Tannaka-Krein reconstruction for coactions of finite quantum groupoids

HAL Id: hal-02143671

https://hal.archives-ouvertes.fr/hal-02143671

Submitted on 4 Jun 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Tannaka-Krein reconstruction for coactions of finite quantum groupoids

Leonid Vainerman, Jean-Michel Vallin

To cite this version:

Leonid Vainerman, Jean-Michel Vallin. Tannaka-Krein reconstruction for coactions of finite quantum groupoids. METHODS OF FUNCTIONAL ANALYSIS AND TOPOLOGY, 2017, 23 (1), pp.76-107.

�hal-02143671�

Tannaka-Krein reconstruction for coactions of finite quantum groupoids

Leonid Vainerman Jean-Michel Vallin 30 June 2016

Abstract

We study coactions of finite quantum groupoids on unital C^˚- algebras and obtain the Tannaka-Krein reconstruction theorem for them.

Contents

1 Introduction 2

2 Finite quantum groupoids, their representations, comodules

and corepresentations 4

3 Coactions of finite quantum groupoids on unital C*-algebras 20 4 From coactions to module categories over U CoreppGq 28 5 From module categories over U CoreppGq to coactions 35

6 Equivalence of categories 41

1 2

1AMS Subject Classification [2010]: 18D10, 16T05, 46L05.

2Keywords : Coactions and corepresentations of quantum groupoids, C^˚-categories, reconstruction theorem.

1 Introduction

As shown in [20], [21], weak Hopf C^˚-algebra in the sense of [3] and their coidealC^˚-subalgebras play important role in the description of Jones’s tower of II₁-subfactors with finite index and finite depth. It is also known that any fusion category can be realized as a representation category of a weak Hopf algebra [9]. This explains the interest in the construction of concrete examples of these objects and in the classification of their coideal subalgebras.

In this paper we use the term ”a finite quantum groupoid” instead of

”a weak Hopf C^˚-algebra” because a groupoid C^˚-algebra and an algebra of functions on a usual finite groupoid carry this type of a structure. Some particular constructions of finite quantum groupoids were proposed in [31]

and [19] (see also the survey [22] and references therein), but the general way to construct them is the application of the Hayashi’s reconstruction theorem of Tannaka-Krein type [10] to concrete tensor categories.

The above approach was used in [13], where a series of concrete finite quantum groupoids was constructed using Tambara-Yamagami categories [27]. These categories belong to the much wider family of Z{2Z-extensions of pointed fusion categories classified in [28]. The problem of the description of coideal C^˚-subalgebras of a given finite quantum groupoid is even harder, and until now only two concrete families of such subalgebras constructed in [13] ”by hand” are known.

On the other hand, a similar problem exists in the theory of compact quantum groups (CQG), where a lot of concrete examples were constructed using well-known Woronowicz’s reconstruction theorem of Tannaka-Krein type [32]. As for the description of coideal C^˚-subalgebras of a given CQG, the recent papers on categorical duality for (co)actions of CQG on C^˚- algebras proved to be very useful - see [6], [7], [14], [17].

Our aim, here and in a subsequent work, is to develop a similar approach for the study of finite quantum groupoid coactions on C^˚-algebras and to apply it to the description of coideal C^˚-subalgebras of concrete finite quan- tum groupoids. The first step in this project is to formulate and to prove the general duality statement - Theorem 1.1 below.

Let us describe the structure of the paper. In Section 2 we recall basic definitions and results on finite quantum groupoids following [3] and [22]. We also translate the representation theory of these objects treated in [4] and [18] into the language of unitary corepresentations andC^˚-tensor categories suitable for the construction of the categorical duality. Finally, we translate

into this language the reconstruction theorem proved in [10] and [26].

In Section 3 we develop the theory parallel to the one of CQG coactions [2]. Doing this, we simplify significantly, in our particular case, some con- structions related to coactions of general measured quantum groupoids - see [8], [31]. Let a be a coaction of a finite quantum groupoid G on a unital C^˚-algebra A (called a G-C^˚-algebra) we get the canonical implementation of a and study the properties of the spectral subspaces (isotypical compo- nents) of A. Note that the subalgebra of fixed points of A with respect to a can be strictly smaller than the spectral subspace corresponding to the trivial corepresentation of G (in the CQG case they are equal). This creates specific problems that we solve in Sections 4,5 and 6 devoted to the proof of our main result which is parallel to [6], Theorem 6.4 and [14], Theorem 3.3:

Theorem 1.1 LetGbe a regular coconnected finite quantum groupoid. Then the following two categories are equivalent:

(i) The category of unital G-C^˚-algebras with unital G-equivariant ˚- homomorphisms as morphisms.

(ii) The category of pairspM, Mq, where Mis a left module C^˚-category over C^˚-tensor category UCoreppGq of unitary corepresentations of G and M is a generator in M, with equivalence classes of unitary module functors respecting the prescribed generators as morphisms.

This proof divides into three parts. First, given a unitalG-C^˚-algebra A, we show in Section 4 that the category D_A of finitely generated equivariant C^˚-correspondences whose morphisms are equivariant maps, is a strict left module category over UCoreppGq. The algebra A itself is a generator in D_A. The idea of such a construction in the CQG case was proposed in [6].

Vice versa, it is shown in [14] that any pair pM, Mq as above generates so-called weak tensor functor. Using this functor, we construct in Section 5 an algebra whose C^˚-completion is a unital G-C^˚-algebra. Finally, we show in Section 6 that the two above mentioned constructions are mutually inverse which gives the equivalence of the categories in question.

It was shown in [17] in the CQG case thatUCoreppGq-module categories parameterized by unitary tensor (not weak tensor !) functors correspond to Yetter-Drinfeld G-C^˚-algebras. In a subsequent work we expect to get a similar result for finite quantum groupoids and to apply it to the description of coideal C^˚-subalgebras of quotient type.

Our standard references are: [12] for general categories, [9] for tensor categories, [16] forC^˚- andC^˚-tensor categories, [11] for HilbertC^˚-modules, and [22] for finite quantum groupoids.

2 Finite quantum groupoids, their represen- tations, comodules and corepresentations

1. Finite quantum groupoidsAweak HopfC^˚-algebra G“ pB,∆, S, εqis a finite dimensionalC^˚-algebra B with the comultiplication ∆ :B ÑBbB, counitε:B ÑC, and antipodeS :B ÑB such thatpB,∆, εqis a coalgebra and the following axioms hold for all b, c, dPB :

(1) ∆ is a (not necessarily unital) ˚-homomorphism :

∆pbcq “ ∆pbq∆pcq, ∆pb^˚q “∆pbq^˚,

(2) The unit and counit satisfy the identities (we use the Sweedler leg notation ∆pcq “ c₁ bc₂, p∆bid_Bq∆pcq “ c₁ bc₂bc₃ etc.):

εpbc₁qεpc₂dq “ εpbcdq,

p∆p1q b1qp1b∆p1qq “ p∆bid_Bq∆p1q, (3) S is an anti-algebra and anti-coalgebra map such that

mpid_BbSq∆pbq “ pεbid_Bqp∆p1qpbb1qq, mpSbid_Bq∆pbq “ pid_Bbεqpp1bbq∆p1qq, where m denotes the multiplication.

The right hand sides of two last formulas are called target and source counital mapsε_tandε_s, respectively. Their images are unitalC^˚-subalgebras of B called target and source counital subalgebras B_t and B_s, respectively.

The dual vector space ˆB has a natural structure of a weak Hopf C^˚- algebra ˆG“ pB,ˆ ∆,ˆ S,ˆ εˆqgiven by dualizing the structure operations of B:

ăϕψ, bą “ ăϕbψ, ∆pbq ą, ă∆pϕq, bˆ bcą “ ăϕ, bcą,

ăSpϕq, bˆ ą “ ăϕ, Spbq ą, ăφ^˚, bą “ ăϕ, Spbq^˚ ą,

for all b, cP B and ϕ, ψP B. The unit of ˆˆ B isε and the counit is 1.

The counital subalgebras commute elementwise, we have S˝ε_s “ε_t˝S and SpBtq “ Bs. We say that B is connectedif BtXZpBq “C (whereZpBq is the center of B), coconnected if B_t XB_s “ C, and biconnected if both conditions are satisfied.

The antipode S is unique, invertible, and satisfies pS ˝ ˚q² “ id_B. We will only consider regular quantum groupoids, i.e., such that S²|Bt “ id. In this case, there exists a canonical positive element H in the center of B_t such that S² is an inner automorphism implemented by G“HSpHq^´1, i.e., S²pbq “GbG^´1for allbPB. The elementGis called the canonical group-like element of B, it satisfies the relation ∆pGq “ pGbGq∆p1q “∆p1qpGbGq. There exists a unique positive functional h on B, called a normalized Haar measure such that

pid_Bbhq∆“ pε_tbhq∆, h˝S “h, h˝ε_t “ε, pid_Bbhq∆p1_Bq “1_B. We will dehote by H_h the GNS Hilbert space generated by B and h and by Λ_h :B ÑH_h the corresponding GNS map.

2. Unitary representations

By definition, the objects of the category U ReppGq of unitary represen- tations ofGare leftB-modules of finite rank such that the underlying vector space is a Hilbert space H with a scalar product ă ¨,¨ ą such that

ăb¨v, wą“ăv, b^˚¨wą, for all v, wP H, bPB,

and morphisms are B-linear maps. It is a semisimple linear category whose simple objects are irreducibleB-modules. It is also a tensor category: for ob- jects H₁, H₂ PU ReppGq, define their tensor product as the Hilbert subspace

∆p1_Bq ¨ pH₁bH₂q of the usual tensor product together with the action of B given by ∆. Here we use the fact that ∆p1_Bqis an orthogonal projection.

The tensor product of morphisms is the restriction of the usual tensor product of B-module morphisms. Let us note that any H P U ReppGq is automatically aBt-bimodule viaz¨v¨t:“zSptq¨v, @z, tP Bt, v PE, and that the above tensor product is in fact bBt, moreover theB_t-bimodule structure for H₁bBt H₂ is given by z¨ξ¨t “ pzbSptqq ¨ξ, @z, t PB_t, ξ PH₁bBt H₂.

One deduces that the above tensor product is associative : pH₁bBtH₂q bBt H₃ “H₁bBt pH₂bBt H₃q,

so the associativity isomorphisms are trivial. The unit object of U ReppGqis B_twith the action of B given byb¨z :“ε_tpbzq, @bP B, z PB_t and the scalar product ăz, tą“hpt^˚zq. The left and right unit morphisms are:

l_EpzbBtvq “z¨v and r_EpvbBt zq “Spzq ¨v, @z PB_t, v PE. (1) For any morphism f : H₁ Ñ H₂, define f^˚ : H₂ Ñ H₁ as the adjoint linear map: ăfpvq, w ą“ăv, f^˚pwq ą, @v PH₁, w PH₂, it is easy to check that f^˚ isB-linear. It is clear thatf^˚˚ “f, that pfbBtgq^˚ “f^˚bBtg^˚, and thatEndpHqis aC^˚-algebra, for any objectH. SoU ReppGqis a strict finite C^˚-multitensor category (i.e., has all the properties of a C^˚-tensor category except for one: 1 is not necessarily simple).

In order to make U ReppGq a rigid C^˚-tensor category in the sense of [16], Definition 2.1.1, we have to define the conjugate for anyH PU ReppGq. Take the dual vector space ˆH which is naturally identified (v ÞÑv) with the conjugate Hilbert space H :ă v, w ą“ă w, v ą, @v, w P H. The action of B on H is defined by b¨v “ G^1{2Spbq^˚G^´1{2¨v, where G is the canonical group-like element of G. Then the rigidity morphisms defined by

R_Hp1_Bq “ Σ_ipG^1{2¨e_ibBt ¨e_iq, R_Hp1_Bq “Σ_ipe_ibBt G^´1{2¨e_iq, (2) where te_iui is any orthogonal basis in H, satisfy all the needed properties - see [5], 3.6. Also, it is known that theB-moduleB_t is irreducible if and only if BsXZpBq “ C1B, i.e., if G is connected. So that, we have

Proposition 2.1 U ReppGq is a strict rigid finite C^˚-multitensor category.

It is C^˚-tensor if and only if G is connected.

3. Unitary comodules

Definition 2.2 A right unitary G-comodule is a pair pH,aq, where H is a Hilbert space with scalar product ă ¨,¨ ą, a:H ÑHbB is a bounded linear map between Hilbert spaces H and HbH_h “HbΛ_hpBq, and such that:

(i) pabid_Bqa“ pid_H b∆qa;

(ii) pidH bεqa“idH;

(iii) ăv¹, w ąv² “ăv, w¹ ąSpw²q^˚, @v, wPH.

A morphism of unitaryG-comodulesH₁ andH₂ is a linear mapT :H₁ Ñ H2 such that aH2 ˝T “ pT bidBqaH1 (i.e., a B-colinear map).

Right unitary G-comodules with finite dimensional underlying Hilbert spaces and their morphisms form a category which we denote byU ComodpGq.

We say that two unitary G-comodules are equivalent (resp., unitarily equivalent) if the space of morphisms between them contains an invertible (resp., unitary) operator.

In what follows, we will use the leg notation apvq “v¹bv², for all v P H.

Example 2.3 Let us equip a right coideal I Ă B with the scalar product ăv, wą:“hpw^˚vq. Then the strong invariance of h gives:

ăv¹, w ąv² “ phbid_Bqppw^˚b1_Bq∆pvqq “

“ phbS^´1qp∆pw^˚qpvb1_Bqq “ăv, w¹ ąSpw²q^˚. Remark 2.4 By (ii) any coaction a is injective

IfpH,aqis a right unitaryG-comodule, thenH is naturally a unitary left G-module viaˆ

ˆb¨v :“v¹ ăˆb, v² ą, @ˆbP B, vˆ PH. (3) The unitarity follows from the calculation

ăˆb¨v, wą“ăv¹ ăˆb, v² ą, w ą“ăˆb,ăv¹, w ąv² ą“

“ăˆb,ăv, w¹ ąSpw²q^˚ ą“ăv, w¹ăˆb, Spw²q^˚ ą ą“ăv,pˆbq^˚¨wą, for all v, w PH and ˆbPBˆ. In particularH is a ˆB_t-bimodule.

Due to the canonical identifications B_t – Bˆ_s and B_s – Bˆ_t given by the maps z ÞÑ zˆ “ εp¨zq and t ÞÑ ˆt “ εpt¨q, H is also a B_s-bimodule via z ¨ v ¨ t “ v¹εpzv²tq, for all z, t P B_s, v P V. The maps α, β : B_s Ñ BpHq defined by αpzqv :“ z ¨v and βpzqv :“ v ¨z, for all z P B_s, v P H are a ˚-algebra homomorphism and antihomomorphism, respectively, with commuting images. Indeed, for instance, for all v, wPH, z PB_s, one has:

ăαpzqv, wą:“ă v¹εpzv²q, w ą“εpă v¹, w ązv²q “

“εpăv, w¹ ązSpw²q^˚q “ăv, w¹ ąεpSpw²qz^˚q “

“ăv, w¹εpSpz^˚qw²q ą“ăv, αpz^˚qw¹εpw²q ą“ăv, αpz^˚qwą.

So that, αpzq^˚ “αpz^˚q, and similarly for the mapβ. We have the following useful relations:

apαpxqβpyqvq “v¹bxv²y @v PH, x, y P Bs. (4) and

αpxqβpyqv¹bv² “v¹bSpxqv²Spyq @v PH, x, y PB_s. (5) The correspondence (3) is bijective as one has the inverse formula: ifpb_iqi

is a basis for B and pˆb_iq is its dual basis in ˆB, then set:

apvq “ÿ

i

pˆb_i¨vq bb_i @v P H. (6) Moreover, formulas (3) and (6) imply also a bijection of morphisms. Thus, we have two functors, F1 : U ComodpGq Ñ U ReppGqˆ and G1 : U ReppGq ш U ComodpGq, which are mutually inverse. So, these categories are isomorphic as linear categories, and we can transport various additional structures from U ReppGqˆ to U ComodpGq.

For instance, let us define tensor product of two unitary G-comodules, pH₁,a_H1q and pH₂,a_H2q. As a vector space, it is

H₁bBˆt H₂ :“∆pˆ ˆ1qpH₁bH₂q “ˆ1₁¨H₁bˆ1₂¨H₂,

and is generated by the elements xbBˆty :“∆pˆ ˆ1q ¨ pxbyq, where xPH₁, y P H2, so it can be identified withH1bBsH2 (see [25], 2.2 or [23], Chapter 4).

Lemma 2.5 If pH₁,aq, pH₂,bq PU ComodpGq, then the projection P :H₁b H₂ ÑH₁bBˆt H₂ defined by Ppvq “∆pˆ ˆ1q ¨v, for all v PH₁bH₂, satisfies

Ppxbyq “x¹by¹εpx²y²q, for all xPH₁, yP H₂.

The proof is the direct calculation using the axiom (2) of a weak Hopf algebra:

ˆ1₁ ¨xbˆ1₂¨y“ px¹by¹qεpx²1₁qεp1₂y²q “

“ px¹ by¹qεpx²y²q.

Corollary 2.6 The linear map abBsb given by:

vbBswÞÑv¹bBs w¹bv²w², @v P H₁, w PH₂,

is a coaction of G on H₁bBs H₂ (i.e., satisfies Definition 2.2, (i), (ii)).

Proof. @v PH₁, wP H₂, one has:

ppab_Bsbq bi_BqpabBs bqpv bBswq “

“ ppabBs bq bi_Bqpv¹bBs w¹bv²w²q

“ p∆ˆpˆ1q b1_BbBq ¨ p∆ˆpˆ1q.papv¹q¹ bbpw¹q¹q bapv¹q²bpw²q²bv²w²q

“ p∆pˆ ˆ1q b1_BbBq ¨ papv¹q¹bbpw¹q¹bapv¹q²bpw²q²bv²w²q

“ p∆ˆpˆ1q b1_BbBq ¨ ppabi_Bqapvqq134pbbi_Bqbpwqq234q

“ p∆pˆ ˆ1q b1_BbBq ¨ ppi_Eb∆qapvqq₁₃pi_F b∆qbpvqq₂₃q

“ p∆ˆpˆ1q b1_BbBqpi_EbF b∆qpp∆ˆpˆ1q b1_Bq.papvqq13bpvqq23q

“ pid_H1bBsH2 b∆qpabBs bqpvbBs wq Moreover, using Lemma 2.5, we have :

pidH1b_BsH2 bεqpabBsbqpvbBs wq “ v¹bw¹εpv²w²q “ Ppvbwq “vbBsw

˝ The direct calculation shows that the tensor product coaction is unitary.

Thus, U ComodpGq is a multitensor category whose associativity morphisms are trivial, the unit object is pB_s,∆|Bsq. It is simple if and only if G is coconnected. The left and right unit isomorphisms are:

l_H :B_sbBsH ÑH, zbBsv ÞÑz¨v, r_H :HbBsB_sÑH, vbBsz ÞÑv¨z. (7) One can check that these isomorphisms are unitary and their inverses are:

l^´1_H pvq “ 1₁ bBs v¹εp1₂v²q and r_H^´1pvq “ v¹bBs ε_spv²q. (8) Let us define the conjugate object for pH,aq PU ComodpGq. The corre- sponding Hilbert space is H. In what follows, we use the Sweedler arrows ˆb áb :“b1 ăˆb, b2 ą, bàˆb:“b2 ăˆb, b1 ą, @bP B,ˆbPBˆ.

Lemma 2.7 The conjugate object forpH,aqinU ComodpGqispH,aq, where˜

˜

apvq “v¹b rGˆ^´1{2 á pv²q^˚ àGˆ^1{2s,

and Gˆ is the canonical group-like element of the dual quantum groupoid G.ˆ

Proof. The unitarity ofG₁pH,˜aqmeans thatăˆb¨v, wą_H“ăv,ˆb^˚¨wą_H, for all v, wPH. The left hand side equals to ăv¹, w ą_Hăˆb, v² ą. And the right hand side equals to

ăv,Gˆ^1{2Spˆ ˆb^˚q^˚Gˆ^´1{2¨wą_H“ăGˆ^1{2Spˆ ˆb^˚q^˚Gˆ^´1{2¨w, v ąH“

“ă w,Gˆ^´1{2Spˆ ˆb^˚qGˆ^1{2¨v ąH“ăw, v¹ ąH ăGˆ^´1{2Spˆ ˆb^˚qGˆ^1{2, v² ą “

“ăv¹, w ą_H ăGˆ^´1{2ˆbGˆ^1{2,pv²q^˚ ą “

“ă v¹, w ą_Hăˆb,rGˆ^´1{2 á pv²q^˚ àGˆ^1{2s ą.

Comparing the above expressions, we have the result. ˝ The rigidity morphisms are given by (2) withB_t replaced byB_s. For any morphism f, f^˚ is the conjugate linear map on the corresponding Hilbert spaces, the colinearity of f implies thatf^˚ is colinear. So that, we have Proposition 2.8 U ComodpGq is a strict rigid finite C^˚-multitensor cate- gory. It is C^˚-tensor if and only if G is coconnected.

4. Unitary corepresentations

Definition 2.9 Aright unitary corepresentationofGon a Hilbert space H is a partial isometry V PBpHq bB such that:

(i) V12V13“ pid_BpHqb∆qpVq.

(ii) pid_BpHqbεqpVq “id_BpHq.

If U and V are two right corepresentations on Hilbert spaces H_U and HV, respectively, a morphism between them is a bounded linear map T P BpH_U, H_Vq such that pT b1_BqU “ VpT b1_Bq. The vector space of such morphisms is denoted by M orpU, Vq. We will denote by U CoreppGq the category whose objects are right unitary corepresentations pH, Vq on finite dimensional vector spaces with morphisms as above.

One says that U and V are equivalent (resp., unitarily equivalent) if M orpU, Vq contains an invertible (resp., unitary) operator.

Proposition 2.10 If pH,aq is a unitary G-comodule, let us define an oper- ator V on HbH_h as follows:

VpxbΛ_hyq:“x¹bΛ_hpx²yqq, for all xPH, yP B.

Then V is a unitary corepresentation of G on H, and one has:

V^˚pxbΛ_hyq:“x¹bΛ_hpSpx²qyqq, for all xPH, yPB.

Proof. LetI_h be an implementation of ∆ (for example,I_h PBpH_hbH_hq: Λ_hbhpy¹byq ÞÑ Λ_hbhp∆pyqpy¹b1_Bqq, see [29], 3.2) for details), then one has for all xP H, y, cPB:

V12V13pxbΛhybΛhcq “ pV b1Bqpx¹bΛhybΛhpx²cqq

“x¹bΛ_hpx²yq bΛ_hpx³cq

“x¹bΛ_hbhp∆px²qpybcqq

“x¹bI_hpx²b1_BqI_h^˚pΛ_hbhpybcqq

“ p1_BpHqbI_hqpx¹b tpx²b1_BqI_h^˚Λ_hbhpybcquq

“ p1_BpHqbI_hqpV b1_BqpxbI_h^˚pΛ_hbhybcqq

“ p1_BpHqbI_hqpV b1_Bqp1_BpHqbI_hq^˚pxbΛ_hbhpybcqq

“ p1_BpHqb∆qpVqpxbΛhybΛhcq.

Next, we have, for any decomposition V “ř

iPI

v_ibb_i (v_i PBpHq, b_i PBq: pid_BpHqbεqpVqpξq “ pid_BpHqbεqpÿ

iPI

vipξq bbiq “

“ÿ

iPI

εpb_iqv_ipξq “ pid_BpHqbεqapξq “ ξ, @ξP H.

In order to show thatV is a partial isometry, consider the separability element e_s “ pid_B bSq∆p1_Bq of the algebra B_s and the idempotents e_β,id “ pβb id_Bqpe_sq P βpB_sq bB_s and e_α,S “ pαbSqpe_sq P αpB_sq bB_s. As α and β are ˚-maps, these idempotents are orthogonal projections on HbH_h. It is straightforward to check, using (4) and (5), that:

• for all x, y P B_s, one has:

Vpαpxqβpyq b1_Bq “ p1_BpHqbxqVp1_BpHqbyq, (9)

pαpxqβpyq b1_BqV “ p1_BpHqbSpxqqqVp1_BpHqbSpyqq. (10)

• V e_β,id “V, e_α,SV “V.

Moreover,V is invertible inBpe_β,idpHbH_hq, e_α,SpHbH_hqq. Indeed, consider an operator W acting on HbH_h defined by

WpvbΛ_hpbq:“v¹bΛ_hpSpv²qbq, @v PH, bPB.

Then we have:

W VpvbΛ_hpbq:“Wpv¹bΛ_hpv²bqq “

“v¹bΛ_hpSpv²qv³bq “v¹bΛ_hpε_spv²qbq.

On the other hand,

e_β,idpvbΛ_hpbqq “ pv¨1₁q bΛ_hpSp1₂qbq “ v¹bΛ_hpSp1₂qˆ ˆεpv²1₁qbq “ v¹bΛ_hp1₁qεpv²Sp1₂qqbq “ v¹bΛ_hpε_spv²qbq.

And similarlyV W “e_α,S, so thatW is the inverse ofV. Finally, we compute, for all v, w PH, b, cP B:

ăVpvbΛ_hpbqq, VpwbΛ_hpcqq ą “ăv¹bΛ_hpv²bq, w¹bΛ_hpw²cq ą

“ăv¹, w¹ ąhpc^˚pw²q^˚v²bq

“ăv, w¹ ąhpc^˚pw³q^˚rSpw²qs^˚bq

“ăv, w¹ ąhpc^˚rε_spw²qs^˚bq.

On the other hand,

ăv bΛ_hpbq, e_β,idpwbΛ_hpcq ą“ăvbΛ_hpbq,pw¨1₁q bΛ_hpSp1₂qcq ą“

“ă v, w¨1₁ ąhpc^˚rSp1₂qs^˚bq “ăv, w¹ ąεpw²1₁qhpc²rSp1₂qs^˚bq. These expressions are equal because ε_spxq :“ 1₁εpx1₂q “ Sp1₂qεpxSp1₁qq “ Sp12qεpx11qq, for all x P B. We used above the equality εpxSpzqq “ εpxzq, for all z P B_t which can be obtained by applying εb ε to both sides of the equality ∆p1_BqpSpzq b1_Bq “ ∆p1_Bqp1_Bbzq. As e_β,id is an orthogonal projection, this means that V is bounded andV^˚V “eβ,id.

Similar reasoning shows that V^˚ equals to the above mentioned W. ˝ We also have a converse statement.

Proposition 2.11 Any unitary corepresentation V of G on a Hilbert space H generates a unitary comodulepH,aq, where apvq “VpvbΛ_hp1_Bqq @v PH.

Proof. The first two conditions of Definition 2.2 follow from the first two conditions of Definition 2.9. The relation between V and the coaction a:v ÞÑv¹bv² is given byVpvbΛhpbqq “v¹bΛhpv²bq. We have seen already that the operator W acting on HbH_h and defined by

WpvbΛhpbqq “v¹bΛhpSpv²qbq, @v PH, bPB,

satisfies the relations V W “e_a,S and W V “e_b,id. AsV is a partial isometry with initial and final Hilbert subspaces e_a,SpH bH_hq and e_b,idpH b H_hq, respectively, we haveW “V^˚. Then for allv, wPH and cPB, the equality

ăVpvbΛ_hp1_Bqq, wbΛ_hpcq ą“ăvbΛ_hp1_Bq, V^˚pwbΛ_hpcqq ą, can be rewritten as

ăv¹, w ąhpc^˚v²q “ăv, w¹ ąhpc^˚rSpw²qs^˚q,

which implies the unitarity of the G-comodule in question. ˝ Let pH₁,aq and pH₂,bq be two unitary G-comodules, and let T be in BpH₁, H₂qintertwining a and b, then one has, for all xPH₁, bP B:

VH2pT b1qpxbΛhpbqq “ pT xq¹bΛhppT xq²bq

“ p1_H2 bπ¹pbqqppT xq¹bΛ_hppT xq²q

“ p1_H2 bπ¹pbqqpid_F bΛ_hqpbpT xqq

“ p1_H2 bπ¹pbqqpid_F bΛ_hqppT b1qapxqq

“ p1_H2 bπ¹pbqqpT b1qpid_H1 bΛ_hqpapxqq

“ pT b1qp1_H1 bπ¹pbqqpid_H1 bΛ_hqpapxqq

“ pT b1qV_H1pxbΛ_hpbqq. Hence, T PM orpV_H1, V_H2q.

Corollary 2.12 The correspondence F₂ defined by F₂pH,aq “ pV, Hq and F₂pTq “ T for all objects pH,aq and morphisms T of U ComodpGq, is a functor from U ComodpGq to U CoreppGq viewed as semisimple linear cate- gories. The correspondence G₂ between unitary corepresentations of G and G-comodules given by Proposition 2.11 clearly extends to morphisms and de- fines a functor inverse toF2, soU ComodpGqandU CoreppGqare isomorphic as linear categories. Then we can equip U CoreppGqwith tensor product and duality by transporting these structures from ComodpGq.

If pU, H_Uq, pV, H_Vq PU CoreppGq, let us define their tensor product.

Lemma 2.13 One has pP bid_BqU₁₃V₂₃ “U₁₃V₂₃pP bid_Bq “ U₁₃V₂₃, where U₁₃V₂₃P BpH_U bH_Vq bB and P was defined in Lemma 2.5.

Proof. There exist finite families tb_ku and tb¹_kuinB_s such that Σ_kb¹_kb_k“ Σ_kb_kb¹_k “1_B, and for all xP H_U and allyP H_V one has:

Ppxbyq “ ∆ˆpˆ1q ¨ pxbyq “ Σ_kβpb_kqxbα¹pb¹_kqy,

where α¹ is the ˚-representation of B_s corresponding to pV, H₂q. Using four times (10), one has:

pP bid_BqU₁₃V₂₃“Σ_kpβpb_kq bα¹pb¹_kq bid_BqU₁₃V₂₃“

“Σ_kpβpb_kq bid_HV bid_BqU₁₃pid_HU bα¹pb¹_kq bid_BqV₂₃

“Σ_kpβpb_kq bid_HV bid_BqU₁₃pid_HU bid_HV bSpb¹_kqV₂₃

“Σkpβpbkqβpb¹_kq bidH_V bidBqU13V23

“Σ_kpβpb¹_kb_kq bid_HV bid_BqU₁₃V₂₃“U₁₃V₂₃

“Σ_kU₁₃pid_HU bid_HV bb_kb¹_kqV₂₃

“Σ_kU₁₃pβpb_kq bid_HV b1_BqV₂₃pid_HU bα¹pb¹_kq bid_Bq

“Σ_kU₁₃V₂₃pβpb_kq bα¹pb¹_kq bid_Bq “U₁₃V₂₃pP bid_Bq.

˝

Lemma2.13 justifies the following:

Definition 2.14 If pU, H_Uq, pV, H_Vq P U CoreppGq, their tensor product is the bounded linear map:

U jV “U13V23“ pP bidBqU13V23pP bidBq viewed as an element of BpH_UbBs H_Vq bB.

Proposition 2.15 U jV P U CoreppGq, it acts on H_U bBs H_V and:

G₂pU, H₁q bBs G₂pV, H₂q “ G₂pUjV, H₁bBs H₂q.

Proof. IfpU, H_Uq, pV, H_Vq PU CoreppGq, letU “ř

i

u_ibb_i,V “ř

j

u_jbb_j be decompositions of U and V. Then U jV “ ř

i,j

ui bvj bbibj, and let us define θ_UjV PBpH_UbBsH_V, H_U bBs H_V bBq by:

θ_UjVpxbBsyq “ ÿ

i,j

u_ibv_jpPpxbyqq bb_ib_j. Then, using Lemma 2.13, one has:

θ_UjVpxbBsyq “ÿ

i,j

u_ibv_jpxbyqq bb_ib_j “ÿ

i,j

Ppu_ipxq bv_jpyqq bb_ib_j

“ pa_U ba_VqpxbBs yq,

and the result follows. ˝

The unit object 1 of U CoreppGq with respect to j acts on B_s and is defined by zbb ÞÑ1₁ b1₂zb, for all z P B_s, b P B. It is simple if and only if G is coconnected. The conjugate object for pV, Hq P U CoreppGq is the unitary corepresentation acting on H via VpxbΛ_hpyqq “ x¹bΛ_hppx²q^˚yq, where ˜apxq is described in Lemma 2.7, and the rigidity morphisms are the same as in U CoreppGq. For any morphism f, again f^˚ is the conjugate bounded linear map on the corresponding Hilbert spaces. So that, we have Proposition 2.16 U CoreppGq is a strict rigid finite C^˚-multitensor cate- gory. It is C^˚-tensor if and only if G is coconnected.

The simple objects of this category are exactly irreducible corepresenta- tions of G. Let us denote by Ω the set of equivalence classes of irreducibles and choose a representative U^x in any class x P Ω. The regular corepresen- tation of G is decomposed as follows:

W “ ‘xPΩdimpxqU^x, (11) where dimpxq is the dimension of the Hilbert space on which U^x acts.

Definition 2.17 Let pU, H_Uq P U CoreppGq and tm_i,juⁿ_i,j“1 be the matrix units of BpH_Uq with respect to some orthonormal basis te_iuⁿ_i“1 in H_U. Then

U “Σⁿ_i,j“1m_i,j bU_i,j,

where U_i,j pi, j “1, ..., nq are called the matrix coefficients of U with respect to te_iu. Put B_U :“SpanpU_i,jqⁿ_i,j“1; in particular, we denote B_U^x byB_x.

Remark 2.18 Let us summarize some properties of matrix coefficients of U^x pxPΩq which can be proved in a standard way.

(i) B_‘^p

k“1Uk “ spantBU1, ..., BUpu for any finite direct sum of unitary corepresentations. In particular, (11) implies that B “ ‘xPΩB_x.

(ii) Decomposition U jV “ ‘zd_zU^z with multiplicities d_z implies that BUBV Ă ‘zBz, where z parameterizes the irreducibles of the above decom- position.

(iii) The definition of a unitary corepresentation written in terms ofU_i,j^x :

∆pU_i,j^xq “ Σ^dimpxq_k“1 U_i,k^x bU_k,j^x , εpU_i,j^x q “δ_i,j, U_i,j^x “SpU_j,i^xq^˚,

for all i, j “ 1, ..., dimpxq, gives: B_x bB_x “ ∆p1_BqpB_x bB_xq, ∆pB_xq Ă

∆p1_BqpB_xbB_xq and B_U “SpB_Uq^˚. We also have B_U “ pB_Uq^˚.

Example 2.19 In the case of the trivial corepresentation of G associated with p∆_|Bs, B_sq, we will use the notation B_ε instead of B_U. Let tb_iu^dimB_i“1 ^s be an orthonormal basis in B_s with respect to the scalar product ă z, t ą“

εpt^˚zq @z, t P B_s. Then one can write ∆p1_Bq “ Σ^dimB_i“1 ^sb^˚_i bSpb_iq (see [22], 2.3.3), which implies: ∆pb^˚_jq “ Σ^dimB_i“1 ^spb^˚_i bSpb_iqb^˚_jq, so U_i,j^ε “ Spb_iqb^˚_j, for all i, j “1, ..., dimB_s. This means that B_ε is the unital C^˚-algebra B_tB_s.

5. Fiber functor and reconstruction theorem

LetQandRbe two unitalC^˚-algebras. By definition, apQ, Rq-correspon- dence is a right HilbertR-moduleE (see [11]) with a unital˚-homomorphism ϕ : Q Ñ LpEq, where LpEq is the C^˚-algebra of all bounded R-linear ad- jointable operators on E. If Q “ R, we call it an R-correspondence. R- correspondences form a C^˚-multitensor category CorrpRq with interior ten- sor product bR and adjointable R-bilinear maps as morphisms.

There exists another definition of a pQ, Rq-correspondence, due to Alain Connes, this is a triple pH, α, βq where H is a Hilbert space equipped with unital ˚-homomorphism α :Q ÑBpHq and ˚-anti-homomorphism β :R Ñ BpHq whose images commute in BpHq. Then H is a pQ, Rq-bimodule via q¨v¨r:“αpqqβprqv, for all qPQ, r PR, vP H.

In this paper, we are especially interested in the particular case, when Q “ R is a finite dimensional C^˚-algebra equipped with a faithful tracial state φ. Below we treat this particular case in detail.

Lemma 2.20 Both definitions of an R-correspondence are equivalent.

Proof. (i) If pH, α, βq PCorrpRq, define, for any ηP H, an operator Πpηq:H_φ ÑH by ΠpηqΛ_φprq:“βprqη, for all r P R, where H_φ is the GNS Hilbert space generated by pR, φq. Then define an R-valued scalar product:

ăξ, ηąR:“Πpξq^˚Πpηq, for all ξ, ηP H.

It is clear that ăξ, η ąR is in fact in π_φpRq. Finally, Πpβpbqηq “Πpηqπ_εpbq, so ă ξ, βpbqη ąR“ă ξ, η ąR π_φpbq, for all ξ, η P H, b P R. Moreover, together with the unital ˚-representationα we have onH the structure of an R-correspondence in the sense of the first definition.

(ii) Vice versa, if H is an R-correspondence in this last sense, then one can define a usual scalar product ă ξ, η ą“ φpă η, ξ ąRq, for all η, ξ P H, and there are clearly a unital˚-homomorphismα :RÑBpHqand a unital˚- anti-homomorphism β :R ÑBpHq whose images commute in BpHq. Thus, pH, α, βq is an R-correspondence in the sense of A. Connes. ˝ A morphism between pH, α, βq and pK, α¹, β¹q is a map T P BpH, Kq intertwiningαand α¹ and alsoβ andβ¹, thenCorrpRqis a semisimple linear category. If pH, α, βq,pK, α¹, β¹q PCorrpRq, we define their tensor product:

pH, α, βq bRpK, α¹, β¹q “ ppβbα¹qpeqpHbKq, αb1K,1H bβ¹q, where e is the symmetric separability idempotent for R, so eβ,α¹ “ pβ b α¹qpeqis an orthogonal projection. For the sake of simplicity we shall denote H₁bRH₂ :“e_β,α¹pHbKq, and vbRw“e_β,α¹pvbwq, for all v PH, w PK.

The unit object is R with the GNS scalar product defined by φ. The unit isomorphisms are as follows:

l_HpzbRvq:“z¨v and r_HpvbRzq:“v¨z, @z PR, v PH.

They are isometric, for example:

||l_HpzbRvq||² :“ ||z¨v||²_H “φp1_Rq||z¨vq||² “ ||1_Rb pz¨vq||² “ ||zbRv||². The conjugate of a morphism T :H₁ ÑH₂ is just the adjoint operatorT^˚ : H2 ÑH1, soCorrpRqis aC^˚-multitensor category. We denote byCorrfpRq its full subcategory with finite dimensional underlying Hilbert spaces. The unit object is simple if and only if R is a full matrix algebra.

For all objects of the three above categories: U ReppGq,U ComodpGq, and U CoreppGq, the underlying Hilbert spaces areB_s-correspondences, so each of these categories has a forgetful C^˚-tensor functor with values in Corr_fpB_sq.

In order to reformulate in suitable terms the reconstruction theorem of Tannaka-Krein type for finite quantum groupoids proved initially in [10], [26], recall the construction of the canonical Hayashi functor H.

LetC be a rigid finiteC^˚-tensor category and Ω“IrrpCqbe an exhaustive set of representatives of equivalence classes of its simple objects. Let R be the C^˚-algebra R “ C^Ω “ À

xPΩCpx, where px “ p^˚_x are mutually orthogonal idempotents: p_xp_y “ δ_x,yp_x, for all x, y P Ω. Then H is a functor from C to CorrfpRq defined by:

Hpxq “H_x “ à

y,zPΩ

Cpz, ybxq, for every xPΩ,

where Cpx, yq is the vector space of morphisms x Ñ y. The R-bimodule structure on H_x is given by:

p_y ¨H_x¨p_z “Cpz, ybxq, for all x, y, z P Ω.

If yPΩ and f P Cpx, yq, then Hpfq:H_xÑH_y is defined by:

Hpfqpgq “ pid_z bfq ˝g, for any z, tPΩ and gP p_z ¨H_x¨p_t. The inverse natural isomorphisms J_x,y^´1 :HxbHy ÑHxb

R Hy are:

J_x,y^´1pvbwq “ az,x,y˝ pvbidyq ˝wPpz¨Hpxbyq ¨pt,

for allv Pp_z¨H_x¨p_t, w Pp_t¨H_y¨p_s, z, s, tPΩ. Herea_z,x,y are the associativity isomorphisms of C.

We define the scalar product on Hx as follows. If x, y, z P Ω and f, g P Cpz, ybxq, then g^˚ P Cpybx, zq and g^˚˝f P Endpzq “ C, so one can put ăf, g ąx“g^˚˝f. The subspaces Cpz, ybxq are declared to be orthogonal, so Hx P CorrfpRq. Dually, Hx PCorrfpRq via z1¨v ¨z2 “ z₂^˚¨v¨z₁^˚, for all z₁, z₂ P R, v P H_x. Now one can check that H : C ÑCorr_fpRq is a unitary tensor functor in the sense of [16] 2.1.3.

Theorem 2.21 Let C be a rigid finite C^˚-tensor category and Ω“ IrrpCq.

Let R be the C^˚-algebra C^Ω and H: C ÑCorr_fpRq be the Hayashi functor.

Then the vector space

B “à

xPΩ

HxbHx, (12)

has a regular biconnected finite quantum groupoid structure G such that C – U CoreppGq as C^˚-tensor categories.

Proof. A rigid finite C^˚-tensor category C is semisimple and spherical, so [25], Theorems 1.1 and 1.2 claims that B has a structure of a selfdual regular biconnected semisimple weak Hopf algebra. The algebra of the dual quantum groupoid ˆG is (see [26], [13]):

Bˆ “à

xPΩ

BpH_xq, (13)

the duality is given, for all x, y PΩ, APBpH_yq, v, w PH_x by:

ăA, wbv ą“δ_x,y ăAv, wąx .

Bˆ is clearly a C^˚-algebra with the obvious matrix product and involution, its coproduct is given (see [13] Theorem 1.3.4) by:

∆pˆ ˆbq “ÿ

iPI

pspriq btppiqqJˆbJ^´1, for any ˆbPB,ˆ where ř

iPI

pr_ibp_iqis the symmetric separability element ofR henceř

iPI

pspr_iq b tppiqq “∆pˆ ˆ1q is an orthogonal projection in ˆB bB; moreoverˆ J “ À

x,yPΩ

Hx,y

is a unitary as a direct sum of unitaries. Then one can easily deduce that

∆pˆ ˆb^˚q “ ∆pˆ ˆbq^˚, so both ˆG and G are finite quantum groupoids.

The explicit structure of G is given in [25], Theorems 1.1 and 1.2. If v, wP H_x, g, hPH_y andte^x_juis an orthogonal basis in H_x p@x, y P Ω), then:

∆pwbvq “à

j

pwbe^x_jqxb pe^x_j bvq_x, (14) εpwbvq “ăv, wąx, (15)

pwbvq_x¨ pgbhq_y “ pJ_x,y^´1pwbgq bJ_x,y^´1pvbhqq_xby PH_xbybH_xby, (16) 1_B “à

xPΩ

pρ_xbρ^´1_x q1, (17) where ρ_x is the unit constraint attached to x, so ρ^´1_x P p_x ¨ H₁ ¨ p_x and ρ_x “ρ^´1_x . In order to define the antipode, consider the natural isomorphisms Φ_x :H_x ÑH_x^˚ and Ψ_x :H_x ÑH_x^˚ given by:

Φ_x “ρ_ypid_ybev_xq˝a_y,x,x^˚˝pvbid_x^˚q,Ψ_x “ pvbid_x^˚q˝a^´1_y,x,x˚˝pid_ybcoev_xq˝ρ^´1_y .

Here ev_x and coev_x pxP Ωq are the rigidity morphisms. Then we define:

Spwbvq “ rΦxpvq bΨxpwqsx^˚. (18) AnyH_x is a right B-comodule via

axpvq “Σ

je^x_j be^x_j bv, where v PHx,

one checks that it is unitary which gives the equivalence C –U CoreppGq. ˝

3 Coactions of finite quantum groupoids on unital C*-algebras

1. Canonical implementation of a coaction

Definition 3.1 A right coaction of a finite quantum groupoid G on a unital

˚-algebra A, is a ˚-homomorphism a:AÑAbB such that:

1) pabiqa“ pid_Ab∆qa.

2) pid_Abεqa“id_A. 3) ap1_Aq PAbB_t.

One also says that pA,aq is a G-˚-algebra.

Remark 3.2 If A is aC^˚-algebra, then a is automatically continuous, even an isometry by 2.4 and [24] 1.5.7.

Proposition 3.3 Any right coaction of G on a unital ˚-algebra A is sim- plifiable: the set apAqp1AbBq “ tapaqp1Abbq | a P A, b P Bu generates ap1_AqpAbBq as a vector space.

Proof. Using Sweedler notations (which makes sense here as B is finite dimensional), one has:

ap1Aqpab1Bq “ pidAbεbidBqpabidBqrap1Aqpab1Bqs

“ pid_Abεbid_Bqrpid_Ab∆qap1_Aqqapaq b1d_Bqs

“ pid_Abεbid_Bqrp1_A¹b∆p1_A²qpa¹ba²b1_Bqs

“ pid_Abεbid_Bqrp1_A¹b∆p1_Bqp1_A²b1_Bqpa¹ba²b1_Bqs

“ pid_Abεbid_Bqrp1_Ab∆p1_Bqqp1_A¹a¹b1_A²a²b1_Bqs

“ pid_Abεbid_Bqrp1_Ab∆p1_Bqqpa¹ba²b1_Bqs

“ pid_Abε_tqapaq.

Definition 2.1.1 (3) of [22] gives that:

ap1_Aqpab1_Bq “ pid_Abmqpid_Abid_BbSqpid_Ab∆qapaq

“ pid_Abmqpid_Abid_BbSqpabid_Bqapaq

“ pid_Abmqpapa¹q bSpa²qq.

Finally, the trivial equality: pidAbmqpxbybzq “ pxbyqp1Abzqimplies:

ap1_Aqpab1_Bq “ apa¹qp1bSpa²qq.

So ap1Aqpab1Bqbelongs to the vector space generated by apAqp1AbBq. ˝ Let us introduce the unital˚-homomorphism α:B_s ÑA:αpxq:“x¨1_A. Equalities (4) and (5) show that, for all xP B_s and aPA :

apαpxqaq “ p1_Abxqapaq, (19)

pαpxq b1Bqapaq “ p1AbSpxqqapaq. (20) It is helpful to note that

ap1_Aq “ pαbid_Bq∆p1_Bq. (21) Indeed :

αp1₁q b1₂ :“1₁¨1_Ab1₂ “ pid_Abεqrp1_Ab1₁qap1_Aqs b1₂ “

“1¹_Ab pεbid_Bq∆p1²_Aq “ap1_Aq.

Lemma 3.4 (cf. [31] 3.1.5, 3.1.6). If pA,aq is a G-˚-algebra A, then:

(i) The set A^a “ ta P A|apaq “ ap1Aqpab1Bqu is a unital ˚-subalgebra of A (it is a unital C^˚-subalgebra of A when A is a C^˚-algebra) commuting pointwise with αpB_sq.

(ii) The mapT^a :“ pidAbhqa (where h is the normalized Haar measure of G) is a conditional expectation from A to A^a; it is faithful when A is a C^˚-algebra.

Proof. (i) For all aPA^a and xPB_s, one has:

apaαpxqq “ap1_Aqpab1_Bqp1_Abxqap1_Aq “ apαpxqaq,