On the Baum--Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg Conjecture
aa r X i v : . [ m a t h . OA ] S e p ON THE BAUM–CONNES CONJECTURE FOR DISCRETE QUANTUMGROUPS WITH TORSION AND THE QUANTUM ROSENBERGCONJECTURE
YUKI ARANO AND ADAM SKALSKI
Abstract.
We give a decomposition of the equivariant Kasparov category for discrete quantumgroup with torsions. As an outcome, we show that the crossed product by a discrete quantumgroup in a certain class preserves the UCT. We then show that quasidiagonality of a reducedC ∗ -algebra of a countable discrete quantum group Γ implies that Γ is amenable, and deducefrom the work of Tikuisis, White and Winter, and the results in the first part of the paper,the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discreteunimodular quantum groups. We also note that the unimodularity is a necessary condition. Introduction
In [RoS], Rosenberg and Schochet have introduced a property of C*-algebras called the UniversalCoefficient Theorem (UCT in short) for K -theory of C*-algebras and have shown that it holdsfor all C*-algebras in the so-called bootstrap class. The UCT gives a formula computing the KK -groups only from the K -groups. This property plays an important role in the classificationof nuclear C*-algebras (see e.g. [TWW] and the last section of this paper).The UCT for group C*-algebras is related to (a variation of) the Baum–Connes conjecture ofgroups. In [Tu], Tu proved that the group C*-algebra of a discrete group with Haagerup propertysatisfies the UCT using the Higson–Kasparov type argument [HiK] for groupoids.The Baum–Connes conjecture for quantum groups first appeared in the series of works ofMeyer and Nest [MN ], [Mey] (after an early paper [GoK]). Even though there is no unifiedmethod proving the Baum–Connes conjecture for fairly general quantum groups, it is proven formany known examples of discrete quantum groups: [FrM], [VeV], [Vo ], [Vo ].In this paper, we study the general theory of the Baum–Connes conjecture for discrete quantumgroup with possible torsion. In particular we give a decomposition of the equivariant category KK G (where G is any compact quantum group), which gives the Baum–Connes assembly map. Asa byproduct of the general theory, we prove that the group C*-algebra of a discrete quantum groupsatisfying the Baum–Connes conjecture satisfies also the UCT. This is applied in the last sectionof the paper to the considerations regarding the quantum version of the Rosenberg Conjecture,connecting amenability of a discrete group to quasidiagonality of its C*-algebra.The detailed plan of the paper is as follows: in the following section we introduce the notationand some background related to discrete/compact quantum groups and triangulated categories.In Section 3 we present a ‘crossed product type’ construction for two C*-algebras equipped re-spectively with left and right action of a given compact quantum group, which is then applied inSection 4 to build an adjoint functor between certain KK -categories. In Section 5 we establishas a consequence a relationship between the h Cof i -Baum–Connes property of a discrete quantumgroup and the Universal Coefficient Theorem for some crossed products. Finally in Section 6 theapplications to quantum Rosenberg Conjecture are discussed. Acknowledgment.
This work was initiated in the workshop “The 6th Workshop on OperatorAlgebras and their Applications” in the School of Mathematics of Institute for Research in Fun-damental Sciences (IPM). The authors would like to thank the organizers and IPM for theirhospitality. Y.I. is supported by JSPS KAKENHI Grant Number JP18K13424. A.S. was partially
Mathematics Subject Classification.
Primary 46L67, Secondary 46L80.
Key words and phrases.
Quantum group, triangulated categories, UCT, Rosenberg conjecture. supported by the National Science Centre (NCN) grant no. 2014/14/E/ST1/00525. He acknowl-edges discussions with Pawe l J´oziak, Piotr So ltan, Stuart White and Joachim Zacharias on thesubject of the last section of the paper.2.
Preliminaries
Quantum groups.
Let Γ be a discrete quantum group (so that b Γ is a compact quantumgroup in the sense of [Wo ] – we refer to that paper for the details of the facts introduced below,and often write simply G for the dual compact quantum group). We study Γ via its algebra offunctions, c ( Γ ). Recall that c ( Γ ) = M α ∈ Irr b Γ M n α , where Irr b Γ denotes the set of equivalence classes of irreducible unitary representations of b Γ ; thespan of coefficients of the latter is a Hopf *-algebra denoted O ( b Γ ), admitting a Haar (bi-invariant)state h . Note that we will also write c c ( Γ ) for the algebraic direct sum: c c ( Γ ) = alg M α ∈ Irr b Γ M n α , The C ∗ -algebra C ∗ r ( Γ ), often written as C( b Γ ), is the C ∗ -completion of O ( b Γ ) in the GNS repre-sentation with respect to h . For each α ∈ Irr b Γ we choose a representative, i.e. a unitary matrix U α = ( u αi,j ) n α i,j =1 ∈ M n α (C ∗ r ( Γ )). We may and do assume that c ( Γ ) is represented on the Hilbertspace ℓ ( Γ ) (viewed here as the GNS space of the Haar state of b Γ , so also the space on whichC ∗ r ( Γ ) acts); this representation will be later denoted by ˆ λ . The matrix units in M n α ⊂ c ( Γ ) willbe denoted by e αi,j .The multiplicative unitary of Γ is the unitary W ∈ B ( ℓ ( Γ ) ⊗ ℓ ( Γ )) given by the formula: W = X α ∈ Irr b Γ n α X i,j =1 e αj,i ⊗ ( u αi,j ) ∗ The von Neumann completion of c ( Γ ) will be denoted by ℓ ∞ ( Γ ). The predual of ℓ ∞ ( Γ ) will bedenoted by ℓ ( Γ ).The coproduct of Γ , a coassociative normal unital ∗ -homomorphism ∆ : ℓ ∞ ( Γ ) → ℓ ∞ ( Γ ) ⊗ ℓ ∞ ( Γ )is implemented by W via the following formula:(2.1) ∆( x ) = W ∗ (1 ⊗ x ) W, x ∈ ℓ ∞ ( Γ ) . Given a functional φ ∈ ℓ ( Γ ) we define the (normal, bounded) maps L φ : ℓ ∞ ( Γ ) → ℓ ∞ ( Γ ) and R φ : ℓ ∞ ( Γ ) → ℓ ∞ ( Γ ) via the formulas L φ = ( φ ⊗ id) ◦ ∆ , R φ = (id ⊗ φ ) ◦ ∆ . A discrete quantum group Γ is said to be finite , if Irr b Γ is finite (equivalently, c ( Γ ) is finite-dimensional), and countable , if Irr b Γ is countable (equivalently, c ( Γ ) is separable); it is unimodular if its left and right Haar weights coincide; equivalently the Haar state h of b Γ is tracial.A discrete quantum group Γ is called amenable if it admits a bi-invariant mean, i.e. a state m ∈ ℓ ∞ ( Γ ) ∗ , such that for all φ ∈ ℓ ( Γ ) there is m ◦ L φ = m ◦ R φ = φ (1) m. By [DQV] a discrete quantum group Γ is amenable if it admits a left invariant mean m ∈ ℓ ∞ ( Γ ) ∗ :a state such that for each φ ∈ ℓ ( Γ ) there is m ◦ L φ = φ (1) m . In fact it suffices to check the lastformula for the functionals of the form c e αi,j , α ∈ Irr b Γ , i, j = 1 , . . . , n α , as the latter are linearlydense in ℓ ( Γ ), and the map φ L φ is a (complete) isometry. Thus we will need the following UANTUM BAUM-CONNES AND ROSENBERG CONJECTURES 3 explicit form of the map L φ for φ = c e αi,j :(2.2) L φ ( x ) = n α X p =1 u αi,p x ( u αj,p ) ∗ (with x ∈ ℓ ∞ ( Γ )).Recall that G denotes the dual compact quantum group of Γ . The left regular representation λ : C ( G ) → B ( L ( G )) is the GNS representation with respect to the Haar state ϕ ; note that L ( G )is canonically isomorphic to ℓ ( Γ ). We also have the right regular representation ρ ( x ) = JR ( x ) ∗ J , x ∈ C ( G ), where J is the modular conjugation and R is the unitary antipode.Via the natural pairing O ( G ) × c c ( Γ ) → C , we put a (multiplier) Hopf algebra structure on c c ( Γ ).For details of quantum group actions and the associated crossed products we refer for exampleto [DC] and [Vae]; note that we always work with reduced/faithful actions. Given a left action α : A → C ( G ) ⊗ A we call A a G -C*- algebra . Such an action induces a right c c ( Γ )-comodulealgebra structure on A : a ⊳ x := ( x ⊗ id) α ( a )for a ∈ A and x ∈ c c ( Γ ). Similarly a right action β : B → B ⊗ C ( G ) induces a left c c ( Γ )-comodulealgebra structure on B : x ⊲ b := (id ⊗ x ) β ( b )for b ∈ B and x ∈ c c ( Γ ).For a finite dimensional C*-algebra D and a left action α of G on D , there always exists a G -invariant state ϕ D on D , which is of the form ϕ D = Tr( ρ · ), where Tr is the trace taking value 1 ateach minimal projection and ρ ∈ D . Let ( λ D , L ( D ) = L ( D, ϕ D ) , Ω D ) be the GNS representationof ϕ D . Then L ( D ) also carries a ∗ -representation ρ D of the opposite C*-algebra D op , given by ρ D ( x op ) λ D ( y )Ω D = λ D ( yρ / xρ − / )Ω D , x, y ∈ D. Furthermore the formula U ∗ ( a ⊗ x Ω) = α ( x )( a ⊗ Ω) , a ∈ C ( G ) , x ∈ D, defines a unitary representation U ∈ C ( G ) ⊗ B ( L ( D )). Definition 2.1. [BS] For a C*-algebra A with a G -action α : A → C ( G ) ⊗ A and a Hilbert A -module E , a G -action on E is a linear map α E : E → C ( G ) ⊗ E such that(1) α E ( xa ) = α E ( x ) α ( a ) for all x ∈ E , a ∈ A ,(2) the linear span α E ( E )( C ( G ) ⊗
1) is dense in C ( G ) ⊗ E and(3) (id ⊗ α E ) α E = (∆ ⊗ id) α E .This is equivalent to say that α E is the corner of an action of G on the linking algebra K ( E ⊕ A ) ≃ (cid:18) K ( E ) EE ∗ A (cid:19) which coincides with α on A ∼ = (cid:18) A (cid:19) . See [BS] for details.Finally for the notion of torsion in the context of compact quantum groups we refer for exampleto [ADC].2.2. Triangulated category.
In [MN ], Meyer and Nest introduced a framework to work on KK -theory in terms of triangulated categories. In this section, we review their work, not goinginto the full generality of triangulated categories but only restricting ourselves to describe thesituation in terms of G -equivariant KK -theory, where G is a fixed compact quantum group witha countable dual.To each equivariant ∗ -homomorphism ϕ : A → B , one can associate an exact sequence calledthe mapping cone exact sequence: 0 → C ϕ ι −→ M ϕ ev −−→ B → YUKI ARANO AND ADAM SKALSKI where M ϕ = { ( f, a ) ∈ ( C [0 , ⊗ B ) ⊕ A } | f (1) = ϕ ( a ) } ⊃ C ϕ = { ( f, a ) ∈ M ϕ | f (0) = 0 } . Noticethat M ϕ is homotopy equivalent to A . We may continue this construction for ι to get anothermapping cone exact sequence: 0 → C ι → M ι → M ϕ → . Then C ι is actually homotopy equivalent to the suspension SB = C ( R ) ⊗ B . Hence in thecategory KK G , we get a diagram · · · → SC ϕ → SA → B → C ϕ → A → B or A ϕ / / B ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ C ϕ ◦ ❆❆❆ ` ` ❆❆❆❆ which gives the six-term exact sequence after taking the K -groups. (Here the circle on the ar-row represents the change of the degree.) A distinguished triangle is a diagram which is KK G -equivalent to some mapping cone triangle as above. Definition 2.2. A localizing subcategory of KK G is a full subcategory which is closed undertaking countable direct sums, suspensions and mapping cones.Let P , N be localizing subcategories of KK G . We say that the pair ( P , N ) is complementary if(1) KK G ( P, N ) = 0 for any P ∈ P and N ∈ N .(2) For any A ∈ KK G , there exists a distinguished triangle P ( A ) / / A } } ④④④④④④④④④ N ( A ) ◦ ●●●● c c ●●●● , where P ( A ) ∈ P and N ( A ) ∈ N . Remark . For arbitrary choice of P ( A ) and N ( A ) as above, the morphism P ( A ) → A isuniversal among all morphisms P → A for P ∈ P . In fact, we write the six-term exact sequenceof KK G : · · · → KK G ( P, SN ( A )) → KK G ( P, P ( A )) → KK G ( P, A ) → KK G ( P, N ( A )) → . . . . Since KK G ( P, SN ( A )) = KK G ( P, N ( A )) = 0 by (1), the map KK G ( P, P ( A )) → KK G ( P, A ) isan isomorphism. This is what we claimed.This in particular shows that the triangle P ( A ) → A → N ( A ) is unique up to isomorphism.The following result holds in a more general setting, namely, when the adjoint is only partiallydefined, but we only use it in the following form. Theorem 2.4. [MN , Theorem 3.31] Let H , G be compact quantum groups with countable dualsand let F i be a countable family of functors KK G → KK H which preserve the distinguished tri-angles. Assume that there exist left adjoint functors F ⊥ i : KK H → KK G , i.e. KK G ( F ⊥ i ( A ) , B ) ≃ KK H ( A, F i ( B )) for all C*-algebras A, B equipped respectively with a G and H action. We set P tobe the smallest thick subcategory containing F ⊥ i ( A ) and N to be the full subcategory whose objectis N ∈ KK G such that F i ( N ) is KK H -contractible (note that N is automatically thick). Then ( P , N ) is localizing. In this case, an explicit construction of P ( A ) is given by the phantom castle construction [MN ,Section 3]. We recall the construction in Section 5. UANTUM BAUM-CONNES AND ROSENBERG CONJECTURES 5 Crossed products
Let G be a compact quantum group. For two C*-algebras with G -actions, there is no generalway of constructing the “product” action on the tensor product. However it is possible to constructa C*-algebra like a “crossed product” with respect to the product action. The construction worksfor any locally compact quantum group actions in an obvious manner, but we restrict ourselvesto work with the compact case.Let A (resp. B ) be a C*-algebra with a right (resp. left) G -action: α : A → A ⊗ C ( G ) , β : B → C ( G ) ⊗ B ;these will be fixed throughout this section. By a covariant representation of ( A, B, G ) on aHilbert space H we understand a triple of representations π A : A → B ( H ), π B : B → B ( H ) and U ∈ M ( C ( G ) ⊗ K ( H )), a unitary representation of G , which satisfies the following. • π A ( A ) and π B ( B ) commute; • U ∗ (1 ⊗ π B ( b )) U = (id ⊗ π B ) β ( b ) for any b ∈ B ; • σ ( U )( π A ( a ) ⊗ σ ( U ) ∗ = ( π A ⊗ id) α ( a ) for any a ∈ A (where σ denotes the tensor flip).Recall that we denote the dual discrete quantum group of G by Γ . Take the algebraic cores [DC,Definition 3.15] A and B of A and B . We define a ∗ -algebra A = A ⋊ alg G ⋉ alg B as follows. • As a vector space, A is isomorphic to A ⊗ alg c c ( Γ ) ⊗ alg B . The element in A correspondingto a ⊗ x ⊗ b is denoted by axb for a ∈ A , x ∈ c c ( Γ ) , b ∈ B . • The product is given by ( axb )( a ′ x ′ b ′ ) = a ( x (1) ⊲ a ′ ) x (2) x ′ (1) ( b ⊳ x ′ (2) ) b ′ for a, a ′ ∈ A , x, x ′ ∈ c c ( Γ ), b, b ′ ∈ B . Notice that the sum in the right hand side is finite since a ′ and b are inthe respective algebraic cores.Let A ⋊ G ⋉ B be the universal C*-completion of A . Remark . In the von Neumann algebra setting, a similar construction arises from Popa’s sym-metric enveloping algebra [Popa] (or the Longo–Rehren inclusion [LoR]) of a subfactor of the form M G ⊂ M for a minimal action of G on a factor M . Proposition 3.2.
We have the following. • The C*-algebras
A, B and c ( Γ ) are nondegenerate C*-subalgebras in the multiplier C*-algebra M ( A ⋊ G ⋉ B ) . • There is a natural one-to-one correspondence between the covariant representations of ( A, B, G ) and ∗ -representations of A ⋊ G ⋉ B .Proof. Let A m (resp. B m ) be the universal C*-envelope of A (resp. B ) equipped with a right(resp. left) universal action of G . We first observe that A m coincides with the maximalization of A in the sense of [Fis, Definition 6.1] (See [EcQ] for the group case). To this end, we only need toshow A ⋊ G ⋊ Γ ≃ A m ⊗ K ( L ( G ))(see the proof of [Fis, Theorem 6.4]). Since id ⊗ ϕ : A m ⊗ C ( G ) → ( A m ) α ⊗ A α ⊗ A → A ⊗ alg O ( G ) extends to α m : A → A m ⊗ C ( G ) . Now the triplet ((id ⊗ ρ ) α m , id A m ⊗ ˆ λ, id A m ⊗ λ ) gives a ∗ -homomorphism A ⋊ G ⋊ Γ → A m ⊗ K ( L ( G )) . Conversely the algebraic crossed product A ⋊ alg G ⋊ alg Γ is isomorphic to A ⊗ alg F ( L ( G ))where F ( L ( G )) = span { xy ∈ x ∈ O ( G ) , y ∈ c c ( Γ ) } ⊂ K ( L ( G )) is the ∗ -algebra of all finiterank operators supported on finitely many components in Irr G . Hence the universal completionof A ⋊ alg G ⋊ alg Γ is naturally isomorphic to A m ⊗ K ( L ( G )) and hence the ∗ -homomorphism A ⋊ alg G ⋊ alg Γ → A ⋊ G ⋊ Γ induces a map A m ⊗ K ( L ( G )) → A ⋊ G ⋊ Γ . Since the two mapsdescribed above are inverse to each other, we get the conclusion. In particular, the natural map A m ⋊ G → A ⋊ G is an isomorphism. YUKI ARANO AND ADAM SKALSKI
Since A and c c ( Γ ) satisfy the commutation relation as in A ⋊ alg G , we obtain a nondegenerate ∗ -homomorphism A ⋊ G ≃ A m ⋊ G → M ( A ⋊ G ⋉ B ). Since there exists a non-degenerate ∗ -homomorphism A ⋊ G ⋉ B → M (( A ⋊ G ) ⊗ ( G ⋉ B )) : axb ( a ⊗
1) ˆ∆( x )(1 ⊗ b ) , the map is injective. Similarly we get a nondegenerate injective ∗ -homomorphism B → M ( A ⋊G ⋉ B ). This proves (1).For the assertion (2), by definition of A , we obtain a ∗ -representation of A from a covariantrepresentation of ( A, B, G ). Conversely the covariant representation of ( A, B, G ) is obtained by(1) from a ∗ -representation of A . (cid:3) Lemma 3.3.
Let ϕ A be a G -invariant state. Then ϕ A induces a conditional expectation A ⋊ G ⋉ B → G ⋉ B : axb ϕ A ( a ) xb, where a ∈ A , x ∈ c c ( Γ ) , b ∈ B .Proof. Take the GNS construction ( L ( A, ϕ A ) , Ω) for ϕ A . Consider the Hilbert G ⋉ B -module E = L ( A, ϕ A ) ⊗ G ⋉ B . Then A ⋊ G ⋉ B admits a representation on E defined by (the continuousextension of) the formula( axb )( a ′ Ω ⊗ x ′ b ′ ) = a ( x (1) ⊲ a ′ )Ω ⊗ x (2) x ′ (1) ( b ⊳ x ′ (2) ) b ′ for a, a ′ ∈ A , x, x ′ ∈ c c ( Γ ), b, b ′ ∈ B , where again A and B denote the respective algebraiccores. Take an approximate unit ( e i ) i ∈ I of G ⋉ B . Then the desired conditional expectation isgiven by x lim i ∈ I (Ω ⊗ e i , x (Ω ⊗ e i )) , hence it is well-defined. (cid:3) Using this lemma, we give an easy structural result on this crossed product for later use.
Proposition 3.4.
Let D be a finite dimensional C*-algebra with a right G -action and B be aseparable C*-algebra with a left G -action. (1) If B is finite dimensional, then the C*-algebra D ⋊G⋉ B is a direct sum of matrix algebras. (2) If B is of type I, then the C*-algebra D ⋊ G ⋉ B is also of type I.Proof. (1) We only need to show that any representation of D ⋊ G ⋉ B decomposes into a directsum of finite dimensional representations. To this end, we take a representation π of D ⋊ G ⋉ B on a Hilbert space H and take the associated covariant representation ( π D , π B , U ). Since G is compact, U decomposes into a direct sum of finite dimensional irreducible representations: H = L i H i . Then for each ξ ∈ H i , its orbit ( D ⋊ G ⋉ B ) ξ = π D ( D ) π B ( B ) H i is finite dimensional.By a simple maximality argument, we get the conclusion.(2) We fix a faithful G -invariant state ϕ D on D . Since D is finite dimensional, there exists λ > ϕ D ( d ∗ d )1 ≥ λd ∗ d. By Lemma 3.3, the map E : D ⋊ G ⋉ B → G ⋉ B : dxa ϕ D ( d ) xa defines a conditional expectation. Then E ( x ∗ x ) ≥ λx ∗ x for any x ∈ D ⋊ G ⋉ B . Therefore theconditional expectation E ∗∗ from ( D ⋊ G ⋉ B ) ∗∗ to ( G ⋉ B ) ∗∗ is of finite index, hence D ⋊ G ⋉ B is of type I, as so is G ⋉ B ⊂ B ⊗ K ( L ( G )) since in the separable context the type I propertypasses to any C*-subalgebra, as explained in the proof of [BrO, Corollary 9.4.5]. (cid:3) Similarly for a G -equivariant Hilbert B -module E , one can define a Hilbert A ⋊ G ⋉ B -module A ⋊ G ⋉ E as a corner of the linking algebra A ⋊ G ⋉ K ( E ⊕ B ). More concretely, the Hilbertmodule A ⋊ G ⋉ E is the completion of the pre-Hilbert module ˜ E defined as follows: • As a vector space, ˜ E is isomorphic to A ⊗ alg c c ( Γ ) ⊗ alg E , where again A denotes therespective algebraic core. Again the element in ˜ E corresponding to a ⊗ x ⊗ b is denotedby axb for a ∈ A , x ∈ c c ( Γ ) , b ∈ E . UANTUM BAUM-CONNES AND ROSENBERG CONJECTURES 7 • The right A ⋊ G ⋉ B -module structure is given by( axb )( a ′ x ′ b ′ ) = a ( x (1) ⊲ a ′ ) x (2) x ′ (1) ( b ⊳ x ′ (2) ) b ′ for a, a ′ ∈ A , x, x ′ ∈ c c ( Γ ), b ∈ E and b ′ ∈ B . • The inner product is given by( bxa, b ′ x ′ a ′ ) = a ∗ x ∗ ( b, b ′ ) x ′ a ′ for a, a ′ ∈ A , x, x ′ ∈ c c ( Γ ), b, b ′ ∈ E . Here bxa expresses an element of ˜ E by the samecommutation relation as in A .It is easy to see that K ( A ⋊ G ⋉ E ) is naturally isomorphic to A ⋊ G ⋉ K ( E ). In particular we getthe following result. Lemma 3.5.
For each G -equivariant Hilbert B -module E , A ⋊ G ⋉ K ( E ) is Morita equivalent to A ⋊ G ⋉ B . Finally we show that the construction preserves the exact sequences in a natural sense.
Lemma 3.6.
Let A be a C*-algebra with a right G -action α . For a C*-algebra B with a left G -action β and a G -invariant ideal I ⊂ B , the sequence → A ⋊ G ⋉ I → A ⋊ G ⋉ B → A ⋊ G ⋉ ( B/I ) → is exact.Proof. Clearly the map A ⋊ G ⋉ B → A ⋊ G ⋉ ( B/I ) is surjective. To see the injectivity of A ⋊ G ⋉ I → A ⋊ G ⋉ B , take a faithful nondegenerate representation π of A ⋊ G ⋉ I on a Hilbertspace H . Consider the associated covariant representation ( π A , π I , U ) and the unique extensionof π I to B , say π B . Since U ∗ (1 ⊗ π I ( b )) U = (id ⊗ π I ) β ( b )for any b ∈ M ( I ), the triple ( π A , π B , U ) is a covariant representation. Since the associatedrepresentation A ⋊ G ⋉ B is an extension of π , we conclude that the map A ⋊ G ⋉ I → A ⋊ G ⋉ B is injective.It remains to prove that the sequence in the lemma is exact at the middle term. Since thecomposition is zero, we can induce a homomorphism( A ⋊ G ⋉ B ) / ( A ⋊ G ⋉ I ) → A ⋊ G ⋉ ( B/I ) . On the other hand, the universality of the crossed product induces the inverse of the map. Hencethe homomorphism is an isomorphism. (cid:3)
Proposition 3.7.
Let G be a compact quantum group with a countable dual. Fix a separable C*-algebra A with a right G -action. The crossed product introduced above gives rise to a triangulatedfunctor on the equivariant Kasparov category A ⋊ G ⋉ · : KK G → KK.
Proof.
This is a direct consequence of [NeV, Theorem 4.4] and the last two lemmas. (cid:3) Adjunction
In this section we develop a construction which will allow us to establish a (natural) isomorphismof certain equivariant and non-equivariant KK -groups.Let D be a finite dimensional C*-algebra with a left G -action, where G is again a compactquantum group (with Γ its discrete dual). The opposite algebra D op admits a right G -action: α op : D op → D op ⊗ C ( G ) : x op (id ⊗ R ) ◦ σ ◦ α ( x ) op . Our goal is to show the functor D op ⋊ G ⋉ · : KK G → KK , developed in the previous section,is the right adjoint functor of D ⊗ · : KK → KK G . This is done by constructing the counit andunit.From now on, we fix a faithful G -invariant state ϕ D = Tr( ρ · ) on D , where Tr is the trace takingvalue 1 at each minimal projection and ρ ∈ D is the ‘density matrix’ of ϕ D . YUKI ARANO AND ADAM SKALSKI
Unit.
The modular group of ϕ D is given by the formula σ ϕ D t ( a ) = ρ it aρ − it , a ∈ D, t ∈ R .By [Eno, Th´eor`eme 2.9], we get( ρ it aρ − it ) ⊳ x = ρ it ( a ⊳ ˆ τ t ( x )) ρ − it , a ∈ D, t ∈ R , where ˆ τ is the scaling group on c ( Γ ). Let p be the support of the counit on c ( Γ ). Lemma 4.1.
Let D be a finite dimensional C*-algebra with a left G -action. There exists X ∈ D op ⊗ D ⊂ M ( D op ⋊ G ⋉ D ) such that (1) X is positive; (2) aX = ( ρ − / aρ / ) op X and Xb = X ( ρ / bρ − / ) op for all a, b ∈ D ; (3) p X = Xp ; (4) ( ϕ op D ⊗ id)( X ) = 1 .Proof. Since D is finite dimensional, we write D as a direct sum of matrix algebras D = M π M n ( π ) and fix a matrix unit ( e πij ) for each matrix algebra. Let X = X π,i,j ( ρ − / e πij ρ − / ) op e πji ∈ D op ⊗ D ⊂ M ( D op ⋊ G ⋉ D ) . The assertions (2) and (4) follow from a straightforward computation.For (1), recall that P π,i,j e πij ⊗ e πij ∈ D ⊗ D is positive. Now using the component-wise transposemap viewed as the an isomorphism t : D → D op : e πij ( e πji ) op , we conclude X = (( ρ − / ) op ⊗ ⊗ id) X π,i,j e πij ⊗ e πij (( ρ − / ) op ⊗ ι : D ⊗ D → End( D ) by ι ( a ⊗ b )( d ) = ϕ D ( db ) a. This is equivariant with respect to the tensor representation of G on D ⊗ D and the adjointrepresentation of G on End( D ). Hence ι − (1 End( D ) ) = P π,i,j ρ − e πij ⊗ e πji is invariant under thetensor representation.Now we observe that for all x ∈ c c ( Γ ) p Xx = p X π,i,j (cid:16) ( ρ − / e πij ρ − / ) ⊳ ˆ τ − i/ ( x (2) ) (cid:17) op (cid:0) e πij ⊳ x (1) (cid:1) = p X π,i,j (cid:16) ρ / ( ρ − e πij ) ⊳ x (2) ρ − / (cid:17) op (cid:0) e πij ⊳ x (1) (cid:1) = p ˆ ε ( x ) X. In particular p Xp = p X . Since X is self-adjoint, we get Xp = p Xp = p X . (cid:3) Thanks to the lemma above, the vector space D admits a right Hilbert D op ⋊ G ⋉ D -modulestructure defined as follows: • The right module structure is given by d ⊳ ( a op xb ) = (( ρ − / aρ / d ) ⊳ x ) b for a, b, d ∈ D , x ∈ c c ( Γ ). • The D op ⋊ G ⋉ D -valued inner product is given by( d, d ′ ) = d ∗ Xp d ′ , d, d ′ ∈ D. We denote D equipped with the right Hilbert D op ⋊ G ⋉ D -module structure above by D . UANTUM BAUM-CONNES AND ROSENBERG CONJECTURES 9
Counit.
Suppose now that we also have a C*-algebra B equipped with a right action β of G ,with the algebraic core B . Recall that we have a conditional expectation D op ⋊ G ⋉ B → G ⋉ B as in Lemma 3.3. Composing D op ⋊ G ⋉ B → G ⋉ B with the natural operator-valued weight from G ⋉ B to B , we get an operator-valued weight E : D op ⋊ G ⋉ B → B . On D op ⋊ alg G ⋉ alg B it isgiven by E : D op ⋊ G ⋉ B → B : axb ϕ op D ( a ) ˆ ψ ( x ) b, with ˆ ψ denoting the right Haar weight of Γ . The corresponding GNS module is isomorphic to E B := L ( D ) ⊗ L ( G ) ⊗ B with the GNS mapΛ : D op ⋊ alg G ⋉ alg B → L ( D ) ⊗ L ( G ) ⊗ B : d op bx d Ω ⊗ Λ ˆ ψ ( x ) ⊗ b for d ∈ D, x ∈ c c ( Γ ) , b ∈ B . Furthermore E B carries a natural G -equivariant D ⊗ ( D op ⋊ G ⋉ B )- B -bimodule structure defined as follows: • The G -action β E B on E B is given by β E B ( x ) = U ∗ V ∗ (id ⊗ id ⊗ β )( x ) , x ∈ E B . • The left action of D ⊗ ( D op ⋊ G ⋉ B ) structure is given by(1) D ⊗ ∋ d ⊗ λ D ( d ) ⊗ ⊗ ⊗ D op ∋ ⊗ d op ( ρ D ⊗ ρ ) α op ( d op ) ⊗ ⊗ c c ( Γ ) ∋ ⊗ x ⊗ ˆ λ ( x ) ⊗ ⊗ B ∋ ⊗ b ⊗ ( λ ⊗ id) β ( b ).From this presentation, it follows that the image of D ⊗ ( D op ⋊ G ⋉ B ) is in K ( E B ).4.3. Adjunction.
We begin by stating a general lemma.
Lemma 4.2.
Let
A, B, C be C*-algebras, E a right Hilbert A -module, D a Hilbert A ⊗ B - C -bimodule. Then we have a natural isomorphism ( E ⊗ B ) ⊗ A ⊗ B F ≃ E ⊗ A F . Proof.
Direct computation. (cid:3)
Lemma 4.3.
There exists a unitary U : D ⊗ D op ⋊G⋉ D E D → D defined by U ( d ⊗ D op ⋊G⋉ D Λ( x )) = ρ ( d ⊳ x ) for d ∈ D , x ∈ D op ⋊ G ⋉ D .Proof. We only need to show that U is an isometry. This is done by a straightforward computation.Indeed, take x, x ′ ∈ c c ( Γ ), a op , a ′ op ∈ D op and b, b ′ ∈ D . Then( d ⊗ D op ⋊G⋉ D Λ( xa op b ) , d ′ ⊗ D op ⋊G⋉ D Λ( x ′ a ′ op b ′ )) = E ( b ∗ ( a op ) ∗ x ∗ d ∗ Xp d ′ x ′ a ′ op b ′ )= E ( b ∗ ( a op ) ∗ ( d ⊳ x ) ∗ Xp ( d ′ ⊳ x ′ ) a ′ op b )= E ( b ∗ ( d ⊳ x ) ∗ ρ / a ∗ ρ − / Xp ρ − / a ′ ρ / ( d ′ ⊳ x ′ ) b )= ( d ⊳ xa op b ) ρ ( d ′ ⊳ x ′ a ′ op b ′ ) . (cid:3) Lemma 4.4.
There exists a unitary V : D ⊗ D op ⋊G⋉ D D op ⋊ G ⋉ E D ≃ D op ⋊ G ⋉ D defined by V ( d ⊗ D op ⋊G⋉ D a op x Λ( b op yc )) = ( d ⊳ a op x ) b op yc Proof.
Since ( d ⊗ ( a op x Λ( b op yc )) = 1 ⊗ Λ( y )( ρ / ( d ⊳ a op xρ / bρ − / ) ρ − / ) op c for a, b, c, d ∈ D and x, y ∈ c c ( Γ ), we only need to show(1 ⊗ Λ( x ) , ⊗ Λ( y )) = x ∗ y. To this end, we compute, using the antipode of c c ( Γ ) denoted ˆ S ,(1 ⊗ Λ( x ) , ⊗ Λ( y )) = (Λ( x ) , Xp Λ( y ))= X π,i,j (Λ( x ) , Λ( ρ / e πji ρ − / ) op y ˆ S − ( p )( ρ / e πij ρ − / ) p )= X π,i,j ϕ D ( ρ / e πij ρ − / ) ˆ ψ ( x ∗ y ˆ S − ( p )) p = x ∗ y. Here we have used the identity p x ⊗ p = p x (1) ⊗ p x (2) ˆ S ( x (3) ) = p ⊗ p ˆ S ( x ) . (cid:3) We have two functors KK → KK G : A D ⊗ A, KK G → KK : A D op ⋊ G ⋉ A. Theorem 4.5.
Let G be a compact quantum group with a countable dual. For A ∈ KK and B ∈ KK G , we have natural isomorphisms KK G ( D ⊗ A, B ) ≃ KK ( A, D op ⋊ G ⋉ B ) . Proof.
We construct the counit-unit adjunction. The unit is given by η A = [ D ] ⊗ A ∈ KK ( A, ( D op ⋊ G ⋉ D ) ⊗ A )and the counit is given by ε B = [ E B ] ∈ KK G ( D ⊗ ( D op ⋊ G ⋉ B ) , B ) . We need to prove ε D ⊗ A (id D ⊗ η A ) = id D ⊗ A , ( D op ⋊ G ⋉ ε B ) η D op ⋊G⋉ B = id D op ⋊G⋉ B . The first identity is due to Lemma 4.3:( D ⊗ D ⊗ A ) ⊗ D ⊗ ( D op ⋊G⋉ D ) ⊗ A ( E D ⊗ A ) ≃ D ⊗ A. This follows from
D ⊗ D op ⋊G⋉ D E D ≃ D . The second identity is due to Lemma 4.4:( D ⊗ D op ⋊ G ⋉ B ) ⊗ ( D op ⋊G⋉ D ) ⊗ ( D op ⋊G⋉ B ) ( D op ⋊ G ⋉ E B ) ≃ D op ⋊ G ⋉ B. (cid:3) Application to UCT
In this section we will apply the last theorem to questions regarding the Universal CoefficientTheorem. Let G be again a compact quantum group.Recall that a G -C*-algebra is said to be cofibrant if it is of the form D ⊗ A where D is a finitedimensional C*-algebra with a G -action α and the G -action on D ⊗ A is given by α ⊗ id. LetCof be the full subcategory of cofibrant objects in KK G and let N be the full subcategory of A ∈ KK G such that D op ⋊ G ⋉ A is KK -contractible for any finite dimensional G -C*-algebra D . UANTUM BAUM-CONNES AND ROSENBERG CONJECTURES 11
Corollary 5.1.
Suppose that the dual of G is countable. The subcategories ( h Cof i , N ) are com-plementary, i.e., for any A ∈ KK G , there exists a unique triangle P ( A ) } } ④④④④④④④④ A / / N ( A ) , ◦ ■■■■ d d ■■■■ where P ( A ) ∈ h Cof i and N ( A ) ∈ N .Proof. By the help of Theorem 2.4 and Theorem 4.5, we only need to show the isomorphism classof finite dimensional G -C*-algebras is at most countable. First from [ADC], the isomorphismclasses of finite dimensional G -C*-algebra are in one-to-one correspondence with the Q -systemsin Rep( G ). There exists only countably many objects in Rep( G ) and each of them has at mostfinitely many structures of a Q -system by [IzK]. (cid:3) Recall now the phantom tower construction.For a separable G -C*-algebra A , define P n , N n ∈ KK G inductively as follows: • Put A = N . • For N n , we set P n +1 = L D :torsion D ⊗ ( D op ⋊ G ⋉ N n ). Then we have the counit morphism L D ε N n : P n +1 → N n . We embed this morphism into a triangle P n +1 → N n → N n +1 → SP n +1 to define N n +1 .With the construction above, we get a diagram: P (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ P ◦ o o ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ ◦ o o (cid:0) (cid:0) ✁✁✁✁✁✁✁✁ A / / N ◦ ❇❇❇❇ ` ` ❇❇❇❇ / / N ◦ ❇❇❇❇ ` ` ❇❇❇❇ / / Now consider the morphism A → N n to fit in a triangle A → N n → ˜ A n . The octahedral axiomshows that ˜ A n also fits to a triangle˜ A n → ˜ A n +1 → P n → S ˜ A n . (5.1)We take the homotopy limit N = ho-lim N n and ˜ A = ho-lim ˜ A n . Then we have a triangle˜ A (cid:0) (cid:0) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) A / / N , ◦ ❈❈❈❈ a a ❈❈❈❈ where ˜ A ∈ h Cof i and N ∈ N . Theorem 5.2.
Let G be a compact quantum group with a countable dual and let A be a separable G -C*-algebra. Suppose that either (i) D op ⋊ G ⋉ A satisfies the UCT for any finite dimensional G -C*-algebra D , or (ii) G has no torsion and G ⋉ A satisfies the UCT.Then P ( A ) satisfies the UCT.Proof. We prove only (i); (ii) follows similarly, using the construction preceding the theorem.First, by induction, we show that D op ⋊ G ⋉ N n satisfies the UCT for any n ∈ N ∪ { } . Thisholds for n = 0 by assumption. Assume D op ⋊ G ⋉ N n satisfies the UCT. Since D op ⋊ G ⋉ D is adirect sum of matrix algebras, D op ⋊ G ⋉ P n +1 ≃ ( D op ⋊ G ⋉ D ) ⊗ ( D op ⋊ G ⋉ N n ) satisfies theUCT. Hence the mapping cone D op ⋊ G ⋉ N n +1 also satisfies the UCT.In particular, P n satisfies the UCT. Again by induction we see that ˜ A n satisfies the UCT forany n ∈ N by ˜ A = P and the triangle (5.1). Passing to the homotopy limit, we get that P ( A )satisfies the UCT. (cid:3) Definition 5.3.
Fix a discrete quantum group Γ with the dual compact quantum group G , andconsider h Cof i and N , the subcategories of KK G introduced above. We say that Γ satisfies the h Cof i -Baum–Connes property if N is KK -contractible for any N ∈ N , or equivalently, P ( A ) → A is a KK -equivalence.We say that Γ satisfies the h Cof i -strong Baum–Connes property if N is KK G -contractible forany N ∈ N , or equivalently, KK G = h Cof i . Remark . When Γ is a classical discrete group, then the h Cof i -Baum–Connes property is equiv-alent to the fact that the strong Baum–Connes conjecture as introduced in [MN , Definition 9.1]holds, by [MN , Theorem 9.3]. We do not know whether KK G = h Cof i even when Γ is a finitegroup.In spite of that, many discrete quantum groups actually satisfy the strong h Cof i -Baum–Connesproperty. In particular, it holds for compact connected groups [MN ], free orthogonal quantumgroups [Vo ], free unitary quantum groups [VeV] and free permutation groups [Vo ]. It passesthrough the monoidal equivalence and is closed under taking free products [VeV], subgroups andfree wreath product [FrM]. Corollary 5.5.
Let A be a separable Γ -C*-algebra. We have the following. (1) For any torsion-free countable discrete quantum group Γ with the h Cof i -Baum–Connesproperty, the C*-algebra Γ ⋉ A satisfies the UCT if A does. (2) For any countable discrete quantum group Γ with the h Cof i -Baum–Connes property, theC*-algebra Γ ⋉ A satisfies the UCT if A is of type I. In particular the reduced groupC*-algebra C ( G ) satisfies the UCT.Proof. Recall that we denote the dual of Γ by G .(1) Consider the G -C*-algebra Γ ⋉ A . Then by the Baaj–Skandalis duality (see [Vae] and recallthat discrete/compact quantum groups are automatically regular), G ⋉ Γ ⋉ A ≃ K ( L ( G )) ⊗ A ,so that G ⋉ Γ ⋉ A satisfies the UCT. Now apply Theorem 5.2 to show P ( Γ ⋉ A ) satisfies the UCT.By h Cof i -Baum–Connes property, the C*-algebra Γ ⋉ A also satisfies the UCT.(2) We denote the G -C*-algebra Γ ⋉ A by B . First we will show that D op ⋊ G ⋉ B is of type I.Again by the Baaj–Skandalis duality, the C*-algebra G ⋉ B ≃ K ( L ( G )) ⊗ A is of type I. Henceby Proposition 3.4, D op ⋊ G ⋉ B is of type I. In particular D op ⋊ G ⋉ B satisfies the UCT. Therest of the proof is the same as (1). (cid:3) Quantum Rosenberg Conjecture
One of very well-known conjectures regarding group C ∗ -algebras, the Rosenberg Conjecture,stating that reduced C ∗ -algebras of a countable amenable discrete groups are quasidiagonal, wasestablished in [TWW] (with the converse implication proved much earlier by Rosenberg in [HaR]).Here we show how as a corollary of that result and the progress on the UCT conjecture forquantum group algebras made in the last section one can obtain a similar statement for a largeclass of (unimodular) discrete quantum groups. A key observation is that the proof of the originalresult of Rosenberg found by Davidson ([Dav] – we refer to this book also for the definition ofquasidiagonality) passes to the quantum case in a straightforward manner. Theorem 6.1.
Assume that Γ is a countable discrete quantum group. Then quasidiagonality of C ∗ r ( Γ ) implies amenability of Γ and if Γ is unimodular, amenable and satisfies the h Cof i -Baum-Connes property then C ∗ r ( Γ ) is quasidiagonal. Finally there exist amenable nonunimodular count-able discrete quantum groups Γ (e.g. \ SU q (2) , q ∈ (0 , ) such that C ∗ r ( Γ ) is not quasidiagonal.Proof. Assume first that C ∗ r ( Γ ) is a quasidiagonal C ∗ -algebra.If Γ is finite, then it is obviously amenable (in that case it is compact and the Haar state yieldsan invariant mean). If Γ is infinite, then the left regular representation of C ∗ r ( Γ ) is essential, i.e.contains no compact operators ([Kal]). This means (via one of the versions of the VoiculescuTheorem) that C ∗ r ( Γ ) is quasidiagonal as the set of operators in B ( ℓ ( Γ )). Let then ( P n ) ∞ n =1 bea sequence of finite rank projections in B ( ℓ ( Γ )) increasing to I and such that for each α ∈ Irr b Γ , UANTUM BAUM-CONNES AND ROSENBERG CONJECTURES 13 i, j = 1 , . . . , n α , we have k P n u αi,j − u αi,j P n k n →∞ −→ . Let then τ n be the normalised trace on the matrix algebra B ( P n ℓ ( Γ )), fix an ultrafilter U on N and define the state ω on B ( ℓ ( Γ )) via the prescription ω ( a ) = lim U τ n ( P n aP n ) , a ∈ B ( ℓ ( Γ )) , and let m = ω | l ∞ ( Γ ) . Fix α ∈ Irr b Γ , i, j ∈ { , . . . , n α } and a non-zero x ∈ l ∞ ( Γ ). Put φ = c e αi,j ∈ l ∞ ( Γ ) ∗ . Note that φ (1) = δ i,j .For each ǫ > N ∈ N such that for all n ≥ N and p = 1 , . . . , n α we have k P n u αi,p − u αi,p P n k ≤ ǫ (3 n α k x k ) − and k P n u αj,p − u αj,p P n k ≤ ǫ (3 n α k x k ) − (note that the latter estimate is valid if one replaces u αj,p by ( u αj,p ) ∗ ). Thus for such n ≥ N (see(2.2)) | τ n ( P n L φ ( x ) P n ) − δ i,j τ n ( P n xP n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ n P n n α X p =1 u αi,p x ( u αj,p ) ∗ ! P n ! − δ i,j τ n ( P n xP n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n α X p =1 τ n ( P n u αi,p P n xP n ( u αj,p ) ∗ P n ) − δ i,j τ n ( P n xP n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n α X p =1 τ n ( P n xP n ( u αj,p ) ∗ P n u αi,p P n ) − δ i,j τ n ( P n xP n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 ǫ ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n α X p =1 τ n ( P n xP n ( u αj,p ) ∗ u αi,p P n ) − δ i,j τ n ( P n xP n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ǫ = ǫ, where in the third equality we used the fact that τ n is a trace and in the last one the unitarity ofthe matrix ( u αi,j ) n α i,j =1 . This implies that in the limit we obtain m ( L φ ( x )) = lim U τ n ( P n L φ ( x ) P n ) = δ i,j m ( x ) = φ (1) m ( x )and the proof of the forward implication is finished.Assume then that Γ is amenable and unimodular. By Theorem 1.1 of [BMT] C ∗ r ( Γ ) is nuclear;by the unimodularity of Γ , C ∗ r ( Γ ) admits a faithful trace. Then the main result of [TWW] showsthat C ∗ r ( Γ ) is quasidiagonal if only it satisfies the UCT. This however follows from the assumptionthat Γ satisfies the h Cof i -Baum-Connes property by Corollary 5.5 (2).It remains to note that Woronowicz’s compact quantum group SU q (2) (with q ∈ (0 , \ SU q (2) is amenable. On the other hand the C ∗ -algebraC ∗ r ( \ SU q (2)), which is in fact independent of q , as observed in Theorem A.2 in [Wo ], contains aproper isometry, so in particular cannot be quasidiagonal. (cid:3) Note that if we knew that all group C ∗ -algebras of discrete amenable unimodular quantumgroups satisfy UCT we could drop the h Cof i -Baum-Connes property assumption in the secondpart of the theorem above. References [ADC] Y. Arano and K. De Commer, Torsion-freeness for fusion rings and tensor C ∗ -categories, J. Noncommut.Geom. (2019), no. 1, 35–58.[BS] S. Baaj and G. Skandalis, C ∗ -alg`ebres de Hopf et th´eorie de Kasparov ´equivariante, K -Theory , (1989),no. 6 , 683–721.[Ban] T. Banica, Representations of compact quantum groups and subfactors, J. Reine Angew. Math. (1999),no. 1, 167–198.[BMT] E.B´edos, G.J. Murphy, and L. Tuset, Amenability and co-amenability of algebraic quantum groups. II.
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