Tensor C*-categories arising as bimodule categories of II_1 factors

T ensor C ˚ -categories arising as bimo dule categories of I I 1 factors by Sébastien F alguières p 1 q and Sven Ra um p 2 q Abstract W e prov e that if C is a tenso r C ˚ -categor y in a certain cla ss, then there exists a n uncount- able family o f pairwise non stably isomorphic II 1 factors p M i q such that the bimodule cat- egory of M i is equiv a lent to C for all i . In particular, w e prov e tha t every ﬁnite tensor C ˚ -categor y is the bimo dule categor y o f a I I 1 factor. As an application we prove the ex- istence of a I I 1 factor for which the set o f indices of ﬁnite index irreducible s ubfactors is ! 1 , 5 ` ? 13 2 , 12 ` 3 ? 13 , 4 ` ? 13 , 11 ` 3 ? 13 2 , 13 ` 3 ? 13 2 , 19 ` 5 ? 13 2 , 7 ` ? 13 2 ) . W e also give the ﬁrs t ex- ample of a I I 1 factor M such that Bimod p M q is explicitly calculated and has an uncoun table n umber of isomorphism classes of irre ducible ob jects. 1 In tr o duction The description of symmetries of a I I 1 factor M , suc h as the fundamental gr oup F p M q of Murra y and v on Neumann and the outer automorphism gr oup Out p M q , is a cen tral and usually very hard problem in the theory of I I 1 factors. Ov er the last ten y ears, Sorin Po pa dev eloped his deformation- rigidity theory [P op02, P op03, P op04] and settled man y long standing op en problems in this di- rection. See [V ae06a, P op06, V ae10] for a surve y . I n particular, he obtained the ﬁrst complete calculations of fundamen tal groups [Pop 02 ] and outer auto morphism groups [IPP05]. H is methods w ere used in further calculations. Without b eing exhaustiv e, see for example [P op03, PV08, Dep10 ] concerning fundamen tal groups and [PV06, V ae07, FV07] for outer automorph ism groups. Bimo dules M H M o v er a I I 1 factor M ha ving ﬁnite left and righ t M -dime nsion are said to b e of ﬁnite Jones index (see [Con94, Po p86]) and they give rise to a category , whic h we denote b y Bimo d p M q . Endo w ed with the Connes tensor pro duct of M - M -bimodules, Bimo d p M q is a compact tensor C ˚ - category , in the sense of Lon go and Rob erts [LR97]. The bimo dule category of a I I 1 factor M ma y b e seen as a gener alize d symmetry gr oup of M . It con tains a lot of structural information on M and enco des sev eral other in v arian ts of M . Indeed, if grp p M q denotes the group-lik e elemen ts in Bimo d p M q , i.e. bimo dules of index 1 , one has the follo wing short exact sequence 1 Ñ O u t p M q Ñ grp p M q Ñ F p M q Ñ 1 . Finite index subfactors N Ă M are also enco ded in a certain sense b y the bimo dule category Bimo d p M q , since, denoting N Ă M Ă M 1 the Jones b asic c onstruction , w e obtain a ﬁnite index bimo dule M L 2 p M 1 q M . As explained ab o ve, in [IPP05] the ﬁrst actual computation of the outer automorphism group of I I 1 factors w as achiev ed, using a com bination of relative property (T) and amalgam ated f ree pro ducts. Extending their metho ds, in [V ae06b], V aes pro ved the existence of a I I 1 factor M with trivial 1 P artially supp orted by the F ond ation des Sciences Mathématiques de P aris 2 P artially supp orted by Marie Curie Researc h T raining N etw ork Non-Comm utative Geometry MR TN-CT-2006- 031962 and by K.U.Leuv en BOF researc h grant OT/08 /032 1 bimo dule category . As a consequence, all the symmetry groups and s ubfactors of M w ere trivial. Also relying on P opa’s metho ds, in [FV08], V aes and the ﬁrst a uthor pro v ed that the represen tation category of an y compact second counta ble group can b e realized as the bimo dule category of a I I 1 factor. More precisely , giv en a compact second count able group G , there exists a I I 1 factor M and a minimal action G ñ M such that, denoting M G the ﬁxed p oin t I I 1 factor, the natural fully faithful tensor functor Rep p G q Ñ Bimo d p M G q is an equiv alence of tensor C ˚ -categories. Both pap ers follow ed closely [IPP05] and th us, they give only exi s tence results. Explicit results on the calculation of bimodule categories are obtained in [V ae07] and [D V10]. Both articles are based on generalization s of P opa’s s eminal papers [Po p02, P op03 ] on Bernoulli crossed pro ducts. In [V ae07], V aes ga ve explicit examples of group-measure space I I 1 factors M for whic h the fusion algeb ra, i.e. isomorphism classes of ﬁnite index bimo dules and fusion rules, w ere calculated. The complete calculation of the category of bimo dules ov er I I 1 factors coming from [V ae07] w as obtained by Deprez and V aes in [D V10]. Even more is pro ve n in [DV10] , since the C ˚ -bicategory of I I 1 factors commensurable with M , i.e. those I I 1 factors N admitting a ﬁnite index N - M -bimo dule, is also comput ed and explicitly arises as the bicatego ry asso ciated with a Hec k e pair of groups. Note that b y [DR 89], represen tation categories of compact second coun table groups can b e c har- acterized abstractly as symmetric compact tensor C ˚ -categories with coun tably man y isomorphism classes of irreducible ob jects. Among compact tensor C ˚ -categories, ﬁnite tensor C ˚ -c ate gories , i.e. those whic h admit only ﬁnitely many isomorph ism classes of irreducible ob jects, form another natural class. In this article, we pro ve that ev ery ﬁnite tensor C ˚ -category arises as the bimo dule category of a I I 1 factor. Theorem A. L et C b e a ﬁnite tensor C ˚ -c ate gory. Then ther e is a II 1 factor M such that Bimo d p M q » C . As an appl ication of the abov e theorem, w e pro v e the existence of a I I 1 factor for whic h the set of indices of irreducible ﬁnite index subfactors can b e explicitly calculated and con tains irrationals. Recall the ama zing theorem of Jon es, pro ving that the index of an inclusion of I I 1 factors N Ă M ranges in the set I “ " 4 cos ´ π n ¯ 2 | n “ 3 , 4 , 5 , . . . * Y r 4 , `8s . Giv en a I I 1 factor M , Jones deﬁnes the in v arian t C p M q “ t λ | there is a ﬁnite index irreducible inclusion N Ă M of index λ u . Jones pro v ed that ev ery elemen t of I arises as the index of a not necessarily irreducible subfactor of the hyperﬁnite I I 1 factor. How ev er, the problem of computing C p R q is s till widely op en. In [V ae07, V ae06b], V aes pro v ed the existence of I I 1 factors M for whic h C p M q “ t 1 u and C p M q “ t n 2 | n P N u . The inv arian t C p M q is also compute d in [FV08] and arises as the set of dimensions of some ﬁnite dimensional v on Neumann algebras. In [DV10], Deprez and V aes constructed concrete group-measure space I I 1 factors M with C p M q ranging ov er all sets of natural nu m b ers that are closed under taking divisors and taking low est common m ultiples. All ab ov e results pro v ide I I 1 factors M for whic h C p M q is a subset of the natural n um b ers. Ho wev er, com bining recen t work on tensor categories [GS11] and our Theorem A , we prov e the follow ing theorem. Theorem B. Ther e exists a II 1 factor M such that C p M q “ " 1 , 5 ` ? 13 2 , 12 ` 3 ? 13 , 4 ` ? 13 , 11 ` 3 ? 13 2 , 13 ` 3 ? 13 2 , 19 ` 5 ? 13 2 , 7 ` ? 13 2 * . 2 In [V ae07], [FV 08] and [D V10 ] only categories with at most coun tably man y isomorphism classes of irreducible ob jects w ere obtained as bimo dule categories of I I 1 factors. In this article we give examples of I I 1 factors M suc h that Bimo d p M q can b e calculated and has uncoun tably man y pairwise non isomorphic irreducible ob jects. F or example, if G is a coun table, discrete group, w e pro ve the existence of a I I 1 factor M suc h that Bimo d p M q » Rep ﬁn p G q . Here, Rep ﬁn p G q denotes the category of ﬁnite dime nsional unitary represen tations of G . Theorem C. L et C denote one of the fol lowing c om p act tensor C ˚ -c ate gories. Either C “ Rep ﬁn p G q for a c ountable discr ete gr oup, or C “ CoRep ﬁn p A q for an amenable or a maximal ly almost p erio di c discr ete Kac algebr a A . Then, ther e is a II 1 factor M such that Bimo d p M q » C . Our construction consists of t w o main steps. (i) Give n any quasi-regular, depth 2 inclusion N Ă Q of I I 1 factors, suc h that N and N 1 X Q are h yp erﬁnite, denote b y N Ă Q Ă Q 1 the Jones basic construction. W e construct a I I 1 factor M and a fully faithful tensor C ˚ -functor F : Bimod p Q Ă Q 1 q Ñ Bimo d p M q (see Section 2.4.3 for the bimodule category asso ciated with an inclusion of I I 1 factors). (ii) Using Ioana, Pet erson and Po pa’s rigidit y results for a malgamated free pro duct von Neumann algebras [IPP0 5], w e pro ve that under suitable assumptions (see Theore m 3.1) the functor F is essentiall y surjectiv e. The ab o ve steps yield a I I 1 factor M such that Bimod p M q » Bimod p Q Ă Q 1 q . Using the setting of [IPP05], as in [FV07, F V 08, V ae06b], this result is not con structiv e. W e only pro v e an existence theorem, whic h in v olves a Baire category argumen t (see Theorem 2.19). M ore precisely , we pro v e the f ollowin g Theor em D and Theore ms A and C are obtained as corollaries. Theorem D. L et N Ă Q b e a quasi-r e gular and depth 2 inclusion of II 1 factors. Assume that N and N 1 X Q ar e hyp erﬁni te and denote by N Ă Q Ă Q 1 the b asic c onstruction. Then, ther e exist unc ountably many p airwi se non-stably isomorphic II 1 factors p M i q such that f or al l i we have Bimo d p M i q » Bimo d p Q Ă Q 1 q as tensor C ˚ -c ate gories. A ckno wledgemen ts The s econd author w ants to thank his advisor Stefaan V aes f or his constan t care and excellen t sup ervision. This w ork w as strongly inﬂuenced b y his ideas. The second author is greatful to the Lab oratoire de Mathématiques Nicolas Oresme at the univ ersit y of Caen for its kind hospitalit y during his s ta y in autumn 2010. Both authors wa n t to thank the I nstitut H enri P oincaré where part of this article has b een written, during the program “V on N eumann algebras and ergo dic theory of group action” in spring 2011. 2 Prelimina ries a nd notations In this pap er, v on Neumann algebras are assumed to act on a separable Hilb ert space. A vo n Neumann algebra p M , τ q endow ed with a f aithful normal tracial state τ is called a tracial v on Neumann algebra. W e deﬁne L 2 p M q as the GNS Hilb ert space with resp ect to τ . Whenev er M is a von Neumann algebra, w e write M n “ M n p C q b M and M 8 “ B p ℓ 2 p N qq b M . Whenev er H is a Hilb ert space, w e also denote H 8 “ ℓ 2 p N q b H . 3 If B Ă M is a tracial inclusion of von N eumann algebras, then w e denote by E B the trace preserving conditional exp ectation of M ont o B . Also if pB n p Ă pM n p is an ampliﬁcation of B Ă M , w e still denote by E B the trace preserving conditional exp ectation on to pB n p . 2.1 Finite index bimo dules Let M , N b e tracial von Neumann algebras. An M - N -bimo dule M H N is a H ilb ert space H equipp ed with a normal represen tation of M and a normal anti -represen tation of N that comm ute. Bimo dules o ver vo n Neumann algebras were studied in [Con94, V .App endix B] and [P op86]. Let H b e an M - N -bimo dule. There exists a pro jection p P N 8 suc h that H N – ` p L 2 p N q 8 ˘ N , and this pro jection p is uniquel y deﬁned up to equiv alence of pro jections in N 8 . There also exists a ˚ -homomorphism ψ : M Ñ pN 8 p s uc h that M H N is is omorphic with the M - N -bimo dule H p ψ q deﬁned as Hilbert space p L 2 p N q 8 and endo w ed with actions given b y a ¨ ξ “ ψ p a q ξ and ξ ¨ b “ ξ b and a P M , b P N , ξ P p L 2 p N q 8 . F urthermore, if ψ : M Ñ p N 8 p and η : M Ñ q N 8 q , then M H p ψ q N – M H p η q N if and only if there exists u P N 8 satisfying uu ˚ “ p , u ˚ u “ q and ψ p a q “ uη p a q u ˚ for all a P M . Note that M - N -bimodules M H N can also be describ ed b y means of righ t actions via ˚ - homomorphi sms ψ : N Ñ pM 8 p as M H N – M ` p ℓ 2 p N q ˚ b L 2 p M qq p ˘ ψ p N q . Let H be a righ t N -mo dule and write H N – ` p L 2 p N q 8 ˘ N , f or a pro jection p P N 8 . Denote dim - N p H q “ p T r b τ qp p q . Observ e that the n um b er d im - N p H q dep ends on the c hoice of the trace τ , if N is not a factor. An M - N -bimo dule M H N is said to b e of ﬁ nite Jones index if dim M - p H q ă `8 and dim - N p H q ă ` 8 . In particular, the Jones index of a subfactor N Ă M is deﬁned as r M : N s “ dim - N p L 2 p M qq , see [Jon83]. Using the ab o ve notations, consider a bimo dule of the form M H p ψ q N with ﬁnite Jones index. Then, one ma y assume that ψ is a ﬁnite inde x inclusion ψ : M Ñ pN n p . 2.2 P opa’s in tertwining-b y-bimodules techn ique In [P op03 , Section 2], P opa in tro duced a v ery p o werful tec hnique to deduce unitary conjugacy of t wo v on Neumann subalgebras A and B of a tracial v on Neumann algebra M f rom their emb e ddi n g A ă M B , using interwining bimo dules . When A, B Ă M are Cartan s ubalgebras of a I I 1 factor M , P opa pro v es [P op02, Theorem A.1] that A ă M B if and only if A and B are actually conjugated b y a unitary in M . W e also recall the notion of f ul l emb e dding A ă f M B of A into B inside M . Deﬁnition 2.1. Let M b e a tracial v on Neumann and A, B Ă M n b e p ossibly non-unital subalge- bras. W e write • A ă M B if 1 A L 2 p M n q 1 B con tains a non-zer o A - B -subbimodule K that satisﬁes dim - B p K q ă 8 . 4 • A ă f M B if Ap ă M B for every non-zero pro jection p P 1 A M n 1 A X A 1 . W e will use the follo wing c haracterization of em b edding of subalgebras. I t can b e found in [P op03, Theorem 2.1 and Corollary 2.3] (see also App endix F in [BO08]). Theorem 2.2 (See [P op03]) . L et M b e a tr acial von Neumann algebr a and A, B Ă M n p ossibly non-unital sub algebr as. The fol lowing ar e e quivalent. • A ă M B , • ther e exist m P N , a ˚ -homomorphism ψ : A Ñ pB m p and a non-zer o p artial i sometry v P 1 A ` M 1 ,m p C q b M n ˘ p satisfying av “ v ψ p a q f or al l a P A , • ther e is no se quenc e of unitaries u k P U p A q such that } E B p xu k y q} 2 Ñ 0 for al l x, y P M n . Note that the en tries of v as in in the previous theorem span an A - B -bimo dule K Ă L 2 p M n q suc h that d im - B p K q ă 8 . W e will mak e use of Theorem 2.4 due to V aes, [V ae07, Theore m 3.11]. W e ﬁrst recall the notion of essen tially ﬁnite index inclusions of I I 1 factors (see [V ae07, Prop osition A.2]) and embedding of von Neumann subalgebras inside a bimo dule. Let N Ă M b e an inclusion of tracial vo n Neumann algeb ras. W e sa y that N Ă M has essential ly ﬁnite index if there exists a sequence of pro jections p n P N 1 X M suc h that p n tends to 1 strongly and N p n Ă p n M p n has ﬁnit e Jones index for all n . Deﬁnition 2.3. Let M , N b e tracial v on Neumann algebras and A Ă M , B Ă N von Neumann subalgebras. Let M H N b e an M - N -bimo dule. W e write • A ă H B if H con tains a non-zero A - B -subbimodule K Ă H with dim - B p K q ă 8 . • A ă f H B if eve ry non-zero A - N -s ubbimodule K Ă H satisﬁes A ă K B . Denote b y τ the trace on M . Let H b e an M - N -bimo dule. Using notations from Section 2.1, write H – H p ψ q where ψ is a ˚ -homomorphism ψ : M Ñ pN 8 p and p a pro jection in N 8 . Supp ose that dim - N p H q ă `8 , i.e p T r b τ qp p q ă ` 8 . Then, as remark ed in [V ae07], one has • A ă H B if and only if ψ p A q ă N B , • A ă f H B if and only if ψ p A q ă f N B . Theorem 2.4 ([V ae07, Theorem 3.11]) . L et N , M b e tr acial von Neumann algebr as, w i th tr ac e τ . L et A Ă M , B Ă N b e von Neumann sub algebr as. Assume the f ol lowing. • Every A - A -subbimo dule K Ă L 2 p M q satisfy ing dim - A p K q ă `8 is include d in L 2 p A q . • Every B - B -subbimo dule K Ă L 2 p N q satisf y ing dim - B p K q ă `8 is include d in L 2 p B q . Supp ose that M H N is a ﬁnite index M - N -bimo dule such that A ă f H B and A ą f H B . Then ther e exists a pr oje ction p P B 8 satisfying p T r b τ qp p q ă `8 and a ˚ -homomorphism ψ : M Ñ pN 8 p such that M H N – M H p ψ q N , ψ p A q Ă p B 8 p and this last inclusion has essential ly ﬁnite index. 5 2.3 Amalgamated free pro ducts of tracial von Neuman n algebras Throughout this s ection w e consider von Neuman n algebras M 0 , M 1 endo w ed with faithful normal tracial states τ 0 , τ 1 . L et N b e a common von Neumann subalgebra of M 0 and M 1 suc h that the traces τ 0 and τ 1 coincide on N . W e denote M “ M 0 ˚ N M 1 the amalgamated f ree pro duct of M 0 and M 1 o ver N with resp ect to the trace preserving conditiona l exp ectations (see [P op93 ] and [VDN92]). Recall that M is endo wed with a conditional exp ectation E : M Ñ N and the pair p M , E q is unique up to E -preserving is omorphism. The von Neumann algebra M 0 ˚ N M 1 is equipp ed with a trace deﬁned b y τ “ τ 0 ˝ E “ τ 1 ˝ E . 2.3.1 Rigid subalgebras Kazhdan’s property (T) w as generalized to tracial von Neumann algebras b y Connes and Jones in [CJ85] and is deﬁned as follo ws. A I I 1 factor M has prop ert y (T) if and only if there ex is ts ǫ ą 0 and a ﬁnite subset F Ă M suc h that ev ery M - M -bimo dule that has a unit v ector ξ satisfying } xξ ´ ξ x } ď ǫ , for all x P F , actually has a non-zero v ector ξ 0 satisfying xξ 0 “ ξ 0 x , for all x P M . Note that a group Γ in whic h ev ery non-trivial conjugacy class is inﬁnite (ICC group) has prop ert y (T) in the sense of Kazhdan if and only if the I I 1 factor L p Γ q has prop ert y (T) in the sense of Connes and Jones. P opa deﬁned a notion of relativ e prop erty (T) for inclusions of tracial von Neumann algebras, see [P op02 , Deﬁnition 4.2]. Suc h an inclusion is also called rigid . I n particular, if N is a I I 1 factor ha ving property (T), then an y inclusion N Ă M in a ﬁnite v on N eumann algebra M is rigid. W e will mak e use of the follo wing c haracterization of relative prop ert y (T). Theorem 2.5 (See [P op02 ] and [PP05]) . An inclusion N Ă M of tr acial von Neumann algebr as is rigid if and only if every se quenc e p ψ n q of tr ac e pr eserving, c ompletely p ositive, unital maps ψ n : M Ñ M c onver ging to the identity p ointwise in } ¨ } 2 , c onver ges uniformly i n } ¨ } 2 on the unital b al l p N q 1 of N . W e recall Ioana, Pete rson and P opa’s Theorem 5.1 f rom [IPP05] whic h con trols the p osition of rigid subalgebras of amalgamat ed free product v on Neumann algebras. W e c ho ose to w ork with matrices o ver amalgamated free pro ducts, whic h is not a more general situation, s ince p M 0 ˚ N M 1 q n can be iden tiﬁed with M n 0 ˚ N n M n 1 . Theorem 2.6 (See [IPP05, Theorem 4.3]) . L et M “ M 0 ˚ N M 1 . L et p P M n b e a pr oje ction and Q Ă pM n p a rigid inclusion. Then ther e exists i P t 0 , 1 u such that Q ă M M i . 2.3.2 Con trol of quasi-normalizers Let M b e a tracial von N eumann algebra and N Ă M a von N eumann subalgebra. The quasi- normalizer of N inside M , denoted QN M p N q , is deﬁned as the set of elemen ts a P M for whic h there exist a 1 , . . . , a n , b 1 , . . . , b m P M suc h that N a Ă n ÿ i “ 1 a i N , , aN Ă m ÿ i “ 1 N b i . The inclusion N Ă M is called quasi-r e gular if QN M p N q 2 “ M . One also deﬁnes the gr oup of normalizing unitaries Norm N p M q of N Ă M as the set of unitaries u P M satisfying uN u ˚ “ N . The normali zer of N in M is Norm M p N q 2 . Note that N 1 X M Ă Norm M p N q 2 Ă QN M p N q 2 . 6 Theorem 2.7 (See [IPP05, Theorem 1.1]) . L et M “ M 0 ˚ N M 1 . L et p P M n 0 b e a pr oje ction and Q Ă pM n 0 p a von Neumann sub algebr a satisfying Q ć M 0 N . Whene ver K Ă p p C n b L 2 p M qq is a Q - M 0 -subbimo dule with dim - M 0 p K q ă `8 , we have K Ă p p C n b L 2 p M 0 qq . In p articular, the quasi-normalizer of Q inside pM n p is c on taine d in pM n 0 p . 2.4 T ensor C ˚ -categories, fusion algebras and bimo dule categories of I I 1 factors W e brieﬂy recall s ome deﬁnitions for tensor C ˚ -categories and refer to [LR97, Sc h97] for more infor- mation and precise statemen ts. A tensor C ˚ -category is a C ˚ -category with a monoidal structur e , suc h that all structure maps are unitary . A tensor C ˚ -category is called r e gular if it has sub ob jects and direct sums and the unit ob ject is strongly irreduci ble. A regular tensor C ˚ -category is calle d c omp act if every ob ject has a conjugate. A compact tensor C ˚ -category is ﬁn ite if it has only ﬁnitely man y isomorphism classes of simple ob jects. Con v ention. Throughout this article w e assume without loss of generalit y that all tensor categories in volv ed are strict. 2.4.1 F usion algebras A fusion algebra A is a free N -mo dule N r G s equipped with • an asso ciativ e and distributiv e pro duct operation, and a multiplica tiv e unit elem en t e P G , • an additiv e, anti-m ultiplicativ e, in voluti v e map x ÞÑ x , called c onjugation , satisfying F rob enius reciprocit y as follo ws. F or x, y , z P G , deﬁne m p x, y ; z q P N suc h that xy “ ÿ z m p x, y ; z q z . Then, one has m p x, y ; z q “ m p x, z ; y q “ m p z , y ; x q for all x, y , z P G . The base G of the fusion algebra A , also called the irr e ducible elements of A , consists of the non-zero elemen ts of A that cannot b e expressed as the sum of t w o non-zero elemen ts. W e ha ve the follo wing example s of fusion alge bras. • Give n a coun table group Γ , one gets the asso ciated fusion algebra A “ N r Γ s . • Let G b e a lo cally compact group and deﬁne the fusion algebra A of Rep ﬁn p G q as the set of equiv alence classes of ﬁnite dimensiona l unitary represen tations of G . The direct sum and tensor pro duct of represen tations in Rep ﬁn p G q yield a fusion algebra structure on A . • More generally , the is omorphism classes of ob jects in a compact tensor C ˚ -category form a fusion algebra. Note that there ex is t non-equiv alen t tensor C ˚ -categories havin g isomorphic fusion algebras. In this article w e are mainly interested in tensor C ˚ -categories and fusion algebras coming from bimo dules o ver II 1 factors. W e recall some deﬁnition s and refer to [Bis97] for bac kground material and results on bimo dules and fusion algebras, in particular in relation with subfactors. 7 2.4.2 The bimo du l e category of a I I 1 factor Let M , N , P b e I I 1 factors. W e denote by H b N K the Connes tensor pro duct of the M - N - bimo dule H and the N - P -bimodule K and refer to [Con94, V.App endix B] f or details. Note that H p ρ q b N H p ψ q – H pp id b ψ q ρ q . W e recall no w the follow ing useful lemma from [FV08] concerning Connes tensor pro duct versus pro duct in a giv en mo dule. The inclusion of I I 1 factors N Ă M considered in [FV08] is assumed to b e irreducible ( N 1 X M “ C 1 ). Instead, w e assume that N Ă M is quasi-regular. W e give a pro of for the con ve nience of the read er. Lemma 2.8 ([FV08, Lemma 2.2]) . L et r N Ă N Ă M b e an in clusion of I I 1 factors and let P b e a II 1 factor. Ass ume that N Ă M is quasi-r e gular and r N Ă N has ﬁn ite i ndex. L et M H P b e an M - P -bimo dule. Supp ose that L Ă H is a close d r N - P -subbimo dule. Supp ose that K Ă L 2 p M q is an N - r N -subbimo dule of ﬁn ite index. Denote by K ¨ L the closur e of p K X M q L insi de H . Then • K ¨ L is an N - P -bimo dule isomorphic to a subbimo dule of K b r N L . • If K ¨ L is non-zer o and K b r N L is irr e ducible then, K ¨ L and K b r N L ar e i somorphic N - P - bimo dules. Whenever P H M is a P - M -bimo dule with close d P - r N -subbimo dule L and K Ă L 2 p M q an r N - N - subbimo dule, w e deﬁne L ¨ K as the closur e of L p K X M q inside H and, by symmetry, we ﬁnd that L ¨ K is isomorphic with a P - N -subbimo dule of L b N K . Pr o of. Let H , K and L b e as in the statemen t of the lemma. Note that K X M is dense in K , since N Ă M is quasi-regular and r N Ă N has ﬁnite index. Moreo ver, all vec tors in K X M are r N -b ounded. So, there exists a ﬁnite index inclusion ψ : N Ñ p r N n p and an N - r N -bimo dular isomorphism T : H p ψ q “ p p C n b L 2 p r N qq Ñ K s uc h that T p p p e i b 1 qq P K X M for all i . W e ha ve K b r N L – p p C n b L q , hence w e can deﬁne an N - P -bimo dular map S : p p C n b L q Ñ K ¨ L b y S p p p e i b ξ qq “ T p p p e i b 1 qq ¨ ξ . The range of T is dense in K ¨ L . After taking the p olar decomp osition of T w e get a coisomet ry K b N L Ñ K ¨ L . The c ontr agr e dient of an M - N -bimo dule M H N is the N - M -bimo dule deﬁned on the conjugate Hilb ert space H with bimo dule actions giv en b y a ¨ ξ “ p ξ a ˚ q and ξ ¨ b “ p b ˚ ξ q . The Connes tensor pro duct and con tragredience induc e a compact tensor C ˚ -category structure on the category of ﬁnite index M - M bimo dules, where morphisms are give n b y bimo dular maps. Deﬁnition 2 . 9. Let M b e a I I 1 factor. W e deﬁne Bimo d p M q to b e the tensor C ˚ -category of ﬁnite index M - M -bimo dules and F alg p M q the asso ciated fusion algebra. W e recall the notio n of pairs of conjugates in strict tensor C ˚ -categories. Deﬁnition 2.10 (See [LR97]) . Let x b e an ob ject in a strict tensor C ˚ -category C . A conjugate for x is an ob ject x in C and morphisms R : 1 C Ñ x b x , R : 1 C Ñ x b x s uch that p R ˚ b id x q ˝ p id x b R q “ id x and p R ˚ b id x q ˝ p id x b R q “ id x . In the follo wing theorem, pairs of conjugates are used to c ha racterize ﬁnite index bimo dules among all bimo dules o ve r a I I 1 factor (see [LR97 ] and also [F al09 , The orem 5.32]). Theorem 2.11. L et M b e a II 1 factor and let M H M an M - M -bimo dule. Then M H M has ﬁn ite index if and only if M H M has a c onjugate in the tensor C ˚ -c ate gory of al l M - M -bimo dules. 8 2.4.3 T ensor C ˚ -categories arising from subfactors Let M b e a I I 1 factor and N Ă M a subfactor. W rite e N for the pro jection L 2 p M q Ñ L 2 p N q . The v on Neumann algebra x M , e N y Ă B p L 2 p M qq generated b y M and e N , called the Jones b asic c ons truction , w as in tro duced in [Jon83] and is denoted M 1 . Note that L 2 p M 1 q is an M - M -bimo dule and it is of ﬁnite Jones index whenev er r M : N s ă `8 . W e will frequen tly use the f act that dim p N 1 X M q ă `8 if r M : N s ă `8 . Deﬁnition 2.12. Let N Ă M b e an inclusion of type I I 1 factors. W e deﬁn e Bimo d p N Ă M q to b e the tensor C ˚ -sub category of Bimo d p N q generated b y all ﬁnite index N - N -bimo dules that app ear in L 2 p M q . W e denot e b y F alg p N Ă M q the asso ciated fusion subalgebr a of F alg p N Ă M q . W e giv e the follow ing deﬁnition of depth 2, as in [EV00]. Deﬁnition 2.13. Let N Ă Q b e an inclusion of I I 1 factors. Let N Ă Q Ă Q 1 Ă Q 2 Ă ¨ ¨ ¨ b e the Jones tow er. Then N Ă Q has dept h 2 if N 1 X Q Ă N 1 X Q 1 Ă N 1 X Q 2 is a basic construction. Iden tify N 1 X Q 2 with the space of b ounded N - Q -bimo dular maps B N - Q p L 2 p Q 1 , T r qq . Deno te b y Hom N - Q p L 2 p Q q , L 2 p Q 1 qq the Hilb ert space completion of N - Q -bimo dular maps f rom L 2 p Q, τ q to L 2 p Q 1 , T r q with resp ect to the scalar pro duct x T , S y “ τ p S ˚ T q . W e recall the follo wing sp ecial case of [EV00, Theorem 3.10]. Theorem 2.14 (See [EV00, Theorem 3.10]) . The inclusion N Ă Q of II 1 factors has depth 2 i f and only if the natur al action of N 1 X Q 2 on Hom N - Q p L 2 p Q q , L 2 p Q 1 qq is faithful. As a consequence, w e obtain the follo wing c haracterization of depth 2 inclusions that w e use in this article. Corollary 2.15. L et N , Q b e II 1 factors. Then, the in clusion N Ă Q has depth 2 if and on ly if N L 2 p Q 1 q Q is isomorphic to an N - Q -subbimo dule of N L 2 p Q q 8 Q . Pr o of. Let N Ă Q b e a depth 2 inclusion of I I 1 factors. Let p P N 1 X Q 2 b e the pro jection on to the orthogonal compleme n t of the maximal N - Q -subbimo dule of L 2 p Q 1 q which is con tained in N L 2 p Q q 8 Q . Then, p acts trivially on Hom N - Q p L 2 p Q q , L 2 p Q 1 qq . Therefore, p “ 0 b y Theo rem 2.14. Assume that N L 2 p Q 1 q Q is is omorphic to a subbimo dule of N L 2 p Q q 8 Q . Let p P N 1 X Q 2 b e a non-zero pro jection. Then p L 2 p Q 1 q is a non-zero N - Q -bimo dule, s o there is a non-trivial N - Q -bimo dular map T : L 2 p Q q Ñ p L 2 p Q 1 q . W e ha ve p ¨ T “ T ‰ 0 , so p acts non-trivially on Hom N - Q p L 2 p Q q , L 2 p Q 1 qq . W e ha ve prov en that N 1 X Q 2 acts faithfully on Hom N - Q p L 2 p Q q , L 2 p Q 1 qq . W e conclud e using again Theorem 2.14. 2.4.4 The fusion algebra of almo st-n ormalizing b imo dul es Let N Ă M b e a regular inclusion, i.e. Norm M p N q 2 “ M . F or an y elemen t u P Norm M p N q the N - N -bimo dule u L 2 p N q has ﬁnite index and lies in L 2 p M q . Suc h bimo dules are generalized b y the notion of bimo dules almost-normal izing the inclusion N Ă M , which was in tro duced b y V aes in [V ae06b]. This notion w as adapted to more general irreducible, quasi-regular inclusions of I I 1 factors N Ă M in [FV 08]. W e recall the deﬁnition. 9 Deﬁnition 2.16. Let N Ă M b e an irreducible and quasi-regular inclusion of t yp e I I 1 factors. A ﬁnite index N - N -bimo dule is said to almost-normaliz e the inclusion N Ă M , inside F alg p N q , if it arises as a ﬁnite index N - N -subbimo dule of a ﬁnite index M - M -bimo dule. W e denote by AF alg p N Ă M q the f us ion algebra generated b y N - N -bimo dules almost-normalizing the inclusion N Ă M . Let N b e a I I 1 factor and Γ a counta ble grou p acting outerly on N . W rite M “ N ¸ Γ and assume that the inclusion N Ă N ¸ Γ is rigid. It is pro ven in [V ae06b, Lemma 4.1] that the fusion algebra AF alg p N Ă N ¸ Γ q is a coun table fusion s ubalgebra of F alg p N q . The next lemma is a straight forw ard adaptation of [V ae06b, Lemma 4.1]. Lemma 2.17. L et N Ă M b e a rigid, irr e ducible and quasi-r e gular inclusion of typ e II 1 factors. Then, the fusion algebr a AF alg p N Ă M q is a c ountable fusion sub algebr a of F alg p N q . 2.4.5 F reeness of fusion algebras The notion of freeness of fusion algebras was in tro duced in [BJ97, Section 1.2], in the study of free comp os ition of subfactors. W e recall the deﬁnit ion. Deﬁnition 2.18 ([BJ97, Section 1.2]) . Let A be a fusion algebra and A 0 , A 1 Ă A fusion subalgebras. W e say that A 0 and A 1 are fr e e inside A if ev ery alternating pro duct of irreducibles in A i zt e u , remains irreducible and diﬀeren t from e . Let M b e a I I 1 factor and M K M a ﬁnite index M - M -bimo dule. Whenev er α P Aut p M q , w e deﬁne the conjugation of K by α as the bimo dule K α “ H p α ´ 1 q b M K b M H p α q . Denote b y R the h yp erﬁnite I I 1 factor. V aes pro v ed in [V ae06b, Theorem 5.1] that coun table fusion subalge bras of F alg p R q can b e made free b y conjugating one of them with an automorph ism of R (see Theorem 2.19 b elo w). Note that the same result has ﬁrst b een pro ven for coun table subgroups of O ut p R q in [IPP05]. In b oth cases, the ke y ingredien ts come from [P op95 ]. Theorem 2.19 ([V ae06b, Theorem 5.1]) . L et R b e the hyp erﬁnite II 1 factor an d A 0 , A 1 two c ount- able fusion sub algebr as of F alg p R q . T hen, t α P Aut p R q | A α 0 and A 1 ar e fr e e u is a dense G δ -subset of Aut p R q . 3 Pro o f of Theorem D W e recall the follo wing construction, from [FV08]. Consider the group Γ “ Q 3 ‘ Q 3 ¸ SL p 3 , Q q , deﬁned b y the action A ¨ p x, y q “ p Ax, p A t q ´ 1 y q of SL p 3 , Q q on Q 3 ‘ Q 3 . T ak e α P R ´ Q , deﬁne Ω α P Z 2 p Q 3 ‘ Q 3 , S 1 q s uc h that Ω α ` p x, y q , p x 1 , y 1 q ˘ “ exp ` 2 π iα px x, y 1 y ´ x y , x 1 yq ˘ for all p x, y q , p x 1 , y 1 q P Q 3 ‘ Q 3 , and extend Ω α to an S 1 -v alued 2 -cocycle on Γ b y SL p 3 , Q q -in v ariance. W rite Λ “ Z 3 ‘ Z 3 . Then, b y [FV08, Lemma 3.3] and [FV 08, Example 3.4], the inclusion s N Ă N 0 Ă P given b y N “ L Ω α p Λ q , N 0 “ L Ω α p Z 3 ‘ Z 3 ¸ SL p 3 , Z qq , P “ L Ω α p Γ q satisfy the follo wing properties. 10 p P 1 q N Ă P is irreducible and quasi-regular, p P 2 q N 0 Ă P is q uasi-regular, p P 3 q N 0 has prop ert y p T q . Note that p P 1 q f ollo ws from the f act that the inclusion Λ Ă Γ is almost-norm al, meaning the commensurator Comm Γ p Λ q deﬁned as Comm Γ p Λ q : “ t g P Γ | g Λ g ´ 1 X Λ has ﬁnite index in g Λ g ´ 1 and in Λ u is the whole of Γ . W e k no w that the group SL p 3 , Q q do es not hav e an y non-trivial ﬁnite dimensional unitary represen tations (see [vN W40]). The smallest normal subgroup of Γ con taining SL p 3 , Q q is Γ itself. This giv es the follo wing propert y . p P 4 q The group Γ has no non-trivial ﬁnit e dimension al unitary represen tations. W e will also need the f ollowin g additional prop ert y , pro ven in [FV08, Example 3.4]. p P 5 q The inclusion L Ω α p Λ 0 q Ă L Ω α p Γ q is irreducible , for every ﬁnite index subgroup Λ 0 ă Λ . Theorem 3.1. L et Q b e a II 1 factor such that N Ă Q . L et B “ N 1 X Q and assume that • N Ă Q is a quasi-r e gular and depth 2 inclusion, • B is hyp erﬁnite, • ther e is no non-trivial ˚ -homomorphism fr om N 0 to any ampliﬁc ation of Q , • the fusion algebr as AF alg p N Ă P q and F alg p N Ă Q q deﬁne d in Se ction 2.4 ar e fr e e i nside F alg p N q . Then, for M “ ` P b B ˘ ˚ N b B Q , we have that Bimo d p M q » Bimo d p Q Ă Q 1 q , as tensor C ˚ - c ate gories, wher e N Ă Q Ă Q 1 is the b asic c on struction. Outline of the pr o of of Theorem D. W e ﬁrst pro ve Theorem 3.1 in t wo steps. In Section 3.1, w e construct a fully faithful tensor C ˚ -functor F : Bimo d p Q Ă Q 1 q Ñ Bimo d p M q . In Section 3.2, w e prov e that F is essential ly surjectiv e, whic h completes the pro of of Theorem 3.1. I n Section 3.3, w e giv e a pro of of Theorem D, relying on Theorem 3.1. In the rest of Sect ion 3 w e will alw ays use the notations of Theorem 3.1. 3.1 A fully faithful functor F : Bimo d p Q Ă Q 1 q Ñ Bimo d p M q Denote b y C the tensor C ˚ -category whose ob jects are ﬁnite index inclusions ψ : Q Ñ pQ 8 p with p P B 8 , p T r b τ qp p q ă 8 and ψ p x q “ xp for all x P N . The tensor pro duct on C is given b y ψ 1 b C ψ 2 “ p id b ψ 2 q ˝ ψ 1 . Morphisms of C are giv en b y Hom C p ψ 1 , ψ 2 q “ t T P q B 8 p | @ x P Q : T ψ 1 p x q “ ψ 2 p x q T u . Prop osition 3.2. The natur al in clusion I : ψ ÞÑ H p ψ q of C into Bimo d p Q q deﬁnes an e quivalenc e of tensor C ˚ -c ate gories C » Bimo d p Q Ă Q 1 q . 11 Pr o of. It is easy to c heck that I is a faithful tensor C ˚ -functor. W e pro ve that I is full and that its essen tial range is Bimod p Q Ă Q 1 q . W e ﬁrst pro ve that I is full. Let T : p L 2 p Q q 8 Ñ q L 2 p Q q 8 b e a Q - Q -bimo dular map b et w een H p ψ 1 q and H p ψ 2 q . Then T P pQ 8 q , since T is righ t Q -mo dular. W e ha ve T xp “ xq T f or all x P N , so it follo ws that T P pB 8 q . This pro ves that I is full. Let us pro v e that the image of I is con tained in Bimo d p Q Ă Q 1 q . T ak e a ﬁnite index inclusion ψ : Q Ñ pQ 8 p with p P B 8 , p T r b τ qp p q ă 8 and ψ p x q “ xp for all x P N and let H “ H p ψ q . W e claim that H is a Q - Q -subbimo dule of L 2 p Q 1 q 8 . Extend ψ to a map L 2 p Q q Ñ L 2 p pQ 8 p q and note that its en tries, considered as op erators on L 2 p Q q , lie in Q 1 . An y non-zero column of ψ deﬁnes a partial isometry v P p p M 8 , 1 p C q b Q 1 q satisf y ing v x “ ψ p x q v , for all x P Q . Note that v v ˚ P ψ p Q q 1 X pQ 8 1 p . If p ‰ v v ˚ , then w e ma y apply the previous pro cedure to the non-zero Q - Q - bimo dule p p ´ v v ˚ q ¨ H – H p ψ p¨qp p ´ v v ˚ qq . T ake a maximal family of non-zero partial is ometries v i inside p p M 8 , 1 p C q b Q 1 q satisfying ψ p x q v i “ v i x f or all x P Q and such that v i v ˚ i are pairwise distinct orthogonal pro jections. Consider the pro jection r “ p ´ ř v i v ˚ i . If r ‰ 0 then w e can apply the previous pro cedure to the non-zero bimodule r ¨ H . As ab ov e, we get a non- zero partial isometry w P r p M 8 , 1 p C q b Q 1 q suc h that ψ p x q w “ w x , for all x P Q . Then, w w ˚ is orthogonal to all of the v i v ˚ i , whic h con tradicts maximalit y of the family . So, p “ ř v i v ˚ i . Putting all these partial isometr ies in a ro w, w e get an elemen t u P p p Q 1 q 8 suc h that ux “ ψ p x q u , for all x P Q and uu ˚ “ ř v i v ˚ i “ p . This pro v es our claim. W e no w pro v e that every bimo dule H of Bimo d p Q Ă Q 1 q is con tained in the essen tial range of I . Assume that H arises as a Q - Q -subbimodule of L 2 p Q 1 q b Q k , for s ome k P N . W e pro v e that H is a subbimodule of L 2 p Q 1 q 8 . By Corollary 2.15, w e ha v e that H is isomorphic, as N - Q -bimo dule, to a subbimodule of L 2 p Q q 8 . W riting H – H p ψ q , for some ﬁnite index inclusion ψ : Q Ñ q Q n q , we ﬁnd a non -zero N -cent ral v ector v P q p M n, 1 p C q b L 2 p Q qq . T aking p olar decompos ition, w e ma y assume that v P q p M n, 1 p C q b Q q is a partial isometry satisfy ing ψ p x q v “ v x , for all x P N . As a consequence, w e ha ve v ˚ v P B . As in the previous paragraph, tak e a maximal family of non-zero partial isometries v i inside q p M n, 1 p C q b Q q satisfying ψ p x q v i “ v i x for all x P N and q “ ř v i v ˚ i . Putting all partial isometries v i in one ro w, w e get an elemen t u P q p M n, 8 p C q b Q q such that ψ p x q u “ ux for all x P N and uu ˚ “ ř v i v ˚ i “ q . Deﬁne p “ u ˚ u and note that p P B 8 . Conjugating with u ˚ from the b eginning y ields a map ψ : Q Ñ p Q 8 p suc h that ψ p x q “ px , for all x P N and still satisf ying H – H p ψ q . T ake a ﬁnite index inclusion ψ : Q Ñ pQ 8 p in C . Then, w e ha v e p P B 8 . Denote b y ι : P b B Ñ p p P b B q 8 p the inclusion map giv en b y x ÞÑ xp on P and b y the restriction ψ | B on B . Since ψ preserv es N , it also preserv es B “ N 1 X Q and w e obtain a map ι ˚ ψ : M Ñ pM 8 p . If T P Hom C p ψ 1 , ψ 2 q , then T P q B 8 p . So, T deﬁnes an M - M -mo dular map from H p ι ˚ ψ 1 q to H p ι ˚ ψ 2 q . W e conclude that the map F 0 : C Ñ Bimo d p M q : ψ ÞÑ H p ι ˚ ψ q is a functor. Prop osition 3.3. F 0 is a ful ly faithful tensor C ˚ -functor. Pr o of. It is clear that F 0 is faithful. W e ﬁrst prov e that F 0 is full. T ake T P Hom M - M p H p ι ˚ ψ 1 q , H p ι ˚ ψ 2 qq . Then T : p L 2 p M q 8 Ñ q L 2 p M q 8 is righ t M -mo dular, hence T P pM 8 q . Since T xp “ xq T for all x P P , w e hav e T P pB 8 q . So T is in the image of F 0 . The functor ˚ on b oth Bimo d p Q Ă Q 1 q and Bimo d p M q is giv en b y T ÞÑ T ˚ , so F is a C ˚ -functor. Since H p ψ 1 q b M H p ψ 2 q – H pp id b ψ 2 q ˝ ψ 1 q , it follo ws immedi ately that F 0 is a tensor C ˚ -functor. 12 No w let G : Bimo d p Q Ă Q 1 q Ñ C b e an in v erse functor for the inclusion I : C Ñ Bimo d p Q Ă Q 1 q . W e deﬁne the fully faithful tensor C ˚ -functor F “ F 0 ˝ G : Bimo d p Q Ă Q 1 q Ñ Bimo d p M q . 3.2 Pro of of Theorem 3.1: essen tial surjectivity of F W e giv e a series of preliminary lemmas b efore pro ving that the functor F constructed in the previous section is essen tially surjectiv e. Lemma 3.4. L et M H M b e a ﬁnite i n dex M - M -bimo dule and P b B K P b B Ă P b B H P b B b e a ﬁnite index P b B - P b B -subbimo dule. Then K c ontains a non-zer o N - N -subbimo dule L such that dim - N p L q ă `8 . Pr o of. Let ψ : M Ñ pM n p and ϕ : P b B Ñ q p P b B q k q b e ﬁnite index inclusions suc h that M H M – M H p ψ q M and P b B K P b B – P b B H p ϕ q P b B . T ak e a non-zero partial isometry v 0 P p p M n,k p C q b M q q suc h that ψ p x q v 0 “ v 0 ϕ p x q , for all x P P b B . W e ha ve v ˚ 0 v 0 P ϕ p P b B q 1 X q M k q , so the supp ort pro jection supp E P b B p v ˚ 0 v 0 q lies in ϕ p P b B q 1 X q p P b B q k q . M oreov er v 0 p supp E P b B p v ˚ 0 v 0 qq “ v 0 . So we can assume that q “ sup p E P b B p v ˚ 0 v 0 q . W e claim that ϕ p N 0 q ă P b B P . Recall that B is h yp erﬁnite, b y assumption. Let Ť n A n b e the dense union of an increasing s equence of ﬁnite dimensional von Neumann subalgebras A n of B . Since P b 1 Ă P b A n is a ﬁnite index inclusion for ev ery n , it suﬃces to sho w that ϕ p N 0 q ă P b B P b A n for some n . Denote by E n the trace-preserving conditional ex p ectation of B on to A n . Then the sequence of unital completely p ositiv e maps id b E n on p P b B q k , still denoted b y E n , con verges p oin twise in } ¨ } 2 to the identit y . Since N 0 has prop ert y (T) (see p P 3 q ), Theorem 2.5 sho ws that p E n q con verges uniformly in } ¨ } 2 on p ϕ p N 0 qq 1 . T ak e n P N suc h that } E n p x q ´ x } 2 ă 1 { 2 for all x P ϕ p N 0 q . Assume that ϕ p N 0 q ć P b B P b A n . By Theorem 2.2, there is a sequence of unitaries u k P U p ϕ p N 0 qq suc h that for all x, y P q p P b B q k q , we hav e } E n p xu k y q} 2 Ñ 0 for k Ñ 8 . In particular, 1 “ } u k } 2 ă 1 { 2 ` } E n p u k q} 2 Ñ 1 { 2 , whic h is a con tradiction. W e ha ve pro v ed our claim. This yields a ˚ -homomorphism π : N 0 Ñ r P l r and a non-zero partial isometry v 1 P q p M k ,l p C q b P b B q r suc h that ϕ p x q v 1 “ v 1 π p x q , for all x P N 0 . Similarly to the ﬁrst paragraph, we can assume that r “ supp E P p v ˚ 1 v 1 q . Note that E P b B p v ˚ 0 v 0 q “ q . So v “ v 0 v 1 P p p M n,l b M q r is a non-zero partial isometry . Moreo ver, w e ha ve vπ p x q “ ψ p x q v , for all x P N 0 . W e claim that π p N q ă P N . W e ﬁrst pro v e that it suﬃces to sho w that π p N q ă P b B N b B . Indeed, if this the case, we get a ˚ -homomorphism θ : N Ñ t p N b B q j t and a non-zero partial isometry u P r p M l,j p C q b P b B q t suc h that π p x q u “ uθ p x q , for all x P N . Denote b y u i the i -th column of u . Then the closed linear span of t u i N b B | i “ 1 , . . . , j u deﬁnes a non-zero π p N q - N b B -subbimo dule of r p C l b L 2 p P b B qq with ﬁnite right dimension. Us ing the N - N -mo dular pro jection ont o r p C l b L 2 p P qq and the action of B , w e ﬁnd a non-zero π p N q - N -bimo dule inside r p C l b L 2 p P qq whic h is ﬁnitely generated as a righ t N -mo dule. Assume no w that π p N q ć P b B N b B . Then, Theorem 2.7 implies that the quasi-normalizer of π p N q in r M l r sits inside r p P b B q l r . As a consequence, v ˚ v P π p N q 1 X r M l r Ă r p P b B q l r . Sinc e the inclusion N Ă M is quasi-regular, w e ha v e that (3.1) v ˚ ψ p M q v Ă r p P b B q l r . 13 Denote by A the von Neumann algebra generated b y ψ p M q and v v ˚ . Then ψ p M q Ă A Ă pM n p and A Ă p M n p has ﬁnite index. U sing (3.1), w e get that v ˚ Av Ă v ˚ v p P b B q l v ˚ v Ă v ˚ M n v , from whic h w e deduce that P b B Ă M has ﬁnite index. W e get a con tradiction. Indeed, M b eing an amalgamated f ree pro duct, w e can ﬁnd in L 2 p M q inﬁnitely man y pairw ise orthogonal P b B - P b B - bimo dules b y means of alternatin g p o w ers of L 2 p P b B q a L 2 p N b B q and L 2 p Q q a L 2 p N b B q . The previous claim yields a ˚ -homomorphism ρ : N Ñ sN m s and a non-zero partial is ometry w P r p M l,m p C q b P q s s uc h that π p x q w “ w ρ p x q for all x P N . Denote by w i the i -th column of w . Deﬁne L as the closed linear s pan of t v 1 w i N | i “ 1 , . . . , m u . Then, L is a non-zero, since E P p w ˚ v ˚ 1 v 1 w q “ w ˚ E P p v ˚ 1 v 1 q w and r “ supp E P p v ˚ 1 v 1 q . So L is a non-zer o ϕ p N q - N -s ubbimodule of K with ﬁnite righ t dimension. Lemma 3.5. L et K b e a ﬁnite index P b B - P b B -subbimo dule of a ﬁnite index M - M -bimo dule H and let N L N Ă N K N b e an i rr e ducible ﬁnite index N - N -subbimo dule. Then N L N is isomorphic to a subbimo dule of N L 2 p P q N . Pr o of. Assume, b y contra diction, that L is not con tained in N L 2 p P q N . T ake some non-trivial ﬁnite index irreducible N - N -bimo dule L Q in L 2 p Q q and some non-trivial ﬁnite index irreducible N - N - bimo dule L P in L 2 p P q b oth with right dimension greater than or equal to 1 . Denote by X 0 the } ¨ } 2 -closure of L ¨ M . Lemma 2.8 implies that X 0 is a non-zero N - M -bimo dule whic h is isomorphic to a subbimodule of H and lies in L b N L 2 M . Deﬁne the N - M -bimo dules X n “ p L P b N L Q q b n b N X 0 . Note that L P AF alg p N Ă P q . By assumption, the fusion algebras F alg p N Ă Q q and AF alg p N Ă P q are free inside F alg p N q . Therefore, as in [FV08], the X n follo w pairwi se disjoin t as N - N -bimo dules and hence pairwise disjoin t as N - M -bimo dules. Decompose X 0 Ă H as a direct sum of irreducible N - N -bimo dules Y i . W rite p L Q q 0 “ L Q X Q and p L P q 0 “ L P X P . Then, p L P q 0 ¨ p L Q q 0 ¨ ¨ ¨ p L Q q 0 ¨ Y i is non-zero. If not, w e had M ¨ Y i ¨ M “ M ¨ p L P q 0 ¨ p L Q q 0 ¨ ¨ ¨ p L Q q 0 ¨ N ¨ Y i ¨ M “ 0 , con tradicting the fact that M is a factor. As abov e, the freeness assumption implies that p L P b N L Q q b n b N Y i is irreducibl e. Then, b y Lemma 2.8, w e hav e that p L P b N L Q q b n b N Y i sits inside H as the } ¨ } 2 -closure of p L P q 0 ¨ p L Q q 0 ¨ ¨ ¨ p L Q q 0 ¨ Y i . W e ha ve pro v en that H con tains a cop y of eac h X n . Note that dim - M p X n q ą dim - M p X 0 q . As a consequence, H , as a righ t M -mo dule, has inﬁnite dimension, whic h is a con tradiction. Lemma 3.6. L et H b e a ﬁnite index M - M -bimo dule and K a ﬁnite index P b B - P b B -subbimo dule. Then, K i s isomorphic to a multiple of the trivial P - P bimo dule. Pr o of. By Lemma 3.4, w e hav e a non-zero N - N -subbimo dule L Ă K with ﬁnite right dimension. Then, for x, y P QN P b B p N q , the closure of N x ¨ L ¨ y N is still an N - N -subbimo dule of K with ﬁnite righ t dimension. Since N Ă P b B is quasi-regular, the linear span of all N - N -subbimo dules of K with ﬁnite righ t dimension is dense in K . Then, a maximalit y argumen t sho ws that K can b e decomp osed as the direct sum N - N -subbimo dules with ﬁnite righ t dimension. By symmetry , K decomposes as the direct sum of N - N -bimodules with ﬁnite left dimension. As a consequence, K ma y b e written as the direct sum of ﬁnite index N - N -subbimo dules. 14 Let L be an irreducible ﬁnite index N - N -subbimo dule of K . Lemma 3.5 sho ws that L is con tained in L 2 p P q . Recall that P “ L Ω p Q 3 ‘ Q 3 ¸ SL p 3 , Q qq , N “ L Ω p Z 3 ‘ Z 3 q . Hence, L arises as the } ¨ } 2 -closure of N u g N for some elemen t g P Q 3 ‘ Q 3 ¸ SL p 3 , Q q . By almost- normalit y (see property p P 1 q and the remarks follo wing it), tak e a ﬁnite index subgroup Λ 0 of Z 3 ‘ Z 3 suc h that Ad p g qp Λ 0 q Ă Z 3 ‘ Z 3 . Denote b y L 0 the closure of N u g L Ω p Λ 0 q . Then, L 0 is an irreducible N - L Ω p Λ 0 q -subbimodule of L . N ote that L 0 b L Ω p Λ 0 q L Ω p Λ 0 q u ˚ g N – L 2 p N q . Lemma 2.8 implies that K con tains a cop y of the trivial N - N -bimo dule L 2 p N q , realized as the } ¨ } 2 -closure of L 0 u ˚ g N . W rite K – H p ψ q for some ﬁnite index inclusion ψ : P b B Ñ q p P b B q 8 q , where p T r b τ qp q q ă 8 . By the ab o ve paragraph, w e can tak e a trivial N - N -bimo dule inside K . Then, there is an N - cen tral v ector v P q L 2 p P b B q 8 . T aking p olar decomp osition, we ma y ass ume that v is a partial isometry in q p M 8 , 1 p C q b P b B q satisfying ψ p x q v “ v x , for all x P N . Note that v v ˚ P ψ p N q 1 X q p P b B q 8 q . Hence, p q ´ v v ˚ q ¨ K deﬁnes a N - P b B -subbimodule of K and we ma y apply the previous pro cedure. A s in the pro of of Propos ition 3.2, a maximalit y argumen t yields a family of partial isometries v i P q p M 8 , 1 p C q b P b B q satisfying ψ p x q v i “ v i x , for all x P N and such that ř v i v ˚ i “ q . Putting these partial is ometries in a ro w, w e obtain an elemen t w P q p P b B q 8 satisfying ww ˚ “ ř v i v ˚ i “ q . By irreducibilit y of N Ă P (see p P 1 q ), w e ha ve a pro jection p “ w ˚ w P p N 1 X P b B q 8 “ B 8 . Conjugating ψ with w ˚ from the b eginning, w e obtain a ﬁnite index inclusion ψ : P b B Ñ p p P b B q 8 p , where p P B 8 suc h that p T r b τ qp p q ă 8 and ψ p x q “ xp for all x P N and still satisfying K – H p ψ q . Let g P Γ “ Q 3 ‘ Q 3 ¸ SL p 3 , Q q and Λ 0 ă Z 3 ‘ Z 3 b e ﬁnite index s ubgroup such that Ad p g ´ 1 qp Λ 0 q Ă Λ . Denote b y α g ´ 1 the ˚ -homomorphism L Ω p Λ 0 q Ñ L Ω p Λ q induced b y Ad p g ´ 1 q . F or x P L Ω p Λ 0 q w e ha ve ψ p u g q u ˚ g x “ ψ p u g q α g ´ 1 p x q u ˚ g “ ψ p u g q ψ p α g ´ 1 p x qq u ˚ g “ xψ p u g q u ˚ g . By p P 5 q , we ha ve that L Ω p Λ 0 q Ă P is irreducible. As a consequence, v g “ ψ p u g q u ˚ g P U p pB 8 p q . Note that ψ p B q Ă ψ p N q 1 X p p P b B q 8 p “ pB 8 p and v g P ψ p B q 1 X pB 8 p . W e prov e that the inclusion ψ p B q Ă pB 8 p has ﬁnite index using Theorem 2.11. Consider the conjugate bimo dule K of P b B K P b B . As pro ven ab o ve, we ma y write K – H p ψ q , where ψ : P b B Ñ q p P b B q 8 q is a ﬁnite index inclusion satisfying ψ p x q “ xq , for all x P N and q P B 8 is a pro j ection suc h that p T r b τ qp q q ă 8 . Note that K b P b B K – H pp ψ b id q ˝ ψ q . Hence there is a conjugate map R : L 2 p P b B q Ñ H pp ψ b id q ˝ ψ q . Considering R as an elemen t of p p b q qp M 8 , 1 p C qb P b B q w e ha ve R P p p b q qp M 8 , 1 p C q b P b B q X N 1 “ p p b q qp M 8 , 1 p C qb B q . Deﬁne ψ B : B Ñ pB 8 p and ψ B : B Ñ q B 8 q as the restrictions of ψ and ψ to B and S : L 2 p B q Ñ H pp ψ B b id q ˝ ψ B q as the restrictio n of R . Similarly , w e ﬁnd an in tertwin er S : L 2 p B q Ñ H pp ψ B b id q ˝ ψ B q , giving a pair of conjugate morphisms for H p ψ B q . Then the same argumen t as in [LR97 , Lemma 3.2] implie s that ψ p B q 1 X pB 8 p is of ﬁnite type I. It follo ws that g ÞÑ v g P ψ p B q 1 X pB 8 p is a direct in tegral of ﬁnite dimensional unitary represen t ations of Γ and hence trivial, since Γ has no non-trivial ﬁnite dimensional unitary represen tations (see p P 4 q ). W e conclude that ψ p u g q “ u g p and that P K P is a m ultiple of the trivial P - P -bimo dule. 15 Lemma 3.7. L et ψ : P b B Ñ p p P b B q n p b e a ﬁnite index inclusion such that ψ p P b 1 q ` p ` C n b L 2 p P b B q ˘˘ P b 1 is a multiple of the trivial P - P -bimo dule. Then ther e exists a non -zer o p artial isometry u P M n, 8 p C q b P b B such that uu ˚ “ p , q “ u ˚ u P B 8 and u ˚ ψ p x q u “ q x for al l x P P , wher e we c ons i der P Ă P 8 diagonal ly. Pr o of. Consider the P - P -bimo dule H giv en b y P H P “ ψ p P b 1 q ` p ` C n b L 2 p P b B q ˘˘ P b 1 . Since H is a m ultiple of the trivial P - P -bimodule, there exists a non-zero v ector v P p ` C n b L 2 p P b B q ˘ suc h that ψ p x q v “ v x for all x P P . T aking its p olar decomp osition, w e ma y assume that v is a non-zero partial isometry in p p C n b P b B q . As in the pro of of Proposition 3.2, a maximalit y argumen t pro vides a family of non-zero partial isometries p v i q , inside p p C n b P b B q , satisfying ψ p x q v i “ v i x f or all x P P and suc h that p “ ř v i v ˚ i . Putting all v i in one ro w, w e get u P M n, 8 p C qb P b B . Then ψ p x q u “ ux , for all x P P . W e also hav e that uu ˚ “ ř v i v ˚ i “ p and u ˚ u P p 1 b P b 1 q 1 X p P b B q 8 “ B 8 . Th us, u is the required partial isometry . Pr o of of The or em 3.1. Let M H M b e a ﬁnite index irreducible M - M -bimo dule. W e pro v e that H is isomorphic to a bimo dule in the range of the functor F : Bimo d p Q Ă Q 1 q Ñ Bimo d p M q , constructed in Section 3.1. W e do this in t w o steps. Step 1. There exists a pro jection p P p P b B q 8 with p T r b τ qp p q ă `8 and ˚ -homomo rphism ψ : M Ñ pM 8 p such that • ψ p M q Ă pM 8 p has ﬁnite index, • ψ p P b B q Ă p p P b B q 8 p and this inclusion has essen tially ﬁnite index and • M H M – M H p ψ q M . Pr o of of Step 1. Let ψ : M Ñ pM n p b e a ﬁnite index inclusion such that M H M – M H p ψ q M . By symmetry , Theorem 2.4 and the remarks preceding it, we are left with pro ving the tw o f ollo wing statemen ts. (i) ψ p P b B q q ă M P b B , for ev ery pro jection q P ψ p P b B q 1 X pM n p . (ii) Whenev er K Ă L 2 p M q is a p P b B q - p P b B q -subbimo dule with dim - P b B p K q ă `8 , w e ha v e K Ă L 2 p P b B q . By assumption, there is no non-trivial ˚ -homomorph ism from N 0 to an y ampliﬁcation of Q . It follo ws that ψ p N 0 q ć M Q . Hence, Theorem 2.6 implies that ψ p N 0 q ă M P b B . So there is a ˚ - homomorphi sm ϕ : N 0 Ñ q p P b B q m q and a non-zero partial isometry v P p p M n,m p C q b M q q suc h that ψ p x q v “ v ϕ p x q for all x P N 0 . W e hav e v ˚ v P ϕ p N 0 q 1 X q M m q . So v ˚ v P q p P b B q m q by Theorem 2.7. Then, v ˚ v p QN q M m q p ϕ p N 0 qq 2 q v ˚ v Ă q p P b B q m q , b y Theorem 2.7 again. Since N 0 Ă P is q uasi-regular (see p P 2 q ), w e als o ha ve that v ˚ ψ p P b B q v Ă v ˚ v p QN q M m q p ϕ p N 0 qq 2 q v ˚ v Ă q p P b B q m q . 16 Note that all the previous argumen ts remain true when cutting do wn ψ with a pro jection in ψ p P b B q 1 X pM n p , so w e ha ve pro ven (i). Theorem 2.7 implies (ii) and Step 1 is prov en. Step 2. W e ma y assume that p P B 8 and that the ˚ -homomorph ism ψ satisﬁes • ψ p x q “ px , for all x P P , • ψ p B q Ă B 8 , • ψ p Q q Ă pQ 8 p . Pr o of of Step 2. By Step 1, the inclusion ψ p P b B q Ă p p P b B q 8 p has essentia lly ﬁnite index. Let q b e a pro jection in ψ p P b B q 1 X p p P b B q 8 p suc h that K “ ψ p P b B q ` q L 2 p P b B q 8 ˘ P b B is a ﬁnite index P b B - P b B -subbimo dule of P b B H P b B . Lemma 3.6 implies that P b 1 K P b 1 is isomorphic to a multi ple of the trivial P - P -bimo dule. Lemma 3.7 yields a non-zero partial isometry u P q p M 8 ,m p C q b P b B q satisfying u ˚ ψ p x q u “ u ˚ ux for all x P P and suc h that uu ˚ “ q and u ˚ u P B m . Since ψ p P b B q Ă p p P b B q 8 p has essen tially ﬁnite index, this procedure pro vides a non-zero partial isometry u P p P b B q 8 satisfying u ˚ ψ p x q u “ u ˚ ux f or all x P P with uu ˚ “ p and u ˚ u P B 8 . Conjugating ψ with u ˚ from the b eginning, w e ma y assume that p P B 8 and ψ p x q “ px for all x P P . W e ha ve P 1 X M “ B and ψ p x q “ px for all x P P , with p P B 8 , therefore ψ p B q Ă B 8 . Since p P p N 1 X Q q 8 and ψ p x q “ px for all x P P , the ˚ -homomorp hism ψ extends to an N - N - bimo dular map v : L 2 p M q Ñ L 2 p pM 8 p q . By freeness of F alg p N Ă Q q and F alg p N Ă P q inside F alg p N q , w e ha v e that v p L 2 p Q qq is an N - N -subbimo dule of L 2 p pQ 8 p q . Hence ψ p Q q Ă pQ 8 p , whic h ends the proof of Step 2 . 3.3 Pro of of Theorem D W e use the follo wing v ersion of [NPS04, Theorem 0.2] f or the pro of of Theorem D. Theorem 3.8 (See [NPS04, Theorem 0.2]) . L et Γ b e a pr op erty (T) gr oup an d M a sep ar able II 1 factor. L et J Ă H 2 p Γ , S 1 q b e the set of sc alar 2-c o cycles Ω such that ther e exists a (not ne c essarily unital) non-trivial ˚ -homomorphism fr om L Ω p Γ q to an ampliﬁc ation of M . Then J is c ountable. Pr o of of The or em D. Fix an inclusion of I I 1 factors N Ă Q and assume that N is hyperﬁnite. Supp os e also that N Ă Q is quasi-regular and has depth 2. Deno te by N Ă Q Ă Q 1 the basic construction. Let α P R ´ Q and consider the groups Λ , Γ and the scalar 2 -co cycle Ω α P Z 2 p Γ , S 1 q deﬁned in at the b eginning of Section 3. Since the group Z 3 ‘ Z 3 ¸ SL p 3 , Z q has prop ert y (T), Theorem 3.8 implies that there are uncoun tably many α P R ´ Q such that there is no non-trivial ˚ -homomorp hism from N 0 “ L Ω α ` Z 3 ‘ Z 3 ¸ SL p 3 , Z q ˘ to an y ampliﬁcati on of Q . T ak e one suc h α P R ´ Q . Note that by p P 1 q , p P 3 q and Lemm a 2.17, the f usion algebra F “ AF alg ` L Ω α p Λ q Ă L Ω α p Γ q ˘ is coun table. Observ e that L Ω α p Λ q and N are t w o copies of the h yp erﬁnite I I 1 factor and tak e an isomorphism θ : N Ñ L Ω α p Λ q . Then, the fusion algebra F θ ma y b e viewe d as a fusion subalgebra of F alg p N q . Since F alg p N Ă Q q is a coun table fusion subalgebra of F alg p N q , Theorem 2.19 allo ws us to choose θ such that F θ is f ree with resp ect to F alg p N Ă Q q . W e iden tify N and L Ω α p Λ q through this isomorphism and all assumptions of Theorem 3.1 are satisﬁed. W rite P α “ L Ω α p Γ q and write M α “ p P α b B q ˚ N b B Q , where B “ N 1 X Q . 17 Using Theorem 3.1, we obtain that Bimod p M α q » Bimo d p Q Ă Q 1 q . W e pro ve that stable isomorphism classes of M α , α P R ´ Q , are coun table. Assume b y con tradiction that there exists an uncoun table subset J Ă R ´ Q such that M α j are pairwise stably isomorphic, for j P J . W e ﬁnd k P J and an uncoun table subset I Ă J such that M α i em b eds (not necessarily unitally) in to M α k , for all i P I . In particular, L Ω α i ` Z 3 ‘ Z 3 ¸ SL p 3 , Z q ˘ em b eds in to M α k , for all i P I . Since Z 3 ‘ Z 3 ¸ SL p 3 , Z q is a prop ert y (T) group and cohomology classes of the co cycles ` Ω α ˘ | Z 3 ‘ Z 3 ¸ SL p 3 , Z q , α P R ´ Q are tw o b y t w o non-equal, this contr adicts Theorem 3.8. 4 Applications 4.1 Examples of categories that arrise as Bimo d p M q In this part, we give examples of categories that arise as Bimo d p M q of some I I 1 factor M . Note that the results in [V ae06b] and in [FV08] sho w that the trivial tensor C ˚ -category and the category of ﬁnite dime nsional represen tation of eve ry compact , second coun table group can be realized as a category of bimodules. 4.1.1 Finite tensor C ˚ -categories The f ollo wing reconstruction theorem for ﬁnite tensor C ˚ -categories is well kno wn, but for con ve- nience, we giv e a short pro of. W e use Jones’ planar algebras [Jon99] and Popa ’s reconstruction theorem f or ﬁnite depth standard inv arian ts [P op90]. See also [BMPS09] for a similar statemen t. Theorem 4.1. L et C b e a ﬁnite tensor C ˚ -c ate gory. Then ther e exists a ﬁnite index depth 2 inclusion Q Ă Q 1 of hyp erﬁnite II 1 factors such that Bimo d p Q Ă Q 1 q » C . Pr o of. W e deﬁne a depth 2 subfactor planar algebra P , such that the inclusion of hyperﬁnite I I 1 factors Q Ă Q 1 asso ciated with it b y [P op90 , Jon99] satisﬁes Bimo d p Q Ă Q 1 q » C . Let x P C b e the direct s um of represen tativ es for ev ery isomorphism class of irreducible ob j ects in C . Denote b y x the conjugate ob ject of x . Let P k : “ E n d p x b x b ¨ ¨ ¨ b x lo oooooo omo oooooo on k factors q . W e pro ve that P “ Ť P k is a subfactor planar algebra. Comp os ition of endomorphi sms and the ˚ - functor of C mak e P a ˚ -algebra. The categorical trace of C deﬁnes a p ositiv e trace on P . Moreo ver, the graphical calculus for tensor C ˚ -categories induces an action of the planar op erad on P . W e ha ve dim P 0 “ 1 , since 1 C is irredu cible. Moreov er, for all k w e hav e dim P k ă 8 , since C is ﬁnite. Finally , the closed lo ops represen t the nu m b er dim C x ‰ 0 . So P is a subfactor planar algebra. It has depth 2 , since dim Z p P k q is the num b er of isomorphism classes of irreducible ob jects of C for ev ery k ě 1 and, in partic ular, dim Z p P 1 q “ dim Z p P 3 q . Note that, in the language of [P op90 ], ﬁnite depth s ubfactor planar algebras corresp ond to canonical comm uting squares [Bis97, Jon08]. So, b y [P op90 ], there is an inclusion Q Ă Q 1 of h yp erﬁnite I I 1 factors with asso ciated planar algebra P Q Ă Q 1 – P . Th en x corresp onds to Q L 2 p Q 1 q Q 1 . Let 18 D “ Bimo d p Q Ă Q 1 q and denote by Q Ă Q 1 Ă Q 2 the basic construction. If p, q are minimal pro jections in Q 1 X Q 2 , w e canonically iden tify Hom Q - Q p p L 2 p Q 1 q , q L 2 p Q 1 qq with q p Q 1 X Q 2 q p . This deﬁnes a C ˚ -functor F : D Ñ C sending p L 2 p Q 1 q – p p Q L 2 p Q 1 q b Q 1 L 2 p Q 1 q Q q to p p x b x q and mapping morphisms as giv en b y the iden tiﬁcation P Q Ă Q 1 – P . Then F is fully faithful and essen tially surjective. W e ha v e to pro ve that F preserv es tensor pro ducts. Let p , q b e pro jections in Q 1 X Q 2 . The shift-b y-tw o op erator sh 2 : P 2 Ñ P 4 is deﬁned b y adding tw o strings on the left. By [Bis97], w e hav e p L 2 p Q 1 q b Q q L 2 p Q 1 q – p ¨ sh 2 p q q L 2 p Q 2 q as Q - Q -bimo dules. On the other hand, w e ha ve p p x b x q b q p x b x q – p p b q qp x b x b x b x q in C . Since under the iden tiﬁcation P k – Q 1 X Q k the shift-by-t w o op erator corresponds to q ÞÑ 1 b q , we ha ve C » D as tensor C ˚ -categories. This completes the proof. Pr o of of The or em A. Let C b e a ﬁnite tensor C ˚ -category . By Theore m D it suﬃces to show that there is a ﬁnite index, depth 2 inclusion N Ă Q of hyperﬁnite I I 1 factors, suc h that for the basic construction N Ă Q Ă Q 1 w e ha ve Bimo d p Q Ă Q 1 q » C . Indeed, if N Ă Q is of ﬁnite index, then it is quasi-regula r. By Theorem 4.1, there is a ﬁnite index depth 2 inclusion N ´ 1 Ă N of hyperﬁnite I I 1 factors suc h that Bimo d p N ´ 1 Ă N q » C . Let N ´ 1 Ă N Ă Q Ă Q 1 b e the basic construction. Then N Ă Q is a ﬁnite index, depth 2 inclusion and Bimo d p Q Ă Q 1 q » Bimo d p N ´ 1 Ă N q » C . 4.1.2 Represen tation categories In [FV08] the categories of ﬁnite dimensional represen tations of compact s econd count able groups w ere realized as bimo dule categories of a I I 1 factor. As already men tioned, this forms a natural class of tensor C ˚ -categories, since they can b e abstractly chara cterized as symmetric tensor C ˚ - categories with at most coun tably man y isomorphism classes of irreducible ob jects. W e realize categories of ﬁnite dimensional represen tations of discrete coun table groups and of ﬁnite dimensional corepresen tations of certain discrete Kac algebras as bimo dule categories of a I I 1 factor. Neither do es this class of categories ha ve an abstract c haracterization, nor do es the ﬁnite dimensional corepresen tation theory of a discrete Kac algebra describ e it completely . H o wev er, Corollary 4.4 sho ws that we ha v e in teresting appl ications coming from this class of tensor C ˚ -categories. F or notation concerning quan tum groups, w e refer the reader to the appendix in Section 5 . Deﬁnition 4.2 (See Section 4.5 and Theorem 4.5 of [Soł05 ]) . A discrete Kac algebra A is called maximally almost p erio dic, if there is a family of ﬁnite dimensional corepresen tations U n P A b B p H U n q suc h that A “ span tp id b ω qp U n q | n P N , ω P B p H U n q ˚ u Theorem 4.3 . L et A b e a discr ete Kac algebr a admitting a strictly outer action on the hyp erﬁnite II 1 factor. Then ther e is a II 1 factor M such that Bimo d p M q » C oRep ﬁn p A coop q . Pr o of. Since A acts strictly outerly on the hyperﬁnite I I 1 factor R , the inclusion R Ă A ˙ R Ă p A coop ˙ A ˙ R is a basic constructio n b y [V ae00 , Prop osition 2.5 and Corollary 5.6]. The inclu sion R Ă A ˙ R has depth 2 by [V ae00, Corollary 5.10] and since A is discrete, it is quasi-regular. Moreo ver, w e ha ve Bimo d p A ˙ R Ă p A ˙ A ˙ R q » CoRep ﬁn p A coop q b y Theorem 5.1. So Theorem D yields a I I 1 factor M suc h that Bimo d p M q » CoRep ﬁn p A coop q . Pr o of of The or ems C. By Theorem 4.3 it suﬃces to show that every discrete group G and every amenable and every maximally almost p erio dic Kac algebra A has a strictly outer action on the h yp erﬁnite I I 1 factor R . 19 Let us ﬁrst consider the case of a discrete group. The non-comm utative Bernoulli shift G ñ p M 2 p C q , tr q b G is well know n to b e outer. It is clear that b 8 n “ 1 p M 2 p C q , tr q is isomorphic to R . First note that p A coop q coop “ A for all quantu m groups A . By V aes [V ae02, Theorem 8.2], it suﬃces to sho w that every amenable and every maximally almost p erio dic Kac algebra A has a faithful corepresen tation of A coop in the h yp erﬁnite I I 1 factor. If A is a discrete amenable Kac algebra, then so is A coop . By [V ae02 , Prop osition 8.1], A coop has a faithful corepresen tation into R . If A is a discrete maximally almost p erio dic Kac algebra, then A coop is also maximally almost p erio dic, since A has a b ounded an tip o de. Ther e is a coun table family of core presen tations U n of A coop whose coeﬃcien ts span A densely . Considering ‘ n B p H U n q as a v on Neumann subalgebra of R , the corepresen tation ‘ n U n of A coop is faithful. As a corollary of Theorem C, we get the f ollo wing impro veme n t of [IPP05, Corollary 8.8] and [FV07]. This is the ﬁrst example of an explicitly kno wn bimod ule category with uncoun tably man y isomorphism classes of irreducible ob j ects. Corollary 4.4. L et G b e a se c ond c ountable, c omp act gr oup. Then ther e is a II 1 factor M such that Out p M q – G and every ﬁnite i ndex bimo dule of M i s of the form H p α q f or some α P Aut p M q . I n p articular, the bimo dule c ate gory of M c an b e explicitly c alculate d and has an unc ountable numb er of isomorphism classes of irr e ducible obje cts. The exact sequence 1 Ñ O ut p M q Ñ grp p M q Ñ F p M q Ñ 1 sho ws that the fundamen tal group of M obtained in Corollary 4.4 is trivial. Note, that the f actors constructed in [FV07, IPP05] also ha ve trivial fundamen tal group. Pr o of. Let G b e a second coun table, compact group. By [Soł06, Theorem 4.2], L p G q is maximall y almost p erio dic and its irreducible, ﬁnite dimensional corepresen tations are one dimensional and indexed b y elemen ts of G . Their tensor pro duct is given b y m ultiplication in G . So w e can apply Theorem C to the discrete Kac algebra L p G q in order to obtain M . 4.2 P ossible indices of irreducible subfactors In this section, we in vestigate the s tructure of subfactors of the I I 1 factor M that w e obtained in Theorem A. W e write C p M q “ t λ | there is an irreducible ﬁnite index subfactor of M with index λ u . W e use the fact that the lattice of irreducible subfactors of a I I 1 factor is actually encoded in its bimo dule category . In sp ecial cases, indices of irreducible subfactors corresp ond to F rob enius- P erron dimensions (see [ENO02, Section 8]) of ob jects in the bimo dule category . Using recen t w ork on tensor categories [GS11] and The orem A, w e giv e examples of of I I 1 factors M suc h that C p M q can b e computed explicitl y and con tains irrationals. Deﬁnition 4.5 (See [FS01, Y am04]) . Let C b e a compa ct tensor C ˚ -category with tensor unit 1 C . (i) An algebra p A , m, η q in C is an ob ject A in C with m ultiplication and unit maps m : A b A Ñ A and η : 1 C Ñ A suc h that the follo wing diagrams comm ute A b A b A m b id / / id b m   A b A m   A b A m / / A A b 1 C id b η   A – / / – o o 1 C b A η b i d   A b A m / / A A b A . m o o 20 (ii) A coalgebra p A , ∆ , ǫ q in C is an ob j ect A in C with com ultiplication and counit map ∆ : A Ñ A b A and ǫ : A Ñ 1 C suc h that p A , ∆ ˚ , ǫ ˚ q is an algebra. (iii) A F rob enius algebra p A , m, η , ∆ , ǫ q in C is an ob ject A in C with maps m , η , ∆ , ǫ suc h that p A , m, η q is an algebra, ∆ “ m ˚ , ǫ “ η ˚ and p id b m q ˝ p ∆ b id q “ ∆ ˝ m “ p m b id q ˝ p id b ∆ q . (iv) A F rob enius algebra p A , m, η , ∆ , ǫ q is sp ecial if ∆ and η are isometric . (v) A F rob enius algebra A is irreducible if dim p Hom p 1 C , A qq “ 1 . Remark 4.6. Note that the notion of a sp ecial F rob enius algebra is equiv alen t to the notion of a Q-system [LR97]. The follo wing lemma and prop osition are probably w ell kno wn, but since w e could not ﬁnd a reference, w e giv e a short proof for con venien ce of the reader. Lemma 4.7. L et M Ă M 1 b e a ﬁn ite index inclusion of tr acial von Neumann algebr as. T hen L 2 p M 1 q i s a sp e cial F r ob enius algebr a in Bimo d p M q . The F r ob enius algebr a L 2 p M 1 q i s irr e ducible if and only if M Ă M 1 is irr e ducible. Pr o of. W e pro ve that L 2 p M 1 q is an algebra in Bimod p M q with coisometric mult iplication and iso- metric unit . By [LR97], this shows that L 2 p M 1 q is a sp ecial F rob enius algebra. The m ultiplication on L 2 p M 1 q is giv en b y m p x b M y q “ xy , f or x , y P M . The comm utativ e diagram L 2 p M 1 q b M L 2 p M 1 q m / / –   L 2 p M 1 q L 2 p M 2 q e 6 6 ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ pro ves that m is w ell deﬁned and coisometric. Here we denote b y M Ă M 1 Ă M 2 the basic construction and w e denote b y e the Jones pro jection. The unit map of L 2 p M 1 q is giv en b y the canonical embedding L 2 p M q Ñ L 2 p M 1 q . The inclusion M Ă M 1 is irreducible if and only if M L 2 p M 1 q M 1 is irreducible if and only if M L 2 p M 1 q M – M L 2 p M 1 q b M 1 L 2 p M 1 q M con tains a unique cop y of M L 2 p M q M . Whenev er H is a ﬁnite index M - M -bimo dule ov er a I I 1 factor M , we denote by H 0 the set of b ounded vectors in H . Recall that H 0 is dense in H . Prop osition 4.8. L et M b e a II 1 factor. Then ther e i s a bije ction b etwe en irr e ducible sp e cial F r ob enius algeb r as in Bimod p M q and irr e ducible ﬁn ite index i nclusions M Ă M 1 of II 1 factors. The bije ction is given by H ÞÑ p M Ă H 0 q and p M Ă M 1 q ÞÑ M L 2 p M 1 q M . Pr o of. Lemma 4.7 sho ws that L 2 p M 1 q is an irreducible special F rob enius algeb ra for all irreducible ﬁnite index inclusions M Ă M 1 . Let p H , m, ǫ, ∆ , η q b e an irreducible s p ecial F rob enius algebra in Bimo d p M q . W e hav e to prov e that M Ă H 0 is a ﬁnite index, irreducible inclusion of vo n Neumann algebras. Let M 2 “ Hom - M p H q b e the comm utan t of the righ t M -action. Then 21 H 0 b H / / & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ H b M H m   H yields a map φ : H 0 Ñ M 2 . By considering the restriction m : H 0 b L 2 p M q Ñ H , it is clear that φ is injectiv e. Consider the sp ecial F rob enius algebra p L 2 p M 2 q , m 2 , η 2 , ∆ 2 , ǫ 2 q . W e prov e that φ p H 0 q is a F rob enius subalgebra of L 2 p M 2 q . Indeed, the comp osition H 0 b H 0 Ñ H b M H Ñ H induces a m ultiplication on H 0 , since M - M -bimo dular maps send M -b ounded vect ors to M -b ounded vect ors. Since m is asso ciative, w e ha ve for ξ , ξ 1 P H 0 φ p m p ξ , ξ 1 qq ¨ x 1 M “ m ˝ p m b id qp ξ , ξ 1 , 1 M q “ m ˝ p id b m qp ξ , ξ 1 , 1 M q “ φ p ξ q ¨ z φ p ξ 1 q “ { φ p ξ q ¨ φ p ξ 1 q . So m is the restriction of m 2 . By taking adjoin ts, we see that ∆ is the restriction of ∆ 2 . Nex t, note that m ˝ p η b id q “ id “ m 2 ˝ p η 2 b id q . So φ p η p x qq ¨ ξ “ xξ for all x P M Ă L 2 p M q and all ξ P H . So η agrees with η 2 . Again, b y taking adjoin ts , ǫ is the restriction of ǫ 2 . Let R : L 2 p M q Ñ H b M H denote the s tandard conjugate for H [LR97]. F rob enius algebras are self-dual via ∆ ˝ η , that is ∆ ˝ η : L 2 p M q Ñ H b M H is a conjugate for H . In particula r, there is an M - M -bimodular isomorphism ψ : H Ñ H suc h that H b M H R ˚ / / L 2 p M q H b M H m / / ψ b i d O O H ǫ O O comm utes. Denoting by R 2 : L 2 p M q Ñ L 2 p M 2 q b M L 2 p M 2 q the standard conjugate for L 2 p M 2 q , w e ha ve the comm uting diagram L 2 p M 2 q b M L 2 p M 2 q R ˚ 2 / / L 2 p M q L 2 p M 2 q b M L 2 p M 2 q m 2 / / p ˝˚qb id O O L 2 p M 2 q . ǫ 2 O O Note that, b y the deﬁnitio n of s tandard conjugates, R is the comp osition of R 2 with the orthogo nal pro jection L 2 p M 2 q b M L 2 p M 2 q Ñ H b M H . So ψ is the restriction of ˝ ˚ . No w consider the comm utativ e diagram L 2 p M 2 q b M L 2 p M 2 q id b ∆ 2 / / L 2 p M 2 q b M L 2 p M 2 q b M L 2 p M 2 q R ˚ 2 b id / / L 2 p M q b M L 2 p M 2 q L 2 p M 2 q b M L 2 p M 2 q id b ∆ 2 / / p ˝˚qb id O O L 2 p M 2 q b M L 2 p M 2 q b M L 2 p M 2 q m 2 b id / / L 2 p M 2 q b M L 2 p M 2 q . ǫ 2 b id O O It restricts to the corresp onding diagram with L 2 p M 2 q replaced by H . Deﬁne m “ p R ˚ b id q ˝ p id b ∆ q : H 0 b H Ñ H and m 2 “ p R ˚ 2 b id q ˝ p id b ∆ 2 q . Since m 2 “ p ǫ 2 b id q ˝ p m 2 b id q ˝ p id b ∆ 2 q in the F rob enius algebra L 2 p M 2 q , w e ha ve that M 2 b L 2 p M 2 q m 2 / / L 2 p M 2 q M 2 b L 2 p M 2 q m 2 / / p ˝˚qb id O O L 2 p M 2 q and H 0 b H m / / H H 0 b H m / / ψ b i d O O H . 22 comm ute and the second diagram is a restriction of the ﬁrst one. Denote by φ : H 0 Ñ M 2 the em b edding deﬁned by m . Then φ p x q “ φ p x q ˚ for x P H 0 and φ p H 0 q “ φ p H 0 q . This pro v es that φ p H 0 q is closed under taking adjoin ts. W e already pro ved that φ p H 0 q is a ˚ -subalgebra of M 2 . Since M H has ﬁnite dimension, φ p H 0 q is ﬁnitely generated ov er M . Hence, it is weakly closed in M 2 , s o it is a von Neumann subalgebra. Finally , M L 2 p H 0 q M – M H M , so M Ă H 0 is irreducible and has ﬁnit e index. Remark 4.9. (i) By uniqueness of m ultiplicativ e dimension functions on ﬁnite tensor C ˚ - categories, s ee [CE04], w e ha ve r M 1 : M s min “ FPdim p M L 2 p M 1 q M q , where r M 1 : M s min denotes the minimal index [Ha v90, Lon92] and FPd im denotes the F rob enius P erron dimen- sion [ENO02, Section 8]. So if M Ă M 1 is extremal (for example irreducible), then we ha v e r M 1 : M s “ FPdim p M L 2 p M 1 q M q . (ii) By Proposition 4.8, irreducible sp ecial F rob enius algebras corresp ond to irreducib le inclusions M Ă M 1 , hence to irreducible subfactors N Ă M . In particular, if Bimo d p M q is ﬁnite, then C p M q “ t FPdim p H q | H irreducible sp ecial F rob enius algebra in Bimod p M qu . W e can pro v e Theorem B no w. Pr o of of The or em B. Denote b y C the Haagerup fusion category [AH98]. In [GS11], p ossible prin- ciple graphs of irreducibl e sp ecial F rob enius algebras in C are classiﬁed. Lemma 3.9 in [GS11] giv es a list of p ossible principle graphs of non-trivial simple algebras in C . Note that the list of indices in Theorem B is the same as the indices of graphs in [GS11, Lemma 3.9]. W e will refer with 1), 2), etc. to the graphs in this lemma. W e pro ve that all the indices of these graphs, are actually realized b y some irreducible sp ecial F rob enius algebra in C . Since, b y [GS11, Theorem 3.25], there are three pairwise diﬀeren t catego ries that are Morita equiv- alen t to C , all the p os sible principal graphs of minimal simple algebras are are actually realized by some irreducible sp ecial F rob enius algebra in C . So the graphs 1) and 3) are realized. Using the notation of [GS11] for irreducible ob jects in C , the graphs 4), 6) and 7) are realized by the irreducible sp ecial F rob enius algebras η η , ν ν and µµ . W e are left with the graphs 2) and 5). Theorem 3.25 in [GS11] gives the f usion rules for mo dule categories o ver C . A short calculation sho ws that the square of the dimen sion of the second ob ject in the mo dule category asso ciated with the Haagerup subfactor is the index of the graph 2). This pro v es that the graph 2) is realized. A similar calcula- tion sho ws that the second ob ject in the second non-trivial mo dule category o ver C giv es rise to an irreducible sp ecial F rob enius algebra with principal graph giv en by 5). So all indices in [GS11, Lemma 3.9] are actually attained by s ome irreducible sp ecial F rob enius algebra in C . Accordin g to Theorem A it is p ossible to ﬁnd a I I 1 factor M such that Bimo d p M q » C . W e conclude using Remark 4.9. 5 App endix In this app endix, w e pro ve that the category of ﬁnite dimensional unitary corepresen tations of a discrete Kac algebra A , whose co opp osite A coop acts strictly outerly on the h yp erﬁnite I I 1 factor R , is realized as the the bimo dule category of the inclusion A ˙ R Ă p A ˙ A ˙ R . F or con veni ence of the reader, w e giv e a short in tro duction. 23 5.1 Preliminaries on quan tum groups 5.1.1 Lo cally compact quan tum groups (see [KV00]) A lo cally compact quantu m group in the setting of v on Neumann algebras is a v on Neumann algebra A equipp ed with a normal ˚ -homomorphism ∆ : A Ñ A b A and tw o normal, semi-ﬁnite, faithful w eigh ts φ , ψ satisfying • ∆ is c omultiplic ative: p id b ∆ q ˝ ∆ “ p ∆ b id q ˝ ∆ . • φ is left invariant: φ pp ω b id qp ∆ p x qqq “ φ p x q ω p 1 M q f or all ω P M ` ˚ and all x P M ` . • ψ i s ri ght invariant: ψ pp id b ω qp ∆ p x qqq “ ψ p x q ω p 1 M q f or all ω P M ` ˚ and all x P M ` . W e call ∆ the com ultiplication of A and φ , ψ the left and the righ t Haar w eigh t of A , resp ectiv ely . If φ and ψ are tracial, then A is called a Kac algeb ra. If A is of ﬁnite t yp e I , then w e sa y that A is discrete. If φ and ψ are ﬁnite, w e sa y that A is compact. If Γ is a discrete group, then ℓ 8 p Γ q is a discrete Kac algebra with com ultiplication giv en b y ∆ p f qp g , h q “ f p g h q and the left and righ t Haar weig h t b oth induced b y the coun ting measure on Γ . F or an y locally compact quan tum group p A, ∆ q one can construct a dual lo cally comp act quan tum group p p A, p ∆ q and a co opp osite lo cally compact quan tum group A coop . They b oth are represen ted on the same Hilb ert space as A . Hence, it mak es sense to write form ulas in volving elemen ts of A and p A at the same time . W e ha v e p A, ∆ q – p x x A, p p ∆ q and A is compact if and only if p A is discrete. 5.1.2 Corepresen tations (see [Tim08]) A unitar y corepresen tation in H of a locally compact q uan tum group A is a unitary U P A b B p H q suc h that p ∆ b id qp U q “ U 23 U 13 . In what f ollo ws, w e ref er to unitary corepresen tations simply as corepresen tations. If U P B p H U qb A is a corepresen tation of A , then w e also refer to U 21 P A b B p H U q as a corepresen tation A . A corepresen tation U of A is called ﬁnite dimensional if H U is ﬁnite dimensional. The direct sum of tw o corepresen tations U, V of A is denoted by U ‘ V P A bp B p H U q ‘ B p H V qq – A b B p H U q ‘ A b B p H V q . The tensor product of t w o corepresen tations U and V is giv en b y U b V “ U 12 V 13 P A b B p H U qb B p H V q . An in tertw iner b et w een t w o corepresen tations U and V is a b ounded linear map T : H U Ñ H V satisfying p id b T q U “ V p id b T q . The space of all in tert winers b etw een U and V is denoted b y Hom p U, V q . T o ever y irreducible corepresen tation U P A b B p H U q of A , one asso ciates its conjugate corepresen tation p˚ b qp U q P A b B p H U q . Here H U denotes the conjugate Hilb ert space of H U . With this structure, the corepresen tations of a lo cally compact quan tum group A b ecome a tensor C ˚ -category CoRep p A q . Its maximal compact tensor C ˚ -sub category is the category of ﬁnite dimensional corepresen tations CoRep ﬁn p A q . If A is a compact quan tum group, every irreducible corepresen tation of A is ﬁnite dimensional and ev ery corepresen tation is a direct sum of (p ossibly inﬁnitely man y) irreducible corepresen tations. Co eﬃcien ts of tensor pro ducts of arbitrary length of irreducible corepresen tations of A span it densely . Let A denote a compact quan tum group. Then w e can describ e the ev aluation of the Haar states on coeﬃcien ts of corepre sen tations. In particul ar, p id b ψ qp U q “ p id b φ qp U q “ δ U,ǫ 1 , where δ U,ǫ is 1 if U is the trivial corepresen tation and 0 otherwise. 24 If A is discrete, its dual is compact. W e can write A as à U irr. corep. of p A B p H U q . F or an y elemen t x P A we can c haracterize ∆ p x q as the unique elemen t in A b A that satisﬁes ∆ p x q T “ T x f or all T P Hom p U 1 , U 2 b U 3 q and all irreducible corepresen tations U 1 , U 2 and U 3 of p A . Moreo ver, w e can write an y corepresen tation V P A b B p H V q of A as a direct sum of elemen ts V U P B p H U q b B p H V q where U runs through the irreducible corepresen tations of p A . If ǫ denotes the trivial corepresen tation, then V ǫ “ 1 b 1 . Moreo v er, V U “ p b ˚qp V q . 5.1.3 A ctions of quan tum grou ps (see [V ae02 ]) An action of a lo cally compact quan tum group A on a von Neumann algebra N is a normal ˚ - homomorphi sm α : N Ñ A b N such that p ∆ b id q ˝ α “ p id b α q ˝ α . The crossed pro duct von Neumann algebra of N b y α is then the von Neumann algebra A ˙ N generated b y p A b 1 and α p N q . W e iden tify N and p A with subalgebras of A ˙ N . There is a natural action p α of p A on A ˙ N , whic h is uniquely deﬁn ed b y p α p a q “ p ∆ p a q f or a P p A and p α p x q “ 1 b x for x P N . This action is called the dual action of α . If an action α : N Ñ A b N of a lo cally compact q uantum group on a factor satisﬁes N 1 X A ˙ N “ C 1 , then α is called strictly outer. Let A be a discrete quantu m group that acts via α on a von Neumann algebra N . W e denote A ˙ N b y M and as b efore w e iden tify p A and N with subalgebras of M . If A is a Kac algebra, N is ﬁnite and α preserv es a trace τ N on N , then M is also ﬁnite. A faithful normal trace on M is giv en b y p τ b id qp U p 1 b x qq “ δ U,ǫ τ N p x q , for all x P N and for all irreducible corepresen tations U P p A b B p H U q of p A . F or x P N and U P B p H U q b p A an irreducible corepresen tation of p A , w e write α U p x q for the pro jection of α p x q onto the direct summand B p H U q b N of A b N . F or x P N we ha v e U p 1 b x q U ˚ “ α U p x q . 5.2 Corepresen tation c ategories of Kac algebras Theorem 5.1. L et N b e a II 1 factor, A a discr ete quantum gr oup and α : A Ñ A b N a strictly outer action. Denote by M “ A ˙ N the cr osse d pr o duct of N by α and write p A ˙ M for the cr osse d pr o duct by the dual action. Then Bimo d p M Ă p A ˙ M q » C oRep ﬁn p A coop q as tensor C ˚ -c ate gories, wher e CoRep ﬁn p A coop q denotes the c ate gory of ﬁnite dimensional c or epr esentations of A coop . Pr o of. W e ﬁrst construct a f ully faithful tensor C ˚ -functor F : CoRep ﬁn p A coop q Ñ Bimo d p M Ă p A ˙ M q . Then, w e sho w that it is essen tially surjectiv e. Step 1. Let V P A b M n p C q b e a ﬁnite dimensional corepresen tation of A coop , that is p ∆ b id qp V q “ V 13 V 23 . W e deﬁne a ˚ -homomorphism ψ : M Ñ M n p C q b M suc h that ψ p x q “ 1 b x f or all x P N and p id b ψ qp U q “ U 13 V 12 , where U P B p H U q b p A Ă A b p A is an irreducible corepresen tation of p A . 25 Pr o of of S t ep 1. W e ﬁrst sho w that ψ deﬁnes a ˚ -homomorphism. This is ob vious on N . In order to pro ve that ψ is mu ltiplicativ e on p A Ă M , w e ha ve to c heck for all irreducible corepresen tations U 1 , U 2 , U 3 of p A and for every in tert winer T P Hom p U 1 , U 2 b U 3 q the iden tit y p id b ψ qp U 2 q 134 p id b ψ qp U 3 q 234 p T b id q “ p T b id qp id b ψ qp U 1 q holds. W e hav e p T b id qp id b ψ qp U 1 q “ p T b id q U 1 , 13 V U 1 , 12 “ U 2 , 14 U 3 , 24 p ∆ b id qp V U 1 q 123 p T b id q “ U 2 , 14 U 3 , 24 V U 2 , 13 V U 3 , 23 p T b id q “ U 2 , 14 V U 2 , 13 U 3 , 24 V U 3 , 23 p T b id q “ p id b ψ qp U 2 q 134 p id b ψ qp U 3 q 234 p T b id q . W e pro v e that ψ is a homomo rphism on alg p p A, N q “ ˚ - alg p p A, N q . Using the fact that U p 1 b x q “ α U p x q U for all x P N and all irredu cible corepresen tations U of p A , it suﬃces to note that p id b ψ qp U qp 1 b 1 b x q “ U 13 V 12 p 1 b 1 b x q “ U 13 p 1 b 1 b 1 b x q V 12 “ α U p x q 13 U 13 V 12 “ α U p x q 13 p id b ψ qp U q . Let us sho w that ψ is ˚ -preserving. W e ha v e p id b ψ qpp b ˚qp U qq “ “ p id b ψ qp U q “ U 13 V 12 “ U 13 V U , 12 “ p b ˚qp U q 13 p b ˚qp V U q 12 “ p b ˚ b ˚qp U 13 V 12 q , This show s that ψ is a ˚ -hom omorphism on ˚ - alg p p A, N q . Let us sho w that ψ is trace preserving on ˚ - alg p p A, N q . Denote b y τ the trace on M . F or an irreducible corepresen tation U of p A and x P N we ha v e p id b τ qp U p 1 b x qq “ δ U,ǫ τ p x q 1 A , b y the deﬁnition of τ . On the other han d w e ha v e p id b tr b τ qp id b ψ qp U p 1 b x qq “ p id b tr b τ qp U 13 V 12 p 1 b 1 b x qq “ δ U,ǫ τ p x qp id b tr qp V ǫ, 12 q “ δ U,ǫ τ p x q 1 A . So ψ is trace preserving and hence it extends to a ˚ -homo morphism ψ : M Ñ M n p C q b M . Step 2. Deﬁne a f unctor F : C oRep ﬁn p A coop q Ñ Bimod p Q Ă Q 1 q suc h that if V is a ﬁnite dimen- sional corepresen tation of A coop and ψ the map asso ciated with it in Step 1, w e hav e F p V q “ H p ψ q . If T P Hom p V 1 , V 2 q is an in tert winer, w e set F p T q “ T b id : H V 1 b L 2 p M q Ñ H V 2 b L 2 p M q . Then F is fully faithful tensor C ˚ -functor. 26 Pr o of of S tep 2. It is ob vious that F is f aithful. In order to s ho w that F is full, let V 1 P A b M m p C q , V 2 P A b M n p C q b e ﬁnite dimensional corepresen tations of A coop . Denote by ψ 1 , ψ 2 the maps asso ciated with V 1 and V 2 , resp ectiv ely . Let T : C m b L 2 p M q Ñ C n b L 2 p M q b e an in tert winer from H p F p V 1 qq to H p F p V 2 qq . Then T P B p C m , C n q b M satisﬁes T p 1 b x q “ T ψ 1 p x q “ ψ 2 p x q T “ p 1 b x q T for all x P N . Hence, T P B p C m , C n q b 1 . So, for any irreducible corepresen tation U of p A , w e ha v e V 2 , 12 T 23 “ U ˚ 13 U 13 V 2 , 12 T 23 “ U ˚ 13 ψ 2 p U q T 23 “ U ˚ 13 T 23 ψ 1 p U q “ T 23 U ˚ 13 U 13 V 1 , 12 “ T 23 V 1 , 12 . So T comes from an in tert winer f rom V 1 to V 2 . This sho ws that F is full. F or an intert winer T P Hom p V 1 , V 2 q w e ha v e F p T ˚ q “ F p T q ˚ , so F is a C ˚ -functor. If V 1 , V 2 are ﬁnite dimensional corepresen tations of A coop , ψ 1 , ψ 2 and ψ denote the maps as so ciated with V 1 , V 2 and V 1 , 12 V 2 , 13 “ V 1 b V 2 resp ectiv ely , then p id b ψ qp U q “ U 14 V 1 , 12 V 2 , 13 “ p id b ψ 2 q ˝ ψ 1 p U q , for eve ry irreducible corepresen tation U P B p H U q b p A of p A . So ψ “ p id b ψ 2 q ˝ ψ 1 . W e obtain F p V 1 q b M F p V 2 q – F p V 1 b V 2 q and this unitary isomorphism is natural in V 1 and V 2 . Hence F is a tensor C ˚ -functor. Step 3. F is essen tially surjectiv e. Pr o of of Step 3. Let H b e a ﬁnite index bimodule in Bimo d p M Ă p A ˙ M q . W rite H – H p ψ q for some ψ : M Ñ p p M n p C q b M q p satisfying p P p 1 b N q 1 X p M n p C q b M q and ψ p x q “ p p 1 b x q for all x P N . Since N Ă M is irred ucible, w e ha v e p P M n p C q b 1 , so w e ma y assume that p “ 1 . F or an irreducible corepresen tation U of p A , by the s ame calculation as in Step 1 , w e obtain p id b ψ qp U q U ˚ 13 α U p x q “ α U p x qp id b ψ qp U q U ˚ 13 , for all x P N . Since N is linearly generated b y the co eﬃcien ts of α U p N q , it follow s that p id b ψ qp U q U ˚ 13 “ V U, 12 for some elemen t in V U P B p H U q b M n p C q Ă A b M n p C q . Let V “ ð U irr. corep. of p A V U P A b M n p C q . W e show that V is a corepresen tation of A coop , i.e. that p ∆ b id qp V q “ V 13 V 23 . I t suﬃces to sho w for an y irreducibl e corepresen tations U 1 , U 2 , U 3 and any in tert winer T P Hom p U 1 , U 2 b U 3 q that w e ha ve V U 2 , 13 V U 3 , 23 p T b id q “ p T b id qp V U 1 q . 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Tensor C*-categories arising as bimodule categories of II_1 factors

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