , 1 ≤ i, j ≤ nα and 1 ≤ k, l ≤ nβ,

  nα α ⊗ β α α α θ vij vkl = δαβXjkvil, Xrr =1. r=1 Proof. The “if” part follows as A generates M and θ is normal. Conversely, let x ∈ M and α ∈ A.For1≤ i, j ≤ nα,

    nα   ⊗ α ⊗ ⊗ ⊗ α ⊗ α ⊗ α ∆θ x vij =(θ id)(id ∆) x vij = θ x vir vrj. r=1  ⊗ α ⊗ α ∗ Let aij = θ(x vij ), so that ∆(aij)= r air vrj. As ∆ is a -homomorphism, for 1 ≤ k, l ≤ nα,wehavethat

  nα α ∗ α ∗ ⊗ α α ∗ ∆ aij(vkl) = air(vks) vrj(vsl) . r,s=1 Applying (ι ⊗ ϕ), we see that, by the calculations in Proposition 2.1,   nα   λα nα ϕ a (vα )∗ 1= a (vα )∗ϕ vα (vα )∗ = δ j a (vα )∗. ij kl ir ks rj sl jl Tr ir kr r,s=1 α r=1  α α ∗ α ≤ ≤ As v is a unitary matrix, we see that k(vkr) vks = δrs1for1 r, s nα.Thus nα   α nα α α ∗ α λj α ∗ α λj ϕ aij(vkl) vks = δjl air(vkr) vks = δjl ais. Trα Trα k=1 r,k=1 It follows that Tr nα     a = α ϕ a (vα )∗ vα α ∈ A, 1 ≤ i, j, s ≤ n . is λα ij kj ks α j k=1

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1354 MATTHEW DAWS  α ⊗ α ⊗ Similarly, if we set bij = θ(vij x), then ∆(bij)= r vir brj, and we can show that nα     α α ∗ α ∈ A ≤ ≤ bsj = λi Trα ϕ (vik) bij vsk α , 1 i, j, s nα . k=1 α ⊗ β { α ≤ ≤ } In particular, we see that θ(vij vkl) is in the linear span of vis :1 s nα { β ≤ ≤ } α ⊗ β and the linear span of vrl :1 r nβ . Hence θ(vij vkl)=0ifα = β.Ifα = β, then by linear independence, we see immediately that α ⊗ α α α θ(vij vkl)=Xjkvil,   α α ⊗ α α α for some scalar Xjk. Finally, as k θ(vik vkj)=vij , it follows that k Xkk =1, as required.

Theorem 3.3. Let (A, ∆) be a compact quantum group with associated von Neu- mann algebra M.Letθ∗ : M∗ → M∗⊗
M∗ be a splitting morphism, and suppose ∗ further that θ = θ∗ is an M-bimodule map, in the sense that θ(∆(a)x∆(b)) = aθ(x)b for x ∈ M⊗M and a, b ∈ M. Then the Haar state ϕ is tracial, so (M,∆) is a Kac algebra.

Proof. Let α ∈ A and 1 ≤ i, j, k ≤ nα.Asθ(x∆(b)) = θ(x)b for x ∈ M⊗M and b ∈ M, using the notation of the last proposition, we see that

  nα   α α α ∗ α ⊗ α α ∗ α α ∗ ⊗ α α ∗ (3.1) Xjkvij (vij) = θ vij vkj (vij) = θ vij (vil) vkj(vlj) . l=1 { β } ∗ A Now, as vrs forms a basis for the -algebra ,andasϕ picks out the trivial corepresentation vα0 = 1, by the calculations of Proposition 2.1, we see that λα λα α α ∗ ⊗ α α ∗ j ⊗ j vij (vil) vkj(vlj) = δjl 1 δkl 1+otherterms. Trα Trα { β } A By “other terms”, we are thinking of expanding using the basis vrs for ;in particular, all these elements are in ker ϕ ⊗ ker ϕ. The last proposition shows in particular that ϕθ = ϕ ⊗ ϕ, and hence it follows that n   α   λα 2 α α ∗ ⊗ α α ∗ j ϕθ vij (vil) vkj(vlj) = δjk . Trα l=1 By applying ϕ to (3.1), we conclude that

α  α 2 α α λj λj α λj Xjk = δjk so that Xjk = δjk . Trα Trα Trα We now repeat this argument on the right, so we find that      α α ∗ α α ∗ α ⊗ α α ∗ α ⊗ α ∗ α Xjk(vij) vij =(vij) θ vij vkj = θ (vis) vij (vsj) vkj s  1 1 = δ δ 1 + other terms. sj λα Tr sk λα Tr s i α k α Again, by applying ϕ we see that

α 1 1 1 α 1 Xjk α = δjk α α so that Xjk = δjk α . λi Trα λi Trα λk Trα λk Trα

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≤ ≤ α α α Hence we see that for all α and 1 k nα,wehaveλk =1/λk .Asλk > 0, we α see that λk = 1. In particular, the modular automorphism group σ is trivial, and so ϕ is tracial, as claimed. α α −1 Indeed, if ϕ is tracial, then from Proposition 2.1 we see that λj =(λi ) for α all i, j.Thusλi =1foralli and α. It follows that the automorphisms ρz,w are trivial, and hence also the scaling group is trivial. So the antipode S is bounded; see also [24, Theorem 2.5]. It is now easy to verify the axioms of a compact Kac algebra; see [7, Section 6.2].

We note that an argument of Soltan ([18, Remark A.2]) shows that if a compact quantum group (A, ∆) has a faithful family of tracial states (that is, for non-zero x ∈ A there is a tracial state φ with φ(x∗x) = 0), then (M,∆) is a Kac algebra. Theorem 3.4. Let (A, ∆) be a compact quantum group with associated von Neu- mann algebra M.Letθ∗ : M∗ → M∗⊗
M∗ be a splitting morphism. Suppose that ∗ θ = θ∗ is completely positive or that ∆θ is a contraction. Then (M,∆) is a Kac algebra. Proof. As θ(1) = θ∆(1) = 1, if θ is positive, then θ is contractive, so ∆θ is contrac- tive. We have that ∆θ : M⊗M → M⊗M is contractive and is a projection of M⊗M onto the subalgebra ∆(M). A result of Tomiyama ([22] or [19, Theorem 3.4, Chap- ter III]) tells us that, in particular, ∆θ(∆(a)x∆(b)) = ∆(a)θ(x)∆(b)fora, b ∈ M and x ∈ M⊗M. As ∆ is an injective homomorphism, the above theorem ap- plies.

In the following section, we shall show the converse to this corollary: namely, that for a compact Kac algebra (M,∆), we can choose θ to be a complete contraction; alternatively, see [15] or [1]. It is shown in [5] that if we have a completely bounded map θ : M⊗M → M with θ∆ = id, then there exists a completely bounded map θ1 : M⊗M → M which is an M-bimodule map, in the above sense. However, there is no reason that θ1 need be normal, and no reason that the other conditions on θ will carry over to θ1,so that Proposition 3.2 need not apply to θ1. We can even choose θ1 to be completely positive, which were it also faithful would imply, by [20, Theorem 4.2, Chapter IX], the existence of a weight ω on M⊗M with interesting modular properties. Again, there seems to be no reason to expect that we can choose θ1 in such a way.

4. Completely bounded maps There is a well-known structure theory for completely bounded maps [6, Sec- tion 5.3]. If θ : N → M ⊆B(H) is a completely positive normal map between von Neumann algebras, then the usual proof of the Stinespring theorem (for ex- ample, [19, Chapter IV, Theorem 3.6]) can be adapted to show that there exists a Hilbert space K,anormal ∗-homomorphism π : N →B(K) and a bounded map U : H → K such that θ(x)=U ∗π(x)U for x ∈ N. Showing the same for completely bounded maps is not quite as simple, but the details are worked out in, for example, the proof of [8, Theorem 2.4]. In particular, given θ : N → M ⊆B(H), a completely contractive normal map between von Neu- mann algebras, there exist unital completely positive normal maps φ1,φ2 : N → M

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such that   ∗ ∗ ab φ1(a) θ(b ) σ : M2(N) → M2(M); → cd θ(c) φ2(d) is unital completely positive and normal. One can now follow the presentation in [6, Theorem 5.33] or [14], essentially applying the Stinespring construction to σ. This yields a Hilbert space K,anormal∗-homomorphism ρ : M2(N) →B(K)and 2 ∗ an isometry U : H → K such that σ(x)=U ρ(x)U for x ∈ M2(N). Following the proof of [8, Theorem 2.3], there also exists a normal ∗-homomorphism ρ :     M2(M) → ρ(M2(N)) such that ρ (y)U = Uy for y ∈ M2(M) . Define π : M⊗M →B(K), π : M  →B(K), and R, T : H → K by      x 0   y 0 00 ξ 0 π(x)=ρ ,π(y)=ρ ,T(ξ)=ρ U ,R(ξ)=U , 0 x 0 y 10 0 ξ

for x ∈ M⊗M,y ∈ M  and ξ ∈ H.Soπ and π are normal ∗-homomorphisms, and R and T are contractions. Then, for x ∈ M⊗M and ξ,η ∈ H,         ∗ x 0 00 ξ 0 R π(x)Tξ η = ρ ρ U U 0 x 10 0 η      00 ξ 0   = σ = θ(x)ξ η . x 0 0 η So θ(x)=R∗π(x)T . Then also, for y ∈ M  and ξ ∈ H,      00 y 0 ξ 00  ξ  Tyξ = ρ U = ρ π (y)U = π (y)Tξ. 10 0 y 0 10 0 So Ty = π(y)T and similarly Ry = π(y)R,fory ∈ M . Let M be a von Neumann algebra with a normal faithful state ϕ, leading to the GNS construction (H, Λ)(hereweidentifyM with a subalgebra of B(H)). We can apply Tomita-Takesaki theory to find an anti-linear isometry J : H → H such that  M = JMJ (see [20]). Let (σt)t∈R be the modular automorphism group and let A⊆M be a ∗-subalgebra of elements analytic for (σt) such that σz(a) ∈Afor  ∗ z ∈ C and a ∈A.Fora ∈A,writea = Jσi/2(a) J.Then  ∗ a Λ(1) = Jσi/2(a) JΛ(1) = Λ(a)(a ∈A). Proposition 4.1. Let M be a von Neumann algebra as above, and suppose that A = M.LetN be a von Neumann algebra. If θ : N → M is a completely bounded normal map, then we can find a Hilbert space K,normal∗-homomorphisms    π : N →B(K) and π : M → π(N) ,andξ0,ξ1 ∈ K such that the maps     Λ(a) → π (a )ξ0, Λ(a) → π (a )ξ1 (a ∈A) are bounded and     (4.1) ϕ(θ(x)a)= π(x)π (a )ξ0 ξ1 (x ∈ N,a ∈A).  Conversely, given such K, π, π ,ξ0 and ξ1, there exists a completely bounded normal map θ : N → M satisfying (4.1). Furthermore, θ is completely positive if and only if we can choose ξ0 = ξ1.

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Proof. As A = M, it follows that A is strongly dense in M and hence that Λ(A)is norm dense in H.Ifθ is of the form claimed, then the map T :Λ(A) → K;Λ(a) →   π (a )ξ0 is bounded and so extends to a bounded linear map T : H → K. Similarly,   there exists R ∈B(H, K) with RΛ(a)=π (a )ξ1. Then, for a, b ∈Aand x ∈ N,       R∗π(x)T Λ(a) Λ(b) = π(x)π(a)ξ π(b)ξ = π(x)π((b)∗a)ξ ξ  0  1 0 1 ∗ = ϕ θ(x)aσ−i(b ) ,  ∗ ∗ ∗ ∗  ∗  as (b ) =(Jσi/2(b) J) = Jσi/2(b)J = Jσi/2(c) J = c if c = σ−i(b ), and d → d is an anti-homomorphism. By the KMS condition, we see that       R∗π(x)T Λ(a) Λ(b) = ϕ b∗θ(x)a = θ(x)Λ(a) Λ(b) . ∗ Hence θ is completely bounded, as θ(x)=R π(x)T for x ∈ N.Ifξ0 = ξ1,then R = T and θ is completely positive. Conversely, given θ, from the discussion above, we can find normal ∗-homo- morphisms π : N →B(K)andπ : M  → π(N), and bounded maps R, T : H → K with θ(x)=R∗π(x)T for x ∈ N and Ry = π(y)R, T y = π(y)T for y ∈ M .Thus, for x ∈ N and a ∈A,     ϕ(θ(x)a)= R∗π(x)T Λ(a) Λ(1) = R∗π(x)π(a)T Λ(1) Λ(1) ,

so the proof is complete by setting ξ0 = T Λ(1) and ξ1 = RΛ(1). If θ is completely positive, then we can set R = T , and hence ξ0 = ξ1.
Notice that by the KMS condition, the calculations above also show that if x, y ∈ M are such that ϕ(xa)=ϕ(ya) for all a ∈A,thenx = y. The following is proved using different methods in [15] and [1]. Our proof makes explicit how ϕ being tracial, for a Kac algebra, is central to the proof and indicates that understanding the modular properties of ϕ for a general compact quantum group will be important in finding a completely bounded analogue of the following. Theorem 4.2. Let (M,∆) be a compact Kac algebra. Then there exists a splitting
∗ morphism θ∗ : M∗ → M∗⊗M∗ such that θ = θ∗ is completely positive. Proof. We have that ϕ is tracial. Let π : M⊗M → M⊗M ⊆B(H ⊗ H)bethe  trivial representation, let ξ0 = ξ1 =Λ(1)⊗ Λ(1), and define π by π(y)=(J ⊗ J)∆(JyJ)(J ⊗ J)(y ∈ M ). This formula is simply the natural coproduct on M ; see [12, Section 4]. Let A be the Hopf ∗-algebra associated to (M,∆), as before. Then we can apply the above proposition to see that there exists a completely positive normal map θ : M⊗M → M such that   ϕ(θ(x)a)= x(J ⊗ J)∆(a∗)(J ⊗ J)Λ(1) ⊗ Λ(1) Λ(1) ⊗ Λ(1) , where we use that σ is trivial, as ϕ is tracial. Then JΛ(a)=Λ(a)∗ for a ∈A,and so, as ∆(a∗) ∈A⊗A,   ϕ(θ(x)a)= x(J ⊗ J)∆(a∗)Λ(1) ⊗ Λ(1) Λ(1) ⊗ Λ(1)     = x(Λ ⊗ Λ)∆(a) Λ(1) ⊗ Λ(1) =(ϕ ⊗ ϕ) x∆(a) .

In particular,   ϕ(θ∆(x)a)=(ϕ ⊗ ϕ) ∆(xa) = ϕ(xa),

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so by the observation above, θ∆ = id. Indeed, one may calculate (thinking about Proposition 3.2) that α ⊗ α 1 α θ(vij vkl)= δjkvil, nα α ⊗ ⊗ ⊗ ⊗ using that λi =1forallα and i.Thusalso∆θ =(θ id)(id ∆) = (id θ)(∆ id), and so θ∗, the preadjoint to θ, is a splitting morphism, as required.
If ϕ is not tracial, then the above proof fails, as for a ∈A,   ∗ ∗ ∆(σi/2(a) )= (τi/2 ⊗ σi/2)∆(a) , ∗ and hence, as JΛ(b)=Λ(σi/2(b) )forb ∈A,   ∗ (J ⊗ J)∆(σi/2(a) )(J ⊗ J)(Λ(1) ⊗ Λ(1)) = (Λ ⊗ Λ) (τi/2σ−i/2 ⊗ id)∆(a) . If we continue to form θ as above, then we find that   α ⊗ β α δjk ∈ A ≤ ≤ ≤ ≤ θ vij vkl = δαβvil (α, β , 1 i, j nα, 1 k, l nβ). Trα This is nearly of the correct form, but we find that   α nα α ∈ A ≤ ≤ θ∆ vij = vij (α , 1 i, j nα). Trα        α 1/2 α −1/2 ≤ α 1/2 α −1 1/2 Notice that as nα = i(λi ) (λi ) i λ i(λ ) =Trα,it α follows that θ∆ = id if and only if λi =1forallα, i; that is, again, ϕ is tracial. It is curious that exactly the same formula, in a seemingly rather different context, arises in [4, Theorem 4.5]. The remarks in [4] after this theorem are also of interest and suggest that a simple “rescaling” argument is unlikely to be able to turn θ into a map with the required properties.

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School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom E-mail address: [email protected]

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