A CLASS OF C -ALGEBRAIC LOCALLY COMPACT QUANTUM GROUPOIDS PART II. MAIN THEORY

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1 A CLASS OF C -ALGEBRAIC LOCALLY COMPACT QUANTUM GROUPOIDS PART II. MAIN THEORY BYUNG-JAY KAHNG AND ALFONS VAN DAELE Abstract. This is Part II in our multi-part series of papers developing the theory of a class of locally compact quantum groupoids, based on the purely algebraic notion of weak multiplier Hopf algebras. Definition was given in Part I. The existence of a certain canonical idempotent element E plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map. Introduction. In Part I of this series [13], we proposed a definition for a class of locally compact quantum groupoids. This definition see Definition 1. below) is motivated by the purely algebraic notion of weak multiplier Hopf algebras, introduced by one of us Van Daele) and Wang in [9], [30]. A fundamental example of a weak multiplier Hopf algebra is the algebra A = KG), where G is a groupoid and KG) is the set of all complex-valued functions on G having finite support. Here, the comultiplication map is not necessarily non-degenerate, while the existence of a certain canonical idempotent element E MA A) is assumed, which coincides with 1) in the unital case. See Introduction of the previous paper Part I) for further motivation. In our C -algebraic framework, building on the authors previous work at the purely algebraic level [11], we consider a class of locally compact quantum groupoids determined by the data A,, E, B, ν, ϕ, ψ), where i) A is a C - algebra, taking the role of the groupoid G; ii) : A MA A) is the comultiplication, corresponding to the multiplication on G; iii) B is a sub- C -algebra of MA), taking the role of the unit space G 0) ; iv) ν is a weight on B; v) E is the canonical idempotent element motivated by the theory of weak multiplier Hopf algebras; and finally, vi) ϕ and ψ are the left and right invariant Haar weights, respectively. This framework is closely related to the notion of measured quantum groupoids, developed in the von Neumann algebra setting [18], [9], though not quite as general. It is because the requiring the existence of a canonical 010 Mathematics Subject Classification. 46L65, 46L51, 81R50, 16T05, A. Key words and phrases. Locally compact quantum groupoid, Weak multiplier Hopf algebra, Separability idempotent. 1

2 BYUNG-JAY KAHNG AND ALFONS VAN DAELE idempotent element restricts the choice of the subalgebra B and the weight ν on it. On the other hand, our framework is less burdened with the technical difficulties that accompany the measured quantum groupoids, which is based on the notion of a fiber product of von Neumann algebras over a relative tensor product of Hilbert spaces. There is really no corresponding notion for a fiber product in the C -algebra theory, and therefore, there has been no C -algebraic theory of quantum groupoids developed for non-unital noncompact) cases. Our framework, while restrictive, achieves this. In addition, as can be seen in what follows, the theory is rich in nature. The authors believe that the current framework will provide a nice bridge, until a more general theory of C -algebraic locally compact quantum groupoids can be developed in the future. Here is how this paper is organized. In Section 1, we review the notations and state our main definition, revisiting Definition 4.8 of Part I [13]. As this paper is a continuation of the previous paper Part I), we refer to that paper for details and proofs. In Section, we further investigate the consequences of the right/left invariance conditions of our Haar weights ψ, ϕ. We construct four unbounded maps, Q R, Q ρ, Q L, Q λ, which will play useful roles throughout the paper. In Section 3, we construct the partial isometries V and W, using the invariance properties developed in Section. These operators are essentially the right and the left regular representations, and they play similar roles as the multiplicative unitary operators in the quantum group case. In Section 4, we carry out the construction of the antipode map S. We first construct a Hilbert space operator K implementing the antipode. Both the right and the left Haar weights play significant roles. Similar to the quantum group case, our antipode map S would be defined in terms of its polar decomposition. As the operators V and W are no longer unitaries, however, the arguments need to be modified and generalized accordingly. The discussion on the antipode map is continued in Section 5, where we collect some useful formulas and properties involving S antipode), σ modular group for ϕ), τ scaling group ). Along the way, we obtain some alternative characterizations of the antipode map, which do not explicitly involve the weights ψ or ϕ. We also explore the restrictions of S, σ, σ ψ, τ to the base algebra level, gathering some useful results. As a consequence, we establish that the weight ν on B is quasi-invariant. With the construction of the regular representations and the antipode, we can say that what we have is indeed a valid framework for locally compact quantum groupoids. This framework contains all locally compact quantum groups [16], [17], [19], [8]), and extends the notions like weak Hopf C -algebras [], [3]), generalized Kac algebras [3]), and finite quantum groupoids [0], [7]). Some other examples include the face algebras [10]), the linking quantum groupoids [5], [6]), or the partial compact quantum

3 LOCALLY COMPACT QUANTUM GROUPOIDS 3 groups [8], [7]). See section 5 of Part I for some brief discussions on these examples. The series will continue in Part III [14]. In that paper, we plan to construct the dual object, which is also a locally compact quantum groupoid of our type. We aim to show there that our class of quantum groupoids is self-dual, by obtaining a Pontryagin-type duality result. Acknowledgments. The current work began during the first named author Kahng) s sabbatical leave visit to the University of Leuven during 01/013. He is very much grateful to his coauthor Alfons Van Daele) and the mathematics department at University of Leuven for their warm support and hospitality during his stay. He also wishes to thank Michel Enock for the hospitality he received while he was visiting Jussieu. The discussions on measured quantum groupoids, together with his encouraging words, were all very inspiring. Discussions with Erik Koelink Radboud) was helpful as he suggested a possible future application toward dynamical quantum groupoids. Two occasions to visit Albert Sheu Kansas) were also helpful, who was very supportive from the beginning of the project. Finally, he also acknowledges Thomas Timmermann Münster), as an on-going partner in pursuing the generalizations and applications of quantum groupoids. The numerous discussions helped to polish the current project as well as shaping the future ones. This work was partially supported by the Simons Fundation grant and the Polish Government MNiSW matching fund. 1. Definition of a locally compact quantum groupoid 1.1. Separability idempotent. The notion of a separability idempotent plays a central role in our theory. We gather here some results and properties concerning such an element, which we will use frequently in later sections. We skip the proofs. Instead, we refer the reader to sections and 3 of Part I [13], as well as the authors earlier paper on the subject [1]. Let B be a C -algebra, and ν a KMS weight on B. This means that ν is faithful, lower semi-continuous, and semi-finite, together with a certain 1-parameter group of automorphisms σ ν t ) t R, the modular automorphism group, satisfying the KMS property. See [4]. See also section 1 of Part I. Consider another C -algebra C = B op, with a -anti-isomorphism R = R BC : B C. A self-adjoint idempotent element E MB C) is referred to as a separability idempotent, if ν id)e) = 1 and ν id) Eb 1) ) = R σ ν i/ b)), b Dσ ν i/ ). See Definition.1 in Part I. Using the -anti-isomorphism R, we can also define a KMS weight µ on C, by µ = ν R 1. We have σ µ t = R σν t R 1, and id µ)e) = 1.

4 4 BYUNG-JAY KAHNG AND ALFONS VAN DAELE Consider a densely-defined map γ B : B C, such that Dγ B ) = Dσi/ ν ) and defined by γ B b) := R σi/ ν )b) = σµ i/ R)b). It is a closed injective map, has a dense range, and is anti-multiplicative: γ B bb ) = γ B b )γ B b). Its inverse map is γ 1 B : C B, such that Dγ 1 B ) = Ranγ B) = Dσ µ i/ ) and given by γ 1 B = σν i/ R 1 = R 1 σ µ i/. It is also a closed, densely-defined, injective map, has a dense range, and is anti-multiplicative. Similarly, there exists a closed, densely-defined, injective map γ C from C into B, such that Dγ C ) = Dσ µ i/ ) and given by γ C = R 1 σ µ i/ = σi/ ν R 1. It also has a dense range and is anti-multiplicative. Its inverse map is γ 1 C = σµ i/ R = R σν i/, again closed, densely-defined, injective, having a dense range, and anti-multiplicative. Lemma 1.1. The maps γ B : B C and γ C : C B defined above are closed, densely-defined, injective anti-homomorphisms, having dense ranges. Moreover, 1) The γ B, γ C maps satisfy the following relations: Eb 1) = E 1 γ B b) ), b Dγ B ), E1 c) = E γ 1 B c) 1), c Dγ 1 B ), 1 c)e = γ C c) 1 ) E, c Dγ C ), b 1)E = 1 γ 1 C b)) E, b Dγ 1 C ). ) We have: γ C γb b) ) = b, for b DγB ), and γ B γc c) ) = c, for c Dγ C ). Proof. See Propositions.4,.5,.7 in Part I. For more details, see [1]. It can be shown that E MB C) is full see Propositions 3.3, 3.4, 3.5 in [1]), satisfying certain density conditions. See also Proposition.8 of Part I. This means that the left leg of E is B and the right leg of E is C. There exist a few different characterizations of E. For instance, we have: σt ν σ µ t )E) = E, t see Proposition.10 in Part I). Meanwhile, it can be shown that γ B γ C )E) is bounded, and that γ B γ C )E) = σe and γ C γ B )σe) = E, where σ denotes the flip map on MB C). As a consequence, we also have: R 1 R)σE) = E and R R 1 )E) = σe. See Proposition.1 in Part I. The separability idempotent condition for E is actually a condition on the pair B, ν), as E is uniquely determined by B, ν). See Proposition. in Part I. It can be shown that B must be postliminal, and the choice of ν cannot be arbitrary. For more details on all these, refer to [1]. 1.. Definition of a locally compact quantum groupoid. In what follows, we re-state Definition 4.8 from Part I. See paragraphs following the definition for some additional comments.

5 LOCALLY COMPACT QUANTUM GROUPOIDS 5 Definition 1.. The data A,, E, B, ν, ϕ, ψ) defines a locally compact quantum groupoid, if A is a C -algebra. : A MA A) is a comultiplication on A. B is a non-degenerate C -subalgebra of MA). ν is a KMS weight on B. E is the canonical idempotent of A, ) as in Definition 3.7 of Part I. That is, 1) A)A A) is dense in EA A) and A A) A) is dense in A A)E; ) there exists a C -subalgebra C = B op contained in MA), with a -anti-isomorphism R = R BC : B C, so that E MB C) and the triple E, B, ν) forms a separability triple see subsection 1.1); 3) E 1 and 1 E commute, and we have: id )E) = E 1)1 E) = 1 E)E 1) = id)e). ϕ is a KMS weight, and is left invariant. ψ is a KMS weight, and is right invariant. There exists a unique) one-parameter group of automorphisms θ t ) t R of B such that ν θ t = ν and that σ ϕ t B = θ t, t R. In the above, the comultiplication is a -homomorphism from A into MA A), not necessarily non-degenerate. It is assumed to satisfy a weak coassociativity condition, and there is also a certain density condition is full ). See Definition 3.1 and Lemma 3. in Part I. Without the non-degeneracy, there is no straightforward way of extending to MA). However, it turns out that Condition 1) for the idempotent element E can be used to make the extension possible, and the following coassociativity condition is shown to hold: id) a) = id ) a), a A. 1.1) See Theorem 3.5 in Part I. Among other consequences of Condition 1) on E, we note that E a) = a = a)e, a A. 1.) See Proposition 3.3 in Part I. In fact, E = 1 MA) ), the image of 1 MA) under the extended comultiplication map. However, without the nondegeneracy of the comultiplication map, we have E 1 1 in general. As for Condition ), the separability idempotent condition, refer to the brief discussion given in the previous subsection. Condition 3) for E is referred to as the weak comultiplicativity of the unit. Parts of it can be proved for instance, Proposition 3.6 in Part I) and it is automatically true in the ordinary groupoid case with E = 1), but the condition as a whole does not follow from other axioms.

6 6 BYUNG-JAY KAHNG AND ALFONS VAN DAELE As a consequence of the existence of the canonical idempotent E, we can show that the C -algebras B and C commute, and so do MB) and MC). See Proposition 3.8 in Part I. We also have: y = E1 y) = 1 y)e, y MB), 1.3) x = x 1)E = Ex 1), x MC). 1.4) See Proposition 3.9 and its Corollary in Part I. The canonical idempotent is uniquely determined by the Conditions 1), ), 3) above. Meanwhile, the left/right invariance of ϕ and ψ means the following: For any a M ϕ, we have a M id ϕ and id ϕ) a) MC). For any a M ψ, we have a M ψ id and ψ id) a) MB). See section 4 of Part I for more discussion on the left/right invariance, and some other properties of ϕ and ψ. In particular, we note the following proposition. It shows the relationships between the weights ϕ, ψ with the expressions id ϕ) x) and ψ id) x), which are in fact operator-valued weights: Proposition 1.3. We have: ν ψ id) x) ) = ψx), for x M ψ. µ id ϕ) x) ) = ϕx), for x M ϕ. Proof. See Proposition 4.9 in Part I. Finally, note the last condition of Definition 1., concerning ν. In fact, it is possible to prove using the other axioms that the σ ϕ t leave B invariant and that σ ϕ t B) t R is an automorphism group of B. In the definition, we are further requiring that ν is invariant under σ ϕ t B that is, ν σ ϕ t B = ν). This additional condition is necessary later, when we show that ν is quasiinvariant. More discussion on this will be given in Section 5 below.. Some alternative formulations of the left/right invariance Fix a locally compact quantum groupoid A,, E, B, ν, ϕ, ψ), as given in Definition 1.. In what follows, the left/right invariance properties of ϕ, ψ are needed in a fundamental way. In preparation, it is quite helpful to introduce certain densely-defined maps on A A..1. Definition of the maps Q R, Q ρ, Q L, Q λ. Let us begin with a lemma, which is a consequence of the weights ϕ, ψ being faithful. Lemma.1. The following subspaces are norm-dense in A: span { id ϕ) a )1 x)) : a, x N ϕ }, span { id ϕ)1 x ) a)) : a, x N ϕ }, span { ψ id) a )y 1)) : a, y N ψ }, span { ψ id)y 1) a)) : a, y N ψ }.

7 LOCALLY COMPACT QUANTUM GROUPOIDS 7 Proof. For a N ϕ, we know a N id ϕ by the left invariance of ϕ See Proposition 4. in Part I.). For any x N ϕ, we have a )1 x) M id ϕ, so we can see that id ϕ) a )1 x) ) A. Suppose θ A such that θ id ϕ) a )1 x)) ) = 0, a, x N ϕ. But, θ id ϕ) a )1 x)) ) = ϕ θ id) a )1 x)) ) = ϕ θ id) a ))x ). Since ϕ is faithful, and since x is arbitrary in N ϕ, a dense subspace of A, it follows that θ id) a ) ) x = 0, or θ id) a )1 x) ) = 0, for any a, x N ϕ. So, for any ω A and a, x N ϕ, we will have: θ id ω) a )1 x)) ) = ω θ id) a )1 x)) ) = 0..1) By definition, the comultiplication is full see Definition 3.1 and Lemma 3. in Part I). Together with the fact that N ϕ is dense in A, this means that the elements id ω) a )1 x)), for a, x N ϕ, span a dense subspace in A. Therefore, Equation.1) shows that actually θ 0. By the Hahn Banach argument, we can thus conclude that the elements id ϕ) a )1 x) ), a, x N ϕ, span a dense subspace in A. The other three density results can be proved in the same way. Now let b A be arbitrary, and consider p = id ϕ) a )1 x) ), for a, x N ϕ. We just saw from Lemma.1 that such elements span a dense subset in A. For p b = id id ϕ) 13 a )1 b x) ), define: Q R p b) = Q R id id ϕ) 13 a )1 b x)) ) := id id ϕ) 13 a )1 E)1 b x) )..) By the left invariance of ϕ, we have 13 a) N id id ϕ. Since x N ϕ, the expression in the right side is valid. Nevertheless, at this stage we do not know whether Q R determines a well-defined map. So, to motivate the choice of Q R as well as to show that it is indeed a well-defined map, let us, for the time being, assume that b T ν. Here, T ν is the Tomita subalgebra of B, which is dense in B. Consider the following lemma, where we are using the γ B, γ C maps we reviewed in subsection 1.1. Lemma.. Suppose b T ν. Then b Dγ B ) Dγ 1 C ), and we have: E 13 1 E)1 b 1) = E 13 id γ C id) 1 γ 1 C b) 1)E 1)). Proof. By the property of E, it is known that γ C γ B )σe) = E, where σ denotes the flip map. Since b Dγ B ), and since the maps γ B, γ C are anti-homomorphisms, we thus have: γ C γ B ) 1 b)σe) ) = E 1 γ B b) ) = Eb 1). The last equality is using Lemma 1.1 It follows that E 13 1 E)1 b 1) = E 13 1 γc γ B )1 b)σe)) ). Using again Eb 1) = E 1 γ B b) ), but this time for E 13, the right side of the equation becomes: E 13 id γ C id) b 1 1)E 1) ). Since b Dγ 1 C )

8 8 BYUNG-JAY KAHNG AND ALFONS VAN DAELE as well, we can now use b 1)E = 1 γ 1 C b)) E. We thus have: E 13 1 E)1 b 1) = E 13 id γ C id) 1 γ 1 C b) 1)E 1)). Remark. Formally, the right hand side of the equation in the Lemma may be written as E 13 id γc )E) 1 ) 1 b 1). Since b is arbitrary, the result of the lemma would mean that E 13 1 E) = E 13 F 1 1), where F 1 := id γ C )E). This exact formula appears in the weak multiplier Hopf algebra theory see Proposition 3.17 in [9] and Proposition 4.6 in [30]). In our case, unlike in the purely algebraic framework, the F 1 map would be unbounded because γ C is), and so we needed to treat this with care. If we assume that p A is as above and b T ν, then Equation.) for Q R p b) becomes: Q R p b) = id id ϕ) 13 a )1 E)1 b x) ) = id id ϕ) 13 a )E 13 1 E)1 b 1)1 1 x) ) = id γ C ϕ) 13 a )1 γ 1 C b) 1)E 1)1 1 x)) = id γ C ) id id ϕ) 13 a )1 1 x))1 γ 1 C b))e) = id γ C ) p 1)1 γ 1 C b))e). The third equality is using the previous Lemma. We also used the fact that a = a)e, as given in Equation 1.). Note that formally, with the notation for F 1 as in Remark above, the result can be written as: Q R p b) = p 1)F 1 1 b). This observation explains where the definition of Q R comes from. In addition, it is now easy to see that if p b = 0, then Q R p b) = 0. So, for p of the type given in Equation.) and b T ν, we see that Q R : p b Q R p b) is a well-defined linear map. For any s MA), we can make sense of Q R sp b), by letting it to be s 1) Q R p b). On the other hand, observe that Equation.) can be written as: Q R p b) = id id ϕ) 13 a )1 E)1 1 x) ) 1 b). So for any d A, we can make sense of Q R p bd), by letting it to be Q R p b) 1 d). We summarize what we learned so far in the next proposition: Proposition.3. 1) Equation.) determines a densely-defined map on A A, and the following dense subset of A A forms a core for the map Q R : span { sp b : p = id ϕ) a )1 x)), a, x N ϕ ; s MA); b A }.

9 LOCALLY COMPACT QUANTUM GROUPOIDS 9 ) The following properties hold: Q R sr b) = s 1) Q R r b), for s MA) and r b DQ R ). Q R r bd) = Q R r b) 1 d), for d MA) and r b DQ R ). Proof. By the discussion given above, the definition of Q R is valid for elements of the form sp bd A A, where s, p, are as above, with b T ν and d A. On the other hand, since T ν is dense in B and since B is a non-degenerate subalgebra of MA), any element of A can be approximated by the elements of the form bd, for b T ν, d A. Looking at the way the Q R map is defined, it is evident that it can be naturally defined for elements of the form sp b A A, for any b A. Such elements span a dense subset in A A, which will form a dense core for Q R. It is also clear from the previous paragraph that the two properties in ) hold. Observe now that Q R is an idempotent map: Proposition.4. We have: Q R Q R = Q R. Proof. Let p = id ϕ) a )1 x)), for a, x N ϕ, and let b A. Then Q R Q R p b) = Q R id id ϕ) 13 a )1 E)1 b x)) ) = Q R id id ϕ) 13 a )1 E)1 E)1 b x)) ) = Q R p b), because E = E. It follows that Q R Q R = Q R. In the next proposition, we write down a characterization of the map Q R, which will be useful later: Proposition.5. If a, x N ϕ and b A, then 1) 13 a )1 b x) DQ R id); ) Q R id) 13 a )1 b x) ) = 13 a )1 E)1 b x). Proof. By Proposition.3, we know that id id ϕ) 13 a )1 b x) ) is contained in DQ R ). But, this immediately means that 13 a )1 b x) DQ R id) and that Q R id) 13 a )1 b x) ) M id id ϕ. And, id id ϕ) Q R id) 13 a )1 b x)) ) = Q R id id ϕ) 13 a )1 b x)) ) = id id ϕ) 13 a )1 E)1 b x) ). This is true for any x N ϕ, while ϕ is a faithful weight. It follows that Q R id) 13 a )1 b x) ) = 13 a )1 E)1 b x). We may modify Equation.) a little to define three other denselydefined idempotent maps Q ρ, Q L, Q λ, which satisfy results analogous to Propositions.3,.4,.5 above. Since the method is essentially no different, we will skip the details and collect the results in three propositions below:

10 10 BYUNG-JAY KAHNG AND ALFONS VAN DAELE Proposition.6. 1) Let p = id ϕ) 1 x ) a) ), for a, x N ϕ, and let b A, s MA). Define: Q ρ ps b) := id id ϕ) 1 b x )1 E) 13 a) ) s 1). This determines a densely-defined map on A A into itself. ) The following properties hold: Q ρ rs b) = Q ρ r b) s 1), for s MA) and r b DQ ρ ). Q ρ r db) = 1 d) Q ρ r b), for d MA) and r b DQ ρ ). 3) Q ρ is an idempotent map. That is, Q ρ Q ρ = Q ρ. 4) If a, x N ϕ and b A, then 1 b x ) 13 a) DQ ρ id), and we have: Q ρ id) 1 b x ) 13 a) ) = 1 b x )1 E) 13 a). Proposition.7. 1) Let q = ψ id) a )y 1) ), for a, y N ψ, and let c A, s MA). Define: Q L c sq) := 1 s) ψ id id) 13 a )E 1)y c 1) ). This determines a densely-defined map on A A into itself. ) The following properties hold: Q L c sr) = 1 s) Q L c r), for s MA) and c r DQ L ). Q L cd r) = Q L c r) d 1), for d MA) and c r DQ L ). 3) Q L is an idempotent map: Q L Q L = Q L. 4) If a, y N ψ and c A, then 13 a )y c 1) Did Q L ), and we have: id Q L ) 13 a )y c 1) ) = 13 a )E 1)y c 1). Proposition.8. 1) Let q = ψ id) y 1) a) ), for a, y N ψ, and let c A, s MA). Define: Q λ c qs) := ψ id id) y c 1)E 1) 13 a) ) 1 s). This determines a densely-defined map on A A into itself. ) The following properties hold: Q λ c rs) = Q λ c r) 1 s), for s MA) and c r DQ λ ). Q λ dc r) = d 1) Q λ c r), for d MA) and c r DQ λ ). 3) Q λ is an idempotent map: Q λ Q λ = Q λ. 4) If a, y N ψ and c A, then y c 1) 13 a) Did Q λ ), and we have: id Q λ ) y c 1) 13 a) ) = y c 1)E 1) 13 a). These maps are very closely related, as one can imagine. The following result will be useful later. Proposition.9. 1) Let p = id ϕ) 1 x ) a) ), for a, x N ϕ, and let b A. Then we have: Q ρ p b) = Q R p b ). ) Let q = ψ id) y 1) a) ), for a, y N ψ, and let c A. Then we have: Q λ c q) = Q L c q ).

11 Proof. Straightforward. LOCALLY COMPACT QUANTUM GROUPOIDS 11 Remark. 1). A different way to look at all these results is that the case of Q ρ is when we instead work with the opposite C -algebra A op ; Q L is when we work with A, cop ) so that the roles of the left/right Haar weights are reversed; Q λ is when we work with A op, cop ). ). The four maps Q R, Q ρ, Q L, Q λ are unbounded maps and they are only densely-defined on A A. For those readers who are familiar with the theory of weak multiplier Hopf algebras [9], [30], we pointed out earlier that the Q R map plays the role of the map p b p 1)F 1 1 b). Similarly, the Q ρ map would correspond to p b 1 b)f 3 p 1); the Q L map would correspond to c q 1 q)f 4 c 1); and the Q λ map would correspond to c q c 1)F 1 q). However, in our case, these expressions are not very useful and we will not mention them except in formal settings, because F 1, F, F 3, F 4 are unbounded elements... Left/Right invariance of ϕ, ψ and the maps Q R, Q ρ, Q L, Q λ. We wish to show that the maps Q R, Q ρ, Q L, Q λ satisfy some very useful properties, as a consequence of the invariance properties of the Haar weights ϕ and ψ. Let a M ϕ. Then by the left invariance of ϕ, we know a M id ϕ and id ϕ) a) MC). Apply here. By Equation 1.4), we have: id ϕ) a) ) = E id ϕ) a) 1 ) = id id ϕ) E 1) 13 a) ). On the other hand, by the coassociativity of extended to the multiplier algebra level), we also have: id ϕ) a) ) = id id ϕ) id) a) ) = id id ϕ) id ) a) ). Combining the two results, we see that id id ϕ) id ) a) ) = id id ϕ) E 1) 13 a) ), a M ϕ. Let c, y A and multiply y c to the equation above, from left. Then, the two sides become: LHS) = id id ϕ) 1 c 1)id )y 1) a)) ), RHS) = id id ϕ) y c 1)E 1) 13 a) ). Now, let θ A be arbitrary and apply θ id to both sides. Then we have: id ϕ) c 1) θ id)[y 1) a)]) ) = id ϕ) θ id id)[y c 1)E 1) 13 a)] )..3) In the right hand side of Equation.3), recall the earlier characterization of the Q λ map given in Proposition.8 4). Assuming the choice of a, c, y are such that y c 1) 13 a) Did Q λ ), this means that the right side of Equation.3) is same as = id ϕ) θ Q λ )[y c 1) 13 a)] ).

12 1 BYUNG-JAY KAHNG AND ALFONS VAN DAELE So Equation.3) now becomes: id ϕ) c 1) θ id)[y 1) a)]) ) = id ϕ) Q λ θ id id)[y c 1) 13 a)]) ) = id ϕ) Q λ c θ id)[y 1) a)]) ). For convenience, write q := θ id)[y 1) a)], an element in A. We can then write the result above as: id ϕ) c 1) q) ) = id ϕ) Q λ c q) )..4) The elements of the form q = θ id)[y 1) a)] span a dense subset in A is full ). So Equation.4) will remain valid as long as the expression makes sense. This would happen when c 1) q) M id ϕ ; c q DQ λ ); and Q λ c q) M id ϕ. See Proposition.10 below: Proposition.10. As a consequence of the left invariance of ϕ, we have: 1) Let c, q A be such that c 1) q) M id ϕ ; c q DQ λ ); and Q λ c q) M id ϕ. Then Equation.4) holds true. Namely, id ϕ) c 1) q) ) = id ϕ) Q λ c q) ). ) In particular, let p = ψ id) x 1) r w) ), for x, w N ψ, r N ϕ, and let s N ϕ. Let c A be arbitrary. Then we have: id ϕ) c 1) ps) ) = id ϕ) Q λ c ps) )..5) Proof. We already discussed 1) in previous paragraphs. As for ), note first that since N ψ is a left ideal in A, so r w N ψ, from which it follows that c ps DQ λ ) by Proposition.8. Meanwhile, since N ϕ is a left ideal in A, we have r w N ϕ. So r w) N id ϕ by the left invariance of ϕ), which in turn means that p N ϕ. Since s N ϕ, we have ps M ϕ. From this observation, we can conclude quickly that c 1) ps) M id ϕ and also that Q λ c ps) M id ϕ. By 1), or by Equation.4), the result follows. The next three propositions are similar in nature, corresponding to maps Q L, Q ρ, Q R, respectively. The proofs are skipped, as they are essentially no different from the above discussion. Proposition.11. As a consequence of the left invariance of ϕ, we have: 1) Let c, q A be such that q)c 1) M id ϕ ; c q DQ L ); and Q L c q) M id ϕ. Then: id ϕ) q)c 1) ) = id ϕ) Q L c q) )..6) ) In particular, let p = ψ id) w r)x 1) ), for x, w N ψ, r N ϕ, and let s N ϕ so s N ϕ). Let c A be arbitrary. Then: id ϕ) s p)c 1) ) = id ϕ) Q L c s p) )..7) Proposition.1. As a consequence of the right invariance of ψ, we have:

13 LOCALLY COMPACT QUANTUM GROUPOIDS 13 1) Let b, p A be such that 1 b) p) M ψ id ; p b DQ ρ ); and Q ρ p b) M ψ id. Then: ψ id) 1 b) p) ) = ψ id) Q ρ p b) )..8) ) In particular, let q = id ϕ) 1 s ) y r) ), for r, s N ϕ, y N ψ, and let w N ψ. Let b A be arbitrary. Then we have: ψ id) 1 b) qw) ) = ψ id) Q ρ qw b) )..9) Proposition.13. As a consequence of the right invariance of ψ, we have: 1) Let b, p A be such that p)1 b) M ψ id ; p b DQ R ); and Q R p b) M ψ id. Then: ψ id) p)1 b) ) = ψ id) Q R p b) )..10) ) In particular, let q = id ϕ) r y)1 s) ), for r, s N ϕ, y N ψ, and let w N ψ so w N ψ ). Let b A be arbitrary. Then: ψ id) w q)1 b) ) = ψ id) Q R w q b) )..11) The results collected in Propositions.10,.11,.1,.13 are all consequences of the left/right invariance properties of ϕ and ψ. They will play useful roles in what follows. Before we wrap up this subsection, we prove the following result, which sharpen the right/left invariance conditions given in Section 1. Proposition.14. We have: 1) B = span { ψ id) k) : k M ψ } ) C = span { id ϕ) k) : k M ϕ } Proof. 1). Consider q = id ϕ) r y)1 s) ), where r, s N ϕ, y M ψ. This is the type of an element considered in Propositions.1 and.13. Since C is a non-degenerate subalgebra in MA), we may, without loss of generality, assume that s = ck, for c C and k N ϕ. In addition, let b λ ) be an approximate unit for A. Then we have: ψ id) q) = lim λ ψ id) q)1 b λ ) ) = lim λ ψ id) Q R q b λ ) ), by Proposition.13. Computing further, now using the definition of the Q R map given in Equation.), we have: ψ id) q) = lim λ ψ id) id id ϕ)[ 13 r y)1 E)1 b λ s)] ) = ψ id) id id ϕ)[ 13 r y)1 E)1 1 ck)] ) = id ϕ) ψ id id) 13 r y))e1 ck) ) = id ϕ) 1 x)e1 ck) ), where in the last line, we wrote x = ψ id) r y) ), which is a valid element contained in N ϕ, because r N ϕ, y M ψ. Next, write ω := ϕx k). This

14 14 BYUNG-JAY KAHNG AND ALFONS VAN DAELE is a linear functional in MA) C, as C is a subalgebra of MA). In this way, we see that ψ id) q) = id ω) E1 c) ). Since c C and ω C, this is an element contained in B Here, we are using the fact that the left leg of E is B. See Proposition.10 in Part I.). By Lemma.1, we know the elements of the form q above are dense in M ψ. It follows that ψ id) k) B, for any k M ψ. For the opposite inclusion, suppose θ B is such that θ ψ id) k) ) = 0, k M ψ..1) Without loss of generality, we may assume that θ = ν b), b M ν, where ν is the KMS weight on B. Note that { ν b) : b M ν } is dense in B. Meanwhile, since b B, we know from Equation 1.3) that b = E1 b). Putting these together, the left side of Equation.1) becomes: = ν ψ id) k)1 b)) ) = ν ψ id) kb)) ) = ψkb). In the last equality, we used the result of Proposition 1.3. In other words, Equation.1) is none other than saying that ψkb) = 0, k M ψ. By the faithfulness of ψ, this would mean that b = 0. So θ 0. This shows that the elements of the form ψ id) k) are dense in B. ). The result about C is similarly proved. 3. The operators V and W : The right and regular representations In the theory of locally compact quantum groups [16], [17], [19], [8], a fundamental role is played by the multiplicative unitary operators in the sense of Baaj and Skandalis [1]; see also [31]), as they are essentially the right/left regular representations of the quantum group. In the setting of measured quantum groupoids [18], [9], similar roles are played by the pseudo-multiplicative unitary operators see also [5]). In our setting, analogous roles will be played by certain partial isometries the multiplicative partial isometries ). However, unlike in the cases of quantum groups or measured quantum groupoids, these operators are in general not unitaries. This causes some subtle issues that are not present in the quantum group theory. The tools developed in Section will be useful Defining the operators V and W. For our discussion, assume the existence of a proper weight η on our C -algebra A and fix a GNS-construction H, π, Λ) corresponding to η. We will identify A = πa) BH). Depending on the context, we may later let η = ψ, η = ϕ, or something else. Let e j ) j J be an orthonormal basis for H. We will also use the standard notation ω ξ,ζ, for ξ, ζ H, to denote the linear form defined by ω ξ,ζ T ) = T ξ, ζ, T BH). Note that any ω BH) can be approximated by the ω ξ,ζ. See Lemma 3.1 below for some useful results:

15 LOCALLY COMPACT QUANTUM GROUPOIDS 15 Lemma 3.1. Let ξ, ζ H. Then: 1) ω ξ,ζ = ω ζ,ξ ; ) id ω ξ,ζ )T ) = id ω ζ,ξ )T ), for T BH H); 3) For S, T BH), we have: ω ej,ζs)ω ξ,ej T ) = ω ξ,ζ ST ); j J 4) For S, T BH H), we have: id ω ej,ζ)s)id ω ξ,ej )T ) = id ω ξ,ζ )ST ). j J Remark. We will skip the proof of the lemma, as these are essentially basic linear algebra results. Here, the complex conjugate ω for ω BH) is given by ωt ) := ωt ), for T BH). Also, the result in 4) is to be understood that the net of finite sums j I id ω e j,ζ)s)id ω ξ,ej )T )) converges, in operator norm, to id ω ξ,ζ )ST ). I F J) Let us first construct the right regular representation, in terms of the operator V. For this, consider the right Haar weight ψ and let H ψ, π ψ, Λ ψ ) be its GNS-triple. See below, where we can recognize the resemblance to the corresponding definition in the case of locally compact quantum groups [16], [17], [8]. Proposition 3.. 1) There exists a bounded operator V BH ψ H) satisfying id ω)v ) ) Λψ p) = Λ ψ id ω) p) ), for p N ψ and ω BH). ) If p N ψ and a N η, then we have: V Λ ψ p) Λa) ) = Λ ψ Λ) p)1 a) ). 3.1) Proof. 1). By the right invariance of ψ, we know id ω) p) N ψ. So the expression makes sense. Now, let ξ be arbitrary, and consider ω ξ,ej BH), where e j ) j J is an orthonormal basis for H. Then: Λ ψ id ω ξ,ej ) p)) = ψ id ω ξ,ej ) p) id ω ξ,ej ) p) ) j J j J = ψ id ω ej,ξ) p ))id ω ξ,ej ) p) ) j J ψ id ω ξ,ξ ) p p)) ), where we are using ), 4) of Lemma 3.1 and the lower semi-continuity of the weight ψ. As ψ id ω ξ,ξ ) p p)) ) = ω ξ,ξ ψ id) p p)) ), this then becomes: Λ ψ id ω ξ,ej ) p)) ψ id) p p)) ξ, ξ ψ id) p p)) ξ. j J

16 16 BYUNG-JAY KAHNG AND ALFONS VAN DAELE Because of the way V was defined, the left side of this equation is actually j J V Λ ψ p) ξ), e j, while ψ id) p p)) in the right side is a bounded element in MB), due to the right invariance of ψ. We can thus see that V : Λ ψ p) ξ V Λ ψ p) ξ) is a bounded operator. ). Let a, b N η be arbitrary and consider ω = ω Λa),Λb). Then, for any p, q N ψ, we have: V Λψ p) Λa)), Λ ψ q) Λb) = id ω Λa),Λb) )V )Λ ψ p), Λ ψ q) = Λ ψ id ω Λa),Λb) ) p)), Λ ψ q) = ψ η) q b ) p)1 a) ) = Λ ψ Λ) p)1 a)), Λ ψ q) Λb). This is true for arbitrary q N ψ and b N η, so we have: V Λ ψ p) Λa) ) = Λ ψ Λ) p)1 a) ). In the below are some immediate properties for V. Here, for convenience of the notation, we just wrote x A to represent π ψ x) BH ψ ). As it will eventually turn out that we can regard H ψ = H see last Remark in Section 5 and see Part III [14]), this casual bookkeeping is not too harmful. Similarly, we regard x = π ψ π) x) and E = π ψ π)e), as contained in BH ψ H). Proposition ) V x 1) = x)v, for any x A; ) EV = V ; 3) RanV ) RanE), in H ψ H. Proof. 1). For any p N ψ and any a N η, we have: V x 1) Λ ψ p) Λa) ) = V Λ ψ xp) Λa) ) = Λ ψ Λ) xp)1 a) ) = x)λ ψ Λ) p)1 a) ) = x)v Λ ψ p) Λa) ), where we used the property of the GNS representation and Equation 3.1). ). We know E p) = p for any p A. So we have, for any p N ψ and any a N η, EV Λ ψ p) Λa) ) = EΛ ψ Λ) p)1 a) ) = Λ ψ Λ) E p)1 a) ) = Λ ψ Λ) p)1 a) ) = V Λ ψ p) Λa) ). 3). Since EV = V, the result is immediate. Note here that the space RanE) is already closed in H ψ H because E is a projection. In an analogous way, we can also construct the left regular representation, in terms of the operator W. For this, consider now the left Haar weight ϕ and let H ϕ, π ϕ, Λ ϕ ) be its GNS-triple. We actually define W first, as is typically done in the quantum group case. See Proposition 3.4 below.

17 LOCALLY COMPACT QUANTUM GROUPOIDS 17 Proposition 3.4. There exists a bounded operator W BH H ϕ ) characterized and defined by θ id)w ) ) Λ ϕ a) = Λ ϕ θ id) a) ), for a N ϕ and θ BH). If a N ϕ and p N η, then we have: W Λp) Λ ϕ a) ) = Λ Λ ϕ ) a)p 1) ). 3.) Proof is essentially no different than in the case of Proposition 3.. We also have the following result, analogous to Proposition 3.3. With a similar comment as before, we are regarding x = π ϕ x) BH ϕ ), while x = π π ϕ ) x) and E = π π ϕ )E), as elements contained in BH H ϕ ). Proposition ) W 1 x) = x)w, for any x A; ) EW = W ; 3) RanW ) RanE), in H H ϕ. In the quantum group case [16], [17], [8], we have E = 1 1 and it is known that V and W are unitaries. In our case E 1 1), we can see from Propositions 3.3 and 3.5 that this is no longer true. It turns out that V and W are partial isometries such that RanV ) = RanE) for V, and RanW ) = RanE) for W. However, the proof of is rather complicated and needs more work. This is what we aim to establish in the following two subsections. 3.. V is a partial isometry. Let us begin with a result that will later help us to understand how the operator V behaves: Proposition 3.6. Consider a = id ϕ) r y)1 s) ), where r, s N ϕ and y N ψ. Then a N ψ. Consider also b N η. Let z = id id ϕ) 13 r y) 3 s) ) N ψ id. Then we have: z1 b) = id id ϕ) 13 r y) 3 s)1 b 1) ). With the notation as above, we have: V Λ ψ Λ)z1 b)) ) = E Λ ψ a) Λb) ). Proof. As y N ψ, it is easy to observe, by the right invariance of ψ, that a N ψ and z N ψ id. We also have z1 b) N ψ η. Suppose w N ψ so w N ψ ) and c N η so c N η). Then by the definition of the operator V, as in Equation 3.1), we have: V Λψ Λ)[z1 b)]), Λ ψ w) Λc) = ψ η) w c )id id ϕ) id)[ r y)] 3 s)1 b 1)) ). By the coassociativity of, the right side becomes: = ψ η) w c )id id ϕ)id )[ r y)] 3 s)1 b 1)) ).

18 18 BYUNG-JAY KAHNG AND ALFONS VAN DAELE This is equal to = ψ η ϕ) w c 1)id )[ r y)1 s)]1 b 1) ) = η ϕ) c 1) ψ id)[w 1) r y)]s)b 1) ) = η ϕ) c 1) ps)b 1) ) = η id ϕ)c 1) ps)) b ), where we wrote p = ψ id)[w 1) r y)]. Apply here the result of Proposition.10, a consequence of the left invariance of ϕ. All the conditions for Equation.5) are met, so we have: id ϕ) c 1) ps) ) = id ϕ) Q λ c ps) ). It follows that we have: V Λψ Λ)[z1 b)]), Λ ψ w) Λc) = η id ϕ)q λ c ps)) b ). But by the definition of the Q λ map at the algebra level as given in Proposition.8, and remembering the definition of p above, we have: Q λ c ps) = ψ id id) w c 1)E 1) 13 r y) ) 1 s). Therefore, combining all these together, we have: V Λψ Λ)[z1 b)]), Λ ψ w) Λc) = η ψ id ϕ)[w c 1)E 1) 13 r y)1 1 s)] b ) = ψ η) w c )E id id ϕ)[ 13 r y)1 1 s)] 1 b) ) = ψ η) w c )Ea b) ) = Λ ψ Λ)Ea b)), Λ ψ w) Λc) = EΛ ψ a) Λb)), Λ ψ w) Λc). Third equality is just remembering that a = id ϕ) r y)1 s) ), and the last equality is the property of the GNS-representation. Since this is true for arbitrary w N ψ and c N η, we have shown the desired result. Remark. In the quantum group theory, a result analogous to the above proposition helps us to see that the map Λ ψ a) Λb) Λ ψ Λ) z1 b) ) determines the adjoint) operator V. It will turn out later that this fact is still true even in our more general setting, but at this stage with E 1 1), this is not so obvious. Define now a subspace K ψ of H ψ, as follows: K ψ := span { Λ ψ id ϕ)[ r y)1 s)]) : y N ψ, r, s N ϕ }. 3.3) Then the result of Proposition 3.6 implies the following: Proposition 3.7. EK ψ H) V K ψ H). Proof. With the notation as in Proposition 3.6, we see that Λ ψ a) K ψ, and such elements span a dense subspace of K ψ. Write z N ψ id as in that proposition, and let b N η.

19 LOCALLY COMPACT QUANTUM GROUPOIDS 19 Note that using basically the same tedious but straightforward) method as in Proposition A.9 in [16], we can show: Λ ψ Λ) z1 b) ) = j J Λ ψ id ωλb),ej )z) ) e j, where e j ) j J is an orthonormal basis for H. With the definition of z, this becomes: = j J Λ ψ id ϕ)[ r y)1 ω Λb),ej id) s))] ) e j. Expressed in this way, we can see that Λ ψ Λ) z1 b) ) K ψ H. By the result E Λ ψ a) Λb) ) = V Λ ψ Λ)z1 b)) ) from Proposition 3.6, we see that EK ψ H) V K ψ H). Remark. While we know from Lemma.1 that the elements of the form id ϕ)[ r y)1 s)] span a dense subspace in A, this does not necessarily mean that the Λ ψ id ϕ)[ r y)1 s)]) span a dense subspace in H ψ. The norms involved operator norm versus Hilbert space norm) are different. So, at this stage, the result of the proposition saying that EK ψ H) V K ψ H) is not yet enough to argue that RanV ) RanE). We will have to give a separate proof that we indeed have K ψ = H ψ. This is what we will achieve in this subsection. Proposition 3.8. Let K ψ H ψ be as defined in Equation 3.3). Then K ψ = span { Λ ψ id ω) x)) : x N ψ, ω A }. Proof. Consider s = aσ ϕ i b ), for a, b T ϕ the Tomita subalgebra). Such elements are dense in T ϕ and M ϕ. Then for any x A, we have: ϕxs) = ϕ xaσ ϕ i b ) ) = ϕb xa), all valid with b xa M ϕ. So, we may consider θ ) := ϕ s) as a functional contained in A. In fact, it is clear that this θ is none other than θ = ω Λa),Λb). Note that any ω A can be approximated by the ω Λa),Λb), for a, b T ϕ. Let x N ψ. Then, for ω A, we know from the definition of V see Proposition 3.) that Λ ψ id ω) x) ) = id ω)v )Λψ x). As id ω)v ) is a bounded operator, the map ω Λ ψ id ω) x) ) is a bounded map. It can be extended, as long as the expression remains valid. It follows that any Λ ψ id ω) x)), for x N ψ, ω A, can be approximated by the Λ ψ id ω Λa),Λb) ) r y)), for y N ψ, r N ϕ. Meanwhile, since ω Λa),Λb) = ϕb a), we have: Λ ψ id ω Λa),Λb) ) r y)) = Λ ψ id ϕ)[1 b ) r y)1 a)] ) = Λ ψ id ϕ)[ r y)1 s)] ).

20 0 BYUNG-JAY KAHNG AND ALFONS VAN DAELE In this way, we just showed that K ψ = span { Λ ψ id ϕ)[ r y)1 s)]) : y N ψ, r, s N ϕ } = span { Λ ψ id ω Λa),Λb) ) x)) : x N ψ, a, b T ϕ } = span { Λ ψ id ω) x)) : x N ψ, ω A }. Corollary. By the new characterization of the subspace K ψ, we can see that RanV ) = V H ψ H) K ψ H. ) Proof. For any x N ψ, recall that id ω)v )Λ ψ x) = Λ ψ id ω) x), which is contained in K ψ by Proposition 3.8. This is true for any ω BH). As the elements Λ ψ x), x N ψ, are dense in H ψ, it follows that V H ψ H) K ψ H. Combine the result of the Corollary, together with our earlier result in Proposition 3.7 that EK ψ H) V K ψ H). We then have: EK ψ H) V K ψ H) V H ψ H) K ψ H. 3.4) Next, apply E from the left, and use the fact that EV = V from Proposition 3.3) and that E = E. Then: EK ψ H) V K ψ H) V H ψ H) EK ψ H). From this, we have the following useful result: V K ψ H) = V H ψ H) = EK ψ H). 3.5) Motivated by Equations 3.4) and 3.5), let us consider U := V Kψ H, which is an operator contained in BK ψ H). Equation 3.5) means that RanU) = EK ψ H). See also the next result: Proposition 3.9. Let U := V Kψ H. Then U BK ψ H) is characterized by Λ ψ a) Λb) Λ ψ Λ) z1 b) ), where a, z, b are as in Proposition 3.6. That is, a = id ϕ) r y)1 s) ), for y N ψ, r, s N ϕ and z = id id ϕ) 13 r y) 3 s) ). Also b N η. Proof. By definition of the subspace K ψ given in Equation 3.3), we know that the elements Λ ψ a) Λb) span a dense subset in K ψ H. We would prove the result if we can show that Λψ a) Λb), V Λ ψ c) Λd)) = Λ ψ Λ)z1 b)), Λ ψ c) Λd), 3.6) for any c N ψ and any d N η. This is actually a stronger result than is necessary because Λ ψ c) Λd) H ψ H, but more convenient because V ζ ξ) = Uζ ξ), if ζ K ψ, ξ H.

21 LOCALLY COMPACT QUANTUM GROUPOIDS 1 For convenience, denote the left side of Equation 3.6) as LHS) and the right side as RHS). We have: RHS) = ψ η) c d )z1 b) ) = ψ η ϕ) c d 1) 13 r y) 3 s)1 b 1) ) = ψ η ϕ) c d 1) 13 r y)1 E) 3 s)1 b 1) ), because s = E s). Meanwhile, id id ϕ)[ 13 r y) 3 s)1 b 1)] DQ R ), because s)b 1) N id ϕ and r y N ϕ. By definition of the Q R map, we can write: RHS) = ψ η) c d ) Q R id id ϕ)[ 13 r y) 3 s)1 b 1)]) ). Recall Proposition.3 and Proposition.13, where we have: ψ id) c 1) Q R p b) ) = ψ id) Q R c p b) ) = ψ id) c p)1 b) ), for an appropriate p b. Applying this result to our case, we obtain: RHS) = η d ψ id ϕ)[ id)c 1) r y)) 3 s)1 b 1)] ). Using the coassociativity of and the fact that is a homomorphism. this becomes: RHS) = η d ψ id ϕ)[ c ) 1)id ) r y)) 3 s)1 b 1)] ) = η d ψ id ϕ) c ) 1)id )[ r y)1 s)])b ). 3.7) Let us now turn our attention to computing the LHS). Note first that by definition of V, we have: LHS) = Λ ψ a) Λb), Λ ψ Λ) c)1 d)) = ψ η) 1 d ) c )a b) ). Using a = id ϕ) r y)1 s) ), this becomes: LHS) = ψ η ϕ) 1 d 1) c ) 1) 13 r y)1 1 s)1 b 1) ). Note here that ψ id id) 1 d 1) c ) 1) 13 r y) ) DQ λ ), because 1 d ) c ) N ψ id and r y N ψ. Applying the Q λ map and using c )E = c ), we can see easily that LHS) now becomes: = η id ϕ)[q λ ψ id id)[1 d 1) c ) 1) 13 r y)])1 s)]b ). Next, use the fact from Proposition.8) that Q λ c q)1 s) = Q λ c qs), and also that id ϕ) Q λ c qs) ) = id ϕ) c 1) qs) ), from Proposition.10. Then we have: LHS) = η ψ id ϕ)1 d 1) c ) 1)id )[ r y)1 s)])b ). 3.8) Comparing Equation 3.7) with Equation 3.8), we can finally see that RHS)=LHS), thereby finishing the proof.

22 BYUNG-JAY KAHNG AND ALFONS VAN DAELE The operators U and U are adjoints of each other, and we now know how they behave. In addition, with the notation as above, we have: UU Λ ψ a) Λb) ) = U Λ ψ Λ)[z1 b)] ) = V Λ ψ Λ)[z1 b)] ) = E Λ ψ a) Λb) ). 3.9) The first equality is the result of Proposition 3.9 and the last equality is using Proposition 3.6. Since the Λ ψ a) Λb) span a dense subset in K ψ H, this shows that we have: UU = E Kψ H. See below: Proposition Recall that U = V Kψ H. We have: 1) UU = E Kψ H; ) UU U = U. This shows that U is a partial isometry in BK ψ H). Proof. 1). Already shown above. See Equation 3.9). ). We know from Proposition 3.3 that EV = V. So EU = U, because we saw from Equation 3.4) that EK ψ H) K ψ H. It follows that UU U = EU = U. It is clear that U is also a partial isometry as U UU = U ). By the general theory on partial isometries, UU is a projection onto RanUU ) = RanU) = KerU ), while U U is a projection onto RanU U) = RanU ) = KerU). These spaces are necessarily closed in K ψ H. This means that we can write: K ψ H = KerU) RanU U), as well as K ψ H = KerU ) RanUU ). In addition, U is an isometry from RanU U) onto RanUU ), and similarly, U is an isometry from RanUU ) onto RanU U). All these are standard results. We next turn our attention to showing that K ψ = H ψ. We need some preparation. Note first that Similarly, H ψ H = KerV ) KerV ) = KerV ) RanV ). 3.10) H ψ H = KerV ) RanV ). 3.11) Let us look more carefully about the subspaces RanV ) and RanV ). First, by Equation 3.5) and Proposition 3.10, we have: RanV ) = V H ψ H) = V K ψ H) = EK ψ H) = RanUU ). Meanwhile, considering Equation 3.11) and the previous equation, we have: RanV ) = V H ψ H) = V RanV )) = V RanUU )), which is actually no different than U RanUU )) = RanU ), by the property of the partial isometry U. In particular, we see that RanV ) K ψ H. We are now ready to prove our main result of this subsection that K ψ = H ψ and that V is a partial isometry in BH ψ H).

23 LOCALLY COMPACT QUANTUM GROUPOIDS 3 Theorem ). Let K ψ be the subspace of H ψ, as in Equation 3.3) and Proposition 3.8. We actually have: K ψ = H ψ. ). V is a partial isometry such that V V = E, where we regard E = π ψ π)e). In particular, we see that RanV ) is closed, and that RanV ) = RanE). Proof. 1). Suppose K ψ H ψ, and let x N ψ be such that Λ ψ x) H ψ K ψ. Then for any ξ H, we have Λ ψ x) ξ H ψ K ψ ) H. Since H ψ H = K ψ H H ψ K ψ ) H, and since we saw above that RanV ) K ψ H, the observation in Equation 3.10) means that we must have H ψ K ψ ) H KerV ). So V Λ ψ x) ξ) = 0 Hψ H. It follows that for any ζ H, we will have: id ωξ,ζ )V ) ) Λ ψ x) = V Λ ψ x) ξ), ζ = 0. By Definition 3., this means that Λ ψ id ωξ,ζ ) x) ) = 0. Since ξ, ζ H is arbitrary, this means that x = 0. This in turn means that H ψ K ψ = {0}, or K ψ = H ψ. ). Since K ψ = H ψ, it is clear that U = V BH ψ H). By Proposition 3.10, we thus conclude that V is a partial isometry such that V V = E. In particular, RanV ) is closed, and RanV ) = RanE). Since V is a partial isometry, it follows that the operator V V is also a projection. To learn more about this projection, let r, s N ϕ, y N ψ, and let p = id ϕ) r y)1 s) ). Then consider Λ ψ p) Λb) H ψ H, where b N η. By Theorem 3.11, such elements span a dense subspace in the Hilbert space H ψ H. Meanwhile, recall from Proposition.3 that p b DQ R ). Now define: G R Λψ p) Λb) ) := Λ ψ Λ) Q R p b) ). 3.1) Here are the properties of the G R map: Proposition 3.1. Let the notation be as in the previous paragraph. Then: 1) G R is self-adjoint and idempotent, so it determines a bounded operator G R BH ψ H); ) The operator G R is characterized by G R id) Λ ψ Λ Λ) 13 a)1 b x)) ) = Λ ψ Λ Λ) 13 a)1 E)1 b x) ), for a N ψ, b, x N η. 3) G R = V V ; 4) RanV ) is a closed subspace in H ψ H, and we have: RanV ) = RanG R ). Proof. 1). G R G R = G R is clear, because we know Q R Q R = Q R Proposition.4). To show that it is self-adjoint, suppose q N ψ is of a similar

24 4 BYUNG-JAY KAHNG AND ALFONS VAN DAELE type as p, that is q = id ϕ) u x)1 v) ), for u, v N ϕ, x N ψ, and consider Λ ψ q) Λd) H ψ H, where d N η. Then: GR Λ ψ p) Λb)), Λ ψ q) Λd) = ψ η) q d )Q R p b) ) = η d ψ id)[q 1)Q R p b)] ). But by Proposition.3 and Equation.11), we have: ψ id) q 1)Q R p b) ) = ψ id) Q R q p b) ) = ψ id) q p)1 b) ). Returning to the earlier equation, we now have: GR Λ ψ p) Λb)), Λ ψ q) Λd) = η d ψ id)[ q p)1 b)] ) = ψ η) 1 d ) q ) p)1 b) ). Using the same idea, we also have: Λψ p) Λb), G R Λ ψ q) Λd)) = G R Λ ψ q) Λd)), Λ ψ p) Λb) = ψ η) 1 b ) p q)1 d) ) = ψ η) 1 d ) q ) p)1 b) ). So G R Λ ψ p) Λb)), Λ ψ q) Λd) = Λ ψ p) Λb), G R Λ ψ q) Λd)). In this way, we observe that G R is symmetric and idempotent, defined on a dense subspace of the Hilbert space H ψ H. This means G R extends to an orthogonal projection, a bounded self-adjoint operator in BH ψ H). ). This follows from the characterization of the Q R map, given in Proposition.5. Unlike the case of Q R, there is no issue about the domain in this case, because G R is a bounded operator in BH ψ H). 3). From the computations in 1) above, and using the definition of the operator V earlier Proposition 3.), we can see that GR Λ ψ p) Λb)), Λ ψ q) Λd) = ψ η) 1 d ) q ) p)1 b) ) = Λ ψ Λ) p)1 b)), Λ ψ Λ) q)1 d)) = V Λ ψ p) Λb)), V Λ ψ q) Λd)) = V V Λ ψ p) Λb)), Λ ψ q) Λd). Since the Λ ψ p) Λb) span a dense subset in the Hilbert space H ψ H, we conclude that G R = V V BH ψ H). 4). This is a property of V being a partial isometry such that V V = G R. See the discussion immediately following Proposition Corollary. In a similar way, we can define a self-adjoint idempotent operator G ρ BH ψ H), given by the Q ρ map at the algebra level.

25 LOCALLY COMPACT QUANTUM GROUPOIDS W is a partial isometry. In Proposition 3.4 and Proposition 3.5, we gave the definition and some basic properties of W BH H ϕ ), which is the operator corresponding to the left regular representation. All the earlier results concerning the operator V will have analogous results for the operator W. The argument is essentially no different. However, for the purpose of clarifying the notation for the later sections, we will gather the relevant results here without proof). Proposition We have the following characterizations for the Hilbert space H ϕ. H ϕ = span { Λ ϕ ψ id)[ z r)y 1)]) : y, z N ψ, r N ϕ } = span { Λ ϕ θ id) x)) : x N ϕ, θ A }. Proposition Let p = ψ id) z r)y 1) ), where y, z N ψ and r N ϕ. Also let c N η. Then W BH H ϕ ) is characterized by W : Λc) Λ ϕ p) Λ Λ ϕ ) wc 1) ), where w = ψ id id) 13 z r) 1 y)1 c 1) ). Theorem W and W are partial isometries in BH H ϕ ), with the following properties: 1) W W = E, where we regard E = π π ϕ )E), as an element in BH H ϕ ). In particular, the subspace RanW ) is closed and we have: RanW ) = RanE) H H ϕ ). ) W W = G L, where G L BH H ϕ ) is a bounded) self-adjoint idempotent operator given by G L : Λc) Λ ϕ p) Λ Λ ϕ ) Q L c p) ). Here, we are using the same notation as in Proposition By definition of the Q L map see Proposition.7), the operator G L is thus characterized by id G L ) Λ Λ Λ ϕ ) 13 a)y c 1)) ) = Λ Λ Λ ϕ ) 13 a)e 1)y c 1) ), for a N ϕ, y, c N η. 3) We also have: RanW ) = RanG L ), as a closed subspace in H H ϕ. 4) In a similar way, we can define a self-adjoint idempotent operator G λ BH H ϕ ), given by the Q λ map at the algebra level Multiplicativity properties of W and V. We indicated earlier that the operators V and W behave much like the multiplicative unitary operators in the case of locally compact quantum groups. In this subsection, we give results that strengthen this observation. Many of these results resemble the ones from the quantum group theory, and all these results will be useful later. For convenience, we give here the results regarding the operator W. Of course, similar results exist for the operator V. We first begin with a result that will be useful down the road.