INDUCTION FOR LOCALLY COMPACT QUANTUM GROUPS REVISITED MEHRDADKALANTAR,PAWEL KASPRZAK,ADAMSKALSKI,ANDPIOTR SOL TAN 7 1 Abstract. In this paper we revisit the theory of induced representations in the setting of 0 locallycompactquantumgroups. Inthecaseofinductionfromopenquantumsubgroups,we 2 show that constructions of Kustermans and Vaes are equivalent to the classical, and much n simpler, construction of Rieffel. Wealso provein general setting thecontinuity ofinduction a in the sense of Vaes with respect to weak containment. J 2 ] A 1. Introduction O The theory of induced representations, allowing one to manufacture in a canonical way a . h representation of a group out of that of a subgroup, dates back to the work of Frobenius for t a finite groups and was later developed by Mackey for locally compact groups. It has played m a very important role in developing the general theory of group representations and has seen [ many applications in various areas of abstract harmonic analysis (we refer to the recent book [7] for a precise description of the construction, the historical background and an exhaustive 1 v list of references). From the modern point of view a very important point in the history of 4 the subject was the paper of Rieffel ([12]), which phrased the construction in the language 4 of Hilbert C∗-modules. This led to later generalizations of similar induction procedures to 3 several operator algebraic contexts. 0 0 Thus it was natural that in the beginning of the 21st century, with the widely accepted 1. notion of locally compact quantum groups emerging in the work of Kustermans and Vaes 0 ([9]), an interest has developed in the study of the theory of induced representations in the 7 quantum context. In fact the two relevant papers were written respectively by Kustermans 1 ([8]) and Vaes ([15]). In the first of these the construction was presented under the assump- : v tion of integrability of the natural action of the quantum subgroup on the larger quantum i X group (automatically satisfied for classical groups – and later shown to be always true also in r the quantum context in [5]), in the second an alternative approach was proposed, explicitly a disposing of the integrability constraint and also allowing for a more general framework of representations on Hilbert C∗-modules, as opposed to Hilbert spaces. Both constructions of Kustermans and Vaes are technically complicated; indeed even the classical procedure of Mackey in general requires overcoming certain measure-theoretic and topological difficulties. The starting point of our paper is a classical observation that in the classical case if the subgroup from which we induce the representation is open, most of the technical problems disappear(seeagain[7]). Moreimportantly,onecanapplythenaverystraightforwardversion of the Rieffel procedure, simply observing that if H is an open subgroup of a locally compact group G, then one has a natural embedding of the group C∗-algebras C∗(H) ⊂ C∗(G), which admits also a canonical conditional expectation E :: C∗(G) → C∗(H). In a recent work [6] 2010 Mathematics Subject Classification. Primary: 22D30, 46L89, Secondary: 22D25, 20G42. Key words and phrases. Locally compact quantumgroup, induction of representations. 1 2 MEHRDADKALANTAR,PAWEL KASPRZAK,ADAMSKALSKI,ANDPIOTRSOL TAN three of the authors of the current paper have introduced the notion of an open quantum subgroup of a locally compact quantum group G. One of the main results of [6] and Section 3 of this paper show that if H is an open quantum subgroup of G, one still has an embedding C∗(H) ⊂ C∗(G) and a canonical conditional expectation E :C∗(G) → C∗(H) (we write these relations as Cu(H) ⊂ Cu(G) and E : Cu(G) → Cu(H)). This opens the way to inducing a 0 0 0 0 unitary represenbtation of Gb(in other wordbs a represebntation of the C∗-algebra C∗(G)) from a unitary representation of H (in other words a representation of the C∗-algebra C∗(H)) using the simple method of Rieffel and raises a natural question whether this leads to an object isomorphic to that obtained from the same starting data via the procedures of Kustermans and Vaes. This is the topic of this paper. Using Kustermans’s induction, we first prove the existence of the canonical embedding C∗(H) ⊂ C∗(G) for any open quantum subgroup H of a locally compact quantum group G, hence in particular dropping the coamenability assumption required in [6, Section 5]. This meansonecaninthatcasealways inducearepresentation ofG(onaHilbertC∗-module)from arepresentationofHviaRieffel’smethod. Wethenprovethatinfactinthiscasetheresulting representationiscanonicallyisomorphictotheonesobtainedeitherviaKustermans’sorVaes’s procedures. The key tool for that is an imprimitivity-type result, Theorem 4.3, allowing us to identify Rieffel-induced representations of the larger quantum group via the existence of a covariant representation of thealgebra of functionson the quantumhomogeneous space G/H. We note that in [15, Section 11] it was asserted that one could show that induction pro- cedures of [15] and [8] are unitary equivalent in general, but that a proof would be highly non-trivial and would need to use the complete machinery of modular theory and properties of the unitary implementation of the action of the larger quantum group on the quantum homogeneous space. Our results provide a complete understanding of the situation in the case of open quantum subgroups. In particular, in this setting, the induction process (in any of the above senses) is rather simple, and does not require the more technical aspects of C∗-algebra or quantum group theories. It is worth recalling that, unsurprisingly, quantum subgroups of discrete quantum groups are always open and note that a simplified picture of the Vaes induction procedure in the discrete case was studied in [16]. The detailed plan of the paper is as follows: after introducing general notation in the last part of this section, in Section 2 we recall the basics of Hilbert C∗-modules and Rieffel’s in- duction, introduceterminology andfundamentalfacts pertainingtolocally compactquantum groups and discuss in reasonable detail the induction procedures developed in this context by Kustermans and Vaes. Section 3 contains a generalization of Theorem 5.2 of [6], droppingthe coamenability assumption for the inclusion Cu(H) ⊂ Cu(G) (with H being an open quantum 0 0 subgroup of G) and thus opening the way to tbhe Rieffebl construction of induced represen- tations in this context. A short Section 4 establishes an imprimitivity type result, offering a method to recognize representations induced in the sense of Rieffel; this theorem is then appliedinSection 5toobtainthemainresultsofthepaper,namely theunitaryequivalence of all the three induction proceduresfor the case of open quantum subgroups. A shortappendix shows the continuity of the Vaes induction in the general case. Allscalarproductsarelinearontheright,i.e.inthesecondvariable. Thesymbol⊗denotes the completed Hilbert-space/Hilbert C∗-module/C∗-algebraic-minimal tensor product, and the algebraic tensor products will be denoted by ⊙. The ultrawewak tensor product of von Neumann algebras will bedenoted by ⊗¯. For a C∗-algebra A we denote its multiplier algebra by M(A) and if B is another C∗-algebra we write Mor(A,B) for the set of all nondegenerate INDUCTION FOR LOCALLY COMPACT QUANTUM GROUPS 3 ∗-homomorphisms from A to M(B). Elements of Mor(A,B) are called morphisms from A to B (cf. [17]). We will often use the leg numbering notation for operators acting on (multiple) tensor products of Hilbert spaces and elements of tensor products of algebras; σ will usually denote the tensor flip. For a set X ⊂ B(H), where H is a Hilbert space, X′ denotes the commutant of X. Given a normal semifinite (n.s.f.) weight θ on a von Neumann algebra N we write N = x ∈ N θ(x∗x) < +∞ , L2(N,θ) for the associated GNS Hilbert space θ (cid:8) (cid:9) and Λ : N → L2(N,θ) for the associated GNS map. Finally for a Hilbert space H and two θ θ vectors ξ,η ∈ H we write ωξ,η for the functional in B(H)∗ given by T 7→ hξ,Tηi and ωξ = ωξ,ξ. Finally a word on the quantum group notation is in order; although in [6] we adopted the conventions of [1], working with right Haar weights and right multiplicative unitaries as primary objects, here, mainly for compatibility with [15] we stick to the left Haar weights, left multiplicative unitaries, etc. Acknowledgement. The third author was partially supported by the NCN (National Sci- ence Centre) grant 2014/14/E/ST1/00525. The second and fourth authors were partially supported by NCN (National Science Centre) grant no. 2015/17/B/ST1/00085. 2. Preliminaries Inthissection weintroducebasicfactsandnotations neededinthemainbodyofthepaper. 2.1. Modules, correspondences and induction in the sense of Rieffel. Let B be a C∗-algebra. By a (Hilbert)C∗-module over B (usually called simply a Hilbert B-module) we understand a right module E over B equipped with a B-valued scalar product satisfying natural requirements (see [10]); by L(E) we denote the C∗-algebra of adjointable operators on E. Every pair of vectors ξ,η ∈ E defines an operator Θ ∈ L(E) by the formula ξ,η Θ (ζ) =ξhη,ζi, ζ ∈E. ξ,η The closure of the span of operators of the form Θ forms a C∗-subalgebra of L(E) called ξ,η the algebra of compact operators on E and denoted by K(E). We then have L(E) = M(K(E)). If in addition we have a non-degenerate ∗-homomorphism from another C∗-algebra A to L(E) (in other words, a representation of A on E) we call E an A-B-correspondence. We will sometimes use without any comment the internal tensor product of C∗-correspondences and the external product of Hilbert modules. In particular if A and B are C∗-algebras and E is a Hilbert B-module then A⊗E denotes the natural completed Hilbert (A⊗B)-module (cf. [10, Chapter 4]). We will also occasionally need the notion of von Neumann modules, replacing C∗-algebras with von Neumann algebras and adding normality conditions for the respective actions (see the Appendix of [15] for the details). A key notion, also introduced in [15], is that of a strict ∗-homomorphism: let MbeavonNeumannalgebraandE aHilbertmoduleover aC∗-algebra B. A map π : M → L(E) is strict if it is continuous with respect to the strong∗ topology on the unit ball of M and strict topology on L(E) (cf. [10, Page 11]). Suppose A is a C∗-algebra, B is a C∗-subalgebra of A and and E :A → B is a conditional expectation (i.e. a norm-one projection – it is automatically completely positive). We will now recall Rieffel’s process of inducing a representation of A from a representation of B (on a Hilbert module) presented in [12]. Let π : B → L(E) be a representation of B on a Hilbert module E over some auxiliary C∗-algebra C. On the B-balanced algebraic tensor product A⊙ E define an A-valued inner B 4 MEHRDADKALANTAR,PAWEL KASPRZAK,ADAMSKALSKI,ANDPIOTRSOL TAN product: ∗ ha⊗v,b⊗wi = hv,E(a b)wi, a,b ∈ A, v,w ∈ E. Then the action of A on A⊙ E defined by a(b⊗v) = ab⊗v gives a representation Ind (π) B R of A on the C∗-C-module Ind (E) obtained from A⊙ E via the usual separation/completion R B procedure. The representation Ind (π) is called the representation induced from π in the R sense of Rieffel. The above construction can be easily phrased in the language of internal tensor product of Hilbert C∗-modules. 2.2. Locally compact quantum groups. As explainedin theintroductionwe willusehere the left conventions of [15]. Thus G denotes a locally compact quantum group in the sense of [9], a virtual object studied via the associated operator algebras: the von Neumann algebra L∞(G) (“essentially bounded measurable functions on G”), the C∗-algebra C (G) ⊂ L∞(G) 0 (“continuous functions on G vanishing at infinity”), and its universal version Cu(G). Each of 0 these is equipped with a coproduct: we have a unital normal coassociative ∗-homomorphism ∆G : L∞(G) → L∞(G)⊗¯ L∞(G) which restricts to ∆G ∈ Mor(C0(G),C0(G)⊗C0(G)) and also a corresponding ∗-homomorphism ∆u ∈ Mor(Cu(G),Cu(G) ⊗ Cu(G)). The canonical G 0 0 0 surjective morphism from Cu0(G) onto C0(G) will be denoted by ΛG. If ΛG is injective, we say that G is coamenable. The left (respectively, right) Haar weight on L∞(G) will be denoted by ϕG (respectively, ψG) and L2(G) will denote the GNS Hilbert space of the left Haar weight. We will always assume that L∞(G) and C (G) are represented on L2(G). The key object 0 carryingalltheinformationaboutGistheleft regular representation WG ∈ B(L2(G)⊗L2(G)) implementing the comultiplication: ∆G(x) = (WG)∗(1⊗x)WG, x ∈L∞(G). This operator is also called the Kac-Takesaki operator or, less formally, the multiplicative unitary of G. It carries also the information about the dual locally compact quantum group G (see [9, Section 8]): on one hand we have C0(G) = (ω ⊗id)(WG) ω ∈ B(L2(G))∗ —k·k, (cid:8) (cid:9) abnd on the other WG ∈ L∞(G) ⊗¯ L∞(G) (or mbore precisely WG ∈ M(C (G) ⊗ C (G))). 0 0 Note that in particular L∞(G) = C (G)′′bis canonically represented on L2(G). Moreoverbthe 0 multiplicative unitary associbated withbG is given by formula WGb = σ(WG)∗. The element WG ∈ M(C (G) ⊗ C (bG)) admits a universal version, VVG ∈ M(Cu(G) ⊗ 0 0 0 Cu0(G)), such that WG = (ΛG ⊗ ΛGb)(VVbG). We may also consider natural “one-sided” re- ducebd/universal versions, WG = (ΛG⊗id)(VVG) and WG = (id⊗ΛGb)(VVG). The predual of L∞(G)isdenotedL1(G);itisaBanachalgebrainacanonicalwayandwehaveacanonicalleft regular representation λu :L1(G)→ Cu(G) given by the formula λu(ω) = (id⊗ω)(WG). The 0 Banach algebra L1(G) admits a dense subbalgebra L1 (G) which carries a natural involution, # and the map λu restricted to the latter becomes a ∗-homomorphism of Banach ∗-algebras. Occasionally wewillalsoneedtheright multiplicative unitary VG ∈ L∞(G)′⊗¯ L∞(G), defined as VG = (J ⊗J)WGˆ(J ⊗J), where J denotes the modular conjugationsbassociated with the pair(L∞(Gb),ϕGb), andbits dbual versionbVGb ∈L∞(G)′⊗¯ L∞(G). A unitary representation of G on a Hilbert module E (usbually shortened to simply “repre- sentation”)isaunitaryU ∈ M(C0(G)⊗K(E)) = L(C0(G)⊗E)suchthat(∆G⊗id)U = U13U23. We then write U ∈ Rep(G,E). Unitary representations of G are in one-to-one correspondence with representations of the C∗-algebra Cu(G): if φ is a representation of Cu(G) on E then 0 0 U = (ibd⊗φ)(WG) b INDUCTION FOR LOCALLY COMPACT QUANTUM GROUPS 5 is a representation of G on E and every U ∈ Rep(G,E) comes from this construction for a unique representation φ of Cu(G). U 0 Given two representations U band V on Hilbert modules E and E (over the same C∗- U V algebra) we say that U is contained in V if there exists a projection P ∈ L(E ) such that V U ∼= (id⊗P)V(id⊗P), where ∼= denotes the self-explanatory relation of unitary equivalence. We say that U is weakly contained in V if kerφ ⊂ kerφ ; we denote it by writing U 4V. V U If M is a von Neumann algebra then by a (right) action of G on M we understand an injective unital normal ∗-homomorphism α : M → M⊗¯ L∞(G) satisfying the action equation (α⊗id)◦α =(id⊗∆G)◦α. If we write M in the form L∞(X) for some classical or “quantum” space X we speak simply about actions of G on X. The crossed product of M by the action α, denoted M ⋊ G, is the von Neumann algebra generated inside M⊗¯ B(L2(G) by α(M) α and 1⊗L∞(G). Actions and crossed products admit also natural left versions. It was shown in [14] that ebvery left action α of a locally compact quantum group G on a von Neumann algebra M admits a canonical unitary implementation Υ∈ L∞(G)⊗¯ B(H), where H is a space on which M acts in the standard form. This means that α(m) = Υ∗(1⊗m)Υ, m ∈ M. Moreover Υ is a representation of G on H We will also need the notion of C∗-algebraic actions of locally compact quantum groups and of (compatible) actions of locally compact quantum groups on Hilbert modules, both in the C∗-algebraic and von Neumann algebraic settings. Here again we refer to the Appendix of [15] for the details, recalling only that by a compatible (right) action of G on a Hilbert B-module E for a C∗-algebra B we understand a pair of maps α : B → M(B ⊗C (G)) and B 0 αE :E → M(E ⊗C0(G)) satisfying natural compatibility conditions ([15, Definition 12.2]. 2.3. Closed and open quantum subgroups. Given two locally compact quantum groups G and H, a morphism Π from H to G (written Π : H → G) is represented via either of the following three objects • a Hopf ∗-homomorphism, i.e. an element πu ∈ Mor(Cu(G),Cu(H)) intertwining the 0 0 respective coproducts: (πu⊗πu)◦∆u = ∆u ◦πu; G H • a bicharacter from H to G, i.e. a unitary V ∈ M(C (H))⊗C (G)) such that 0 0 (id⊗∆Gb)V = V13V12, b (∆H⊗id)V = V13V23; • a right quantum group homomorphism, i.e. an action α : L∞(G) → L∞(G)⊗¯ L∞(H) of H on G such that (∆G⊗id)◦α =(id⊗α)◦∆G. The relationships between Hopf ∗-homomorphisms, bicharacters and right quantum group homomorphisms are described in [11, 3] (note there is a right/left change in the notation in the treatment of bicharacters above). One of these is that V = (ΛH◦π⊗id)(WG), another is α(x) = V∗(1⊗x)V, x ∈ L∞(G). Each homomorphism Π : H → G admits a unique dual Π : G → H. If πu is the Hopf ∗- homomorphism describing Π then the Hopf ∗-homomorphisbm cobrrespbonding to Π is denoted b 6 MEHRDADKALANTAR,PAWEL KASPRZAK,ADAMSKALSKI,ANDPIOTRSOL TAN by πu. It is uniquely determined by the relation b (id⊗πu)(VVH) =(πu ⊗id)(VVG). (2.1) On the level of bicharacters Π is debscribed by V = σ(V)∗. A Hopf ∗-homomorphismbπu ∈ Mor(Cu(G),bCu(H)) may admit a reduced version, i.e. an 0 0 element π ∈ Mor(C0(G),C0(H)) such that ΛH ◦ πu = π ◦ΛG. It may then happen that π admits an extension to a normal ∗-homomorphism L∞(G) → L∞(H). Our notation will not distinguish between π and its extension to L∞(G). Definition 2.1. A homomorphism from H to G described by a Hopf ∗-homomorphism πu ∈ Mor(Cu(G),Cu(H)) identifies H with a closed quantum subgroup of G (in the sense of Vaes) 0 0 if there exists a reduced version π of πu which extends to an injective normal map L∞(H) → L∞(G). We often simply say thabt H ibs a closed quantum subgroup of G. b b In the situation of Definition 2.1 the normal injection L∞(H) → L∞(G) automatically intertwines comultiplications and in fact existence of such an binjection isbequivalent to H being a closed quantum subgroup of G. Moreover on the level of Cu(G) the map πu is then a 0 surjective ∗-homomorphism Cu(G) → Cu(H). 0 0 Let H be a closed quantum subgroup of G. The (measured) quantum homogeneous space G/H is defined by setting L∞(G/H) = x ∈ L∞(G) α(x) = x⊗1 , (cid:8) (cid:9) where α is the right quantum group homomorphism associated to the inclusion H ֒→ G. The coproduct ∆G restricts to a left action of G on G/H. We denote this restriction by ρG/H. Definition 2.2. A homomorphism from H to G corresponding to a Hopf ∗-homomorphism πu ∈ Mor(Cu(G),Cu(H)) identifies H with an open quantum subgroup of G if there exists a 0 0 reduced version π of πu which extends to a surjective normal map π : L∞(G) → L∞(H). We often simply say that H is an open quantum subgroup of G. An open quantum subgroup of a locally compact quantum group is automatically closed ([6, Section 3]). The key object when dealing with open quantum groups is the central support (in the terminology of [6], in literature it is often called the central cover) of the normal surjective morphism π : L∞(G) → L∞(H). It is the smallest central projection mappedto 1 by π and it isdenoted by1H. Itwas shownin[6]that1H isagroup-like projection, i.e.∆G(1H)(1H⊗1)= 1H ⊗1H. Moreover we can then identify L2(H) with 1HL2(G) and L∞(H) with 1HL∞(G); we will do so without further comment. We call 1H simply the support projection of H and note that it belongs to L∞(G/H). 2.4. Induction in the sense of Kustermans. In this subsection we briefly recall Kuster- mans’s notion of induction for unitary representations of closed quantum subgroupsof locally compact quantum groups on Hilbert spaces. We focus on establishing the notation and ter- minology; for the details of the construction we refer the reader to [8]. Let G be a locally compact quantum group and H a closed quantum subgroup of G. Let U ∈ L∞(H) ⊗¯ B(K) be a unitary representation of H on a Hilbert space K. Fix an n.s.f. weight θ on L∞(G/H) and denote by H the corresponding GNS Hilbert space. The θ GNS representation will be denoted by π :L∞(G/H) → B(H ). We often identify L∞(G/H) θ θ with its image under the GNS map and omit π when there is no danger of confusion. θ INDUCTION FOR LOCALLY COMPACT QUANTUM GROUPS 7 Let H be a fixed Hilbert space. Recall from Subsection 2.3 the canonical right action of H on G denoted by α. Following the notation introduced in [8, Definition 4.1], we let PH = (cid:8)X ∈ B(H)⊗¯ L∞(G)⊗¯ B(K) (id⊗α⊗id)X = U3∗4X124(cid:9). When H= C then we shall simply write P for PC. Observe that if X ∈ P and y ∈ L∞(G/H) then (y⊗1)X ∈ P. On the algebraic tensor product P ⊙(H ⊗K) define the (pre-)inner product θ hX ⊗w,Y ⊗vi = hw,(id⊗π ⊗id)(X∗Y)vi, X,Y ∈ P, w,v ∈ H ⊗K. θ θ and let Ind (K) be the Hilbert space obtained via the separation/completion procedure from K this (pre-)inner product. As the notation suggests, Ind (K) will be the Hilbert space on which the induced repre- K sentation of U will act. Proposition 4.6 of [8] shows that to every element X ∈ PH one can associate in a canonical way an operator X∗ ∈ B(H⊗Hθ⊗K,H⊗IndK(K)). We will later use the following properties stated in [8, Results 4.9, Notation 4.7]: ∗ ∗ (X∗) Y∗ = (id⊗πθ ⊗id)(X Y), X,Y ∈ PH, (2.2a) (XZ)∗ = X∗(id⊗πθ ⊗id)(Z), X ∈ PH, Z ∈ B(H)⊗¯ L∞(G/H)⊗¯ B(K). (2.2b) In the special case when H = C the definition of X∗ simplifies: for X ∈ P and w ∈ Hθ ⊗K the element X∗w is the class of X ⊗w in IndK(K) (denoted X⊗˙ w in [8]). The representation Ind (U) induced from U in the sense of Kustermans is defined by the K linear extension of the formula ∗ ∗ IndK(U) (η⊗X∗w) = (cid:0)(∆G⊗id)(X)(cid:1)∗Υ12(η⊗w) for X ∈ P, η ∈ L2(G), and w ∈ H ⊗K, where Υ ∈ L∞(G)⊗¯ B(H ) is the canonical unitary θ θ implementation of the action ρG/H of G on L∞(G/H) (note that (∆G ⊗id)(X) ∈ PL2(G)). Recall that, as stated in the introduction, results of [5] guarantee together with these of [8, Section 7] that Ind (U) is indeed a unitary representation of G. K 2.5. Induction in the sense of Vaes. In this subsection we recall the notion of induction due to Vaes, starting from a unitary representation of a locally compact quantum group H on a Hilbert module E. As before we assume that H is a closed quantum subgroup of G. Here the basic idea comes from identifying representations of G with certain L∞(G)−L∞(G) correspondences equipped in addition with a bicovariant action of the algebra Lb∞(G)′ (sbee [15, Proposition 3.7]). We begin however with defining an auxiliary object, the so-called imprimitivity bimodule. Let H be a closed quantum subgroup of G and let π : L∞(H) → L∞(G) be the corresponding inclusion. We will need its “commutant” version πb′ :L∞(Hb)′ → L∞(bG)′: π′(x) = JGπ JHxJH JG, b xb∈ L∞(H)′,b (cid:0) (cid:1) where JG and JH denotbe the mbodbulabr cobnjubgations associated to pairs L∞(G),ϕGb and (cid:0) (cid:1) L∞(H)b,ϕHb resbpectively. b (cid:0) (cid:1) Thebimprimitivity bimodule is the space I = v ∈ B(L2(H),L2(G)) vx = π′(x)v for all x ∈ L∞(H)′ . (cid:8) (cid:9) With the natural scalar product hv,wi = v∗w abnd left and right actibons I becomes a von Neumann G⋉L∞(G/H)−L∞(H) correspondence. b 8 MEHRDADKALANTAR,PAWEL KASPRZAK,ADAMSKALSKI,ANDPIOTRSOL TAN Furthermore the map αI : I → I ⊗L∞(G) given by αI(v) =VGb(v⊗1) (ibd⊗π) VHb ∗, v ∈ I, (cid:0) (cid:0) (cid:1)(cid:1) together with ∆Hb defines a compatible action obf G on I. Moreover, when we equip the crossed productbG⋉L∞(G/H) with the dual action of G, the left module action of G⋉L∞(G/H) on I becomes covariant in a natural sense. b Now let X ∈ L(C (H)⊗E) be a unitary representation of H on a Hilbert module E over 0 a C∗-algebra B. Lemma 4.5 of [15] yields the existence of a unique strict ∗-homomorphism π : L∞(H) → L(L2(G)⊗E) satisfying l b H H (id⊗π )W = (id⊗π)(W ) X . (2.3) l 12 13 If we now define π : L∞(G)→ L(L2(G)⊗E) byb the formula r πrb(x) = JGx∗JG ⊗1, x ∈ L∞(G) and a map γ :L∞(G)′ → L(L2(G)⊗bE) bby b γ(y) = y⊗1, y ∈ L∞(G)′ we obtain a so-called a bicovariant B-correspondence L∞(G)′ L∞(Hb) L2(G)⊗E L∞(Gb). (2.4) Then the results of the Appendix of [15] imply that one can definethe Hilbert B-moduleF = I ⊗(L2(G)⊗E) and left and right module actions on F such that we get a B-correspondeence πl e G⋉L∞(G/H) F L∞(Gb) which is equipped with the product action αFe of G, constructed out of αI and αL2(Ge)⊗E, where αL2(G)⊗E is the action of G on L2(G)⊗E definbed by αL2(G)⊗E(ξ) = V1Gb3(ξ⊗1), b ξ ∈ L2(G)⊗E. (cf. [15, Paragraph following Lemma 4.5]). The action αFe is given by the formula αFe(v ⊗ ζ)= αI(v) ⊗ αL2(G)⊗E(ζ), v ∈ I, ζ ∈ L2(G)⊗E. πl πl⊗id (cf. [15, Proposition 12.13]). The action αFe yields a representation π′ : L∞(G)′ → L(F) and U ∈ Rep(G,F) such that ′ Gb e b b e U = (π ⊗id)(V ) (2.5) b and U(Ω⊗a)= αFe(Ω)(1⊗a) (2.6) for all Ω ∈ F and a ∈ C (G) (see b[15, Definition A.2 and the following paragraph]). 0 Covariancee of Y with rbespect to π and π in the sense of [15, Remark 3.6] yields the l r bicovariant B-correspondence L∞(G)′ L∞(Gb) F L∞(Gb). e INDUCTION FOR LOCALLY COMPACT QUANTUM GROUPS 9 Now [15, Proposition 3.7] shows the existence of a canonically determined Hilbert B-module Ind(E) together with a unitary representation Ind(X) ∈ Rep(G,Ind(E)) such that L∞(G)′ L∞(G)′ L∞(Gb) F L∞(Gb) ∼= L∞(Gb) L2(G)⊗Ind(E) L∞(Gb) (2.7) e as bicovariant correspondences. In what follows we shall use the symbol Ind (·) to denote V the objects defined above. Note that the fact that F is a left G⋉L∞(G/H)-module implies in particular existence of a map e ρ :L∞(G/H) −→ L(Ind (E)) (2.8) V satisfying the covariance relation: Ind(X)∗ 1 ⊗ ρ(x) Ind(X) = (id ⊗ ρ)∆G(x) for all x ∈ L∞(G/H). (cid:0) (cid:1) TheHilbert B-moduleInd (E) is called the Hilbert B-moduleinduced in the sense of Vaes V from E and the unitary representation Ind (X) is called the representation induced by X V (also in the sense of Vaes). 3. Representation-theoretic characterization of open quantum subgroups InthissectionweprovethatforanopenquantumsubgroupHofalocallycompactquantum groupGwehaveacanonicalC∗-inclusionCu(H)⊂ Cu(G). Weremarkthatthiswaspreviously 0 0 proved by the first three authors under the cobamenabiblity assumption on G (see [6, Theorem 5.2]). The proof given here is completely different from the one in [6]. In fact, in the latter we showed through direct computations that every positive definite function in L∞(H) is also positive definite when regarded as an element in L∞(G). As in the general, non-coamenable, case thecriterion for positive-definiteness usedin [6]does nothold (see [4]), that proofcannot beextended to arbitrary locally compact quantum groups. In the present approach we utilize Kustermans’s theory of induced representations. In view of the 1-1 correspondence between unitary representations of a locally compact quantum group G and representations of Cu(G), the result mentioned above, together with 0 the existence of a suitable conditional expectabtion, allows us to apply classical construction of inducing representations from C∗-subalgebras due to Rieffel [12] to the case of unitary representations of open quantum subgroups. Inwhatfollows G is alocally compact quantum groupandH isan openquantum subgroup of G. Recall that we denote by α :L∞(G) → L∞(G)⊗¯ L∞(H) the canonical (right) action of H on G associated to the inclusion H ֒→ G. Recall also that we write ρG/H for the restriction of ∆G to L∞(G/H), giving the canonical left action of G on G/H. Let θ be the disintegration of ϕG, that is the unique n.s.f. weight on L∞(G/H) satisfying θ (id⊗ϕH)(α(x)) = ϕG(x), x ∈ L∞(G)+ (3.1) (cid:0) (cid:1) (see [8, Proposition 8.7]). Lemma 3.1. The weight θ is invariant, that is (id⊗θ)(ρG/H(x)) = θ(x)1 for all x ∈L∞(G/H)+. Moreover θ(1H) is finite and non-zero. Proof. Let ω ∈ L1(G) be a state. Define a (normal semifinite) weight θ on L∞(G/H) by ω θω(y) = θ (ω⊗id)(ρG/H(y)) , y ∈ L∞(G/H)+. (cid:0) (cid:1) 10 MEHRDADKALANTAR,PAWEL KASPRZAK,ADAMSKALSKI,ANDPIOTRSOL TAN For x ∈ L∞(G)+ we have θω (id⊗ϕH)α(x) = θ(cid:16)(ω⊗id)(ρG/H (id⊗ϕH)(α(x)) )(cid:17) (cid:0) (cid:1) (cid:0) (cid:1) = θ(cid:16)(ω⊗id)(∆G (id⊗ϕH)α(x) )(cid:17) (cid:0) (cid:1) = θ (ω⊗id)((id⊗id⊗ϕH)(∆G⊗id)(α(x))) (cid:0) (cid:1) = θ (ω⊗id)((id⊗id⊗ϕH)(id⊗α)(∆G(x))) (cid:0) (cid:1) = θ (id⊗ϕH)((ω⊗id⊗id)(id⊗α)(∆G(x))) (cid:0) (cid:1) = θ(cid:16)(id⊗ϕH)(α (ω⊗id)(∆G(x))) (cid:17) (cid:0) (cid:1) = ϕG (ω⊗id)(∆G(x)) = ϕG(x). (cid:0) (cid:1) By uniqueness of θ we get θω = θ, which means that θ is invariant under the action ρG/H. For the second assertion, note first that since the support projection 1H of H is minimal in L∞(G/H)by[6,Proposition3.2],itfollowsthatθ(1H)< +∞(otherwiseθwouldnotbesemifi- nite). Suppose now that θ(1H)= 0. Then the invariance of θ yields θ (ω⊗id)(∆G(1H)) = 0 (cid:0) (cid:1) for all ω ∈ L1(G). Since (ω⊗id)(∆G(1H)) ω ∈ L1(G) is a weak∗-dense ideal in L∞(G/H) (cid:8) (cid:9) by [6, Paragraph following Eq. (3.6) in the proof of Theorem 3.3] the last condition implies that θ = 0, which contradicts (3.1). (cid:3) By normalizing the Haar weights, if necessary, we may (and will) assume θ(1H) = 1. We will write L2(G/H) for the GNS Hilbert space of θ (it was denoted H in Section 2.4). The θ weight θ determines aunitaryimplementation Υ of theaction ρG/H of Gon G/H. Theprecise form of the unitary Υ can be deduced as follows. Define an operator Υ∗ :L2(G)⊗L2(G/H) → L2(G)⊗L2(G/H) by Υ∗(cid:0)ΛϕG(x)⊗Λθ(y)(cid:1) = (ΛϕG ⊗Λθ)(cid:0)ρG/H(y)(x⊗1L∞(G/H))(cid:1), x ∈ NϕG, y ∈ Nθ. (3.2) Then Υ∗ is easily seen to be an isometry satisfying the following two conditions: • Υ∗ ∈ L∞(G)⊗¯ B(L2(G/H)), • (∆G⊗id)(Υ∗) = Υ∗ Υ∗ . 23 13 Hence, by [2, Corollary 4.11] Υ∗ is unitary and satisfies the equality ∆G(x) = Υ∗(1⊗x)Υ, x ∈ L∞(G/H). Since θ is invariant (Lemma 3.1), it follows by the same methods as those used in [14, Propo- sition 4.3] that Υ is the canonical implementation of the action ρG/H of G on L∞(G/H). Beforeprovingthemainresultofthissection, Theorem3.2,werecallthefollowingfactfrom [6]. SupposeH ⊂ Gisidentifiedas anopenquantumsubgroupviatheHopf∗-homomorphism πu ∈ Mor(Cu(G),Cu(H)) as in Definition 2.2 and let πu ∈ Mor(Cu(H),Cu(G)) be the dual of 0 0 0 0 πu. Then πu(cid:0)Cu0(H)(cid:1) ⊂ Cu0(G) by [6, Lemma 2.5]. b b b Theoremb3.2. Lbet G be a lbocally compact quantum group and let H ⊂ G be an open quantum subgroup identified via the Hopf ∗-homomorphism πu ∈ Mor(Cu(G),Cu(H)). Then the ∗- 0 0 homomorphism πu :Cu(H)→ Cu(G) is injective. 0 0 Proof. We will cboncludebinjectivitybof πu by proving surjectivity of the dual map πu∗ :bCu(G)∗ −→ Cu(H)∗. 0 0 b b b