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Measured quantum groupoids associated to proper dynamical quantum groups

Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang-Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II_1 factors. In this article, we associate to suitable dynamical quantum groups, which are a purely algebraic objects, Hopf C*-bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical quantum group, we first construct a fundamental unitary which yields Hopf bimodules on the level of C*-algebras and von Neumann algebras. Next, we assume properness of the dynamical quantum group and lift the integrals to the operator algebras. In a subsequent article, this construction shall be applied to the dynamical SU_q(2) studied by Koelink and Rosengren.


Introduction
Dynamical quantum groups were introduced by Etingof and Varchenko as an algebraic framework for the study of the dynamical quantum Yang-Baxter equation [5,6,7], a variant of the Yang-Baxter equation arising in statistical mechanics. Every (rigid) solution of this equation has a naturally associated tensor category of representations which turns out to be equivalent to the category of representations of some dynamical quantum group. In the case of the basic rational or basic trigonometric solution, this dynamical quantum group can be regarded as a quantization of the function algebra on some Poisson-Lie-groupoid. In general, it can be regarded as a quantum groupoid and fits into the theory of Hopf algebroids developed by Böhm and others [1].
Measured quantum groupoids were introduced by Enock, Lesieur and Vallin [2,12] to capture generalized Galois symmetries of certain inclusions of type II 1 factors [4,3,14]. Apart from this fundamental example in von Neumann algebra theory, which was also considered in the algebraic setting [8,18], and from the finite case, only few measured quantum groupoids have been constructed and investigated yet [12,27].
Up to now, connections between algebraic and operator-algebraic approaches to quantum groupoids have only been explored in the finite case [13,16,26] and in the form of a few examples and constructions that exist on both levels. The situation is very different in the area of quantum groups, where Woronowicz's theory of compact quantum groups [31] and van Daele's theory of multiplier Hopf algebras with integrals [11,29] form a bridge between the algebraic and operator-algebraic approaches, combining the computational convenience of the former with the power and richness of the latter.
In this article, we associate to suitable dynamical quantum groups, which are purely algebraic objects, Hopf C˚-bimodules and measured quantum groupoids on the level of von Neumann algebras. The main example of a dynamical group we have in mind for application is the dynamical SU q p2q studied by Koelink and Rosengren [9], and in a subsequent article, we want to study the construction for this example in detail.
On the dynamical quantum groups, we have to impose several assumptions. First, we need a left-and a right-invariant integral, which correspond to fiber-wise integration on a groupoid, and a weight on the basis that is suitably quasi-invariant, such that the resulting total integrals are faithful, positive, and coincide. In the case of the dynamical SU q p2q, the leftand right-invariant integrals can be obtained from a Peter-Weyl decomposition due to Koelink and Rosengren [9], while the quasi-invariant weight on the basis can be chosen quite freely.
Second, we assume the dynamical quantum group to be proper, which is the natural analogue of compactness and unitality for quantum groupoids, and to possess a specific approximate unit in the base algebra. The dynamical SU q p2q even is compact and thus satisfies this condition.
Third, we assume that the quasi-invariant weight on the basis admits a bounded GNS-construction. Like the first condition, this one is very natural. In the case of the dynamical SU q p2q, the base algebra is formed by all meromorphic functions on the plane and does not admit any non-trivial bounded representations. To apply our construction, one therefore has to change the base and check that the Peter-Weyl decomposition persists.
Given these assumptions, the measured quantum groupoid is constructed as follows.
The algebraic GNS-construction, applied to the total integral on the dynamical quantum group, yields a Hilbert space of square-integrable functions on the dynamical quantum group together with a natural representation by densely defined multiplication operators. To obtain a C˚-algebra or von Neumann algebra, one has to show that these multiplication operators are bounded. To prove this and to lift the comultiplication to the resulting C˚-algebra and von Neumann algebra, we proceed as in the case of quantum groups [22] and construct a fundamental unitary which is pseudo-multiplicative on the level of von Neumann algebras and C˚-algebras in the sense of [25] and [23], respectively. The general theory of these unitaries then yields completions of the dynamical quantum group in the form a Hopf C˚-bimodule and a Hopf von-Neumann bimodule, and simultaneously a Pontrjagin dual in the same form. Finally, we extend the invariant integrals to the level of operator algebras, using properness of the dynamical quantum group and standard von Neumann algebra techniques.
This article is organized as follows. Section 1 provides the algebraic basics on dynamical quantum groups and integration that are needed for the construction in Section 2. We first generalize the definition of a dynamical quantum group or h-Hopf algebroid, allowing the base to be non-unital, then consider leftand right-invariant integrals on the total algebra and quasi-invariant weights on the basis, and finally construct a˚-algebra related to the Pontrjagin dual. The main result of this section is the existence of a modular automorphism for the total integral, which follows from a strong invariance property similarly as in the setting of multiplier Hopf algebras [29]. Section 2 presents the construction of the measured quantum groupoid outlined above. It uses Connes spatial theory, in particular the relative tensor product of Hilbert modules, and the C˚-algebraic analogue of that construction [21], and introduces the necessary concepts along the way when they are needed.
We use standard notation and adopt the following conventions. All algebras will be over the ground field C and we do not assume the existence of a unit element. Given a vector space V with a subset X Ď V , we denote by xXy Ď V the linear span and, if V is normed, by rXs Ď V the closed linear span of X. Inner products on Hilbert spaces will be linear in the second and anti-linear in the first variable.

Dynamical quantum groups with integrals on the algebraic level
This section summarizes and develops the basics on dynamical quantum groups and integration used in this article. Before turning to details, let us outline the main concepts.
A dynamical quantum group is a special quantum groupoid and as such consists of an algebra B called the basis, an algebra A, commuting inclusions r, s : B popq Ñ A, and a comultiplication, antipode and counit which are in some sense fibered over r and s. What makes it special is that the basis B is commutative, that rpBq and spBq are central in A up to a twist which is controlled by an action of a group Γ on B and a bigrading of A by Γ, and that the target of the comultiplication is given by a nice monoidal product AbA.
Integration on a quantum groupoid involves several ingredients. The analogue of the leftor right-invariance property of Haar measures on groups, Haar systems on groupoids, and Haar weights on quantum groups can be formulated for maps A Ñ B that are linear with respect to rpBq or spBq, respectively. To obtain a total integration A Ñ C, such a partial integral A Ñ B has to be composed with a suitable functional B Ñ C that has to be compatible with the action of Γ.
Let us now turn to details. We proceed as follows.
From the beginning, we assume all our algebras to possess an involution but not necessarily a unit. We first recall terminology concerning non-unital algebras ( §1.1), then describe the monoidal product AbA ( §1.2), and define dynamical quantum groups or, more precisely, multiplier pB, Γq-Hopf˚-algebroids ( §1.3). Afterwards, we introduce and study integrals ( §1.4- §1.6) and prove the existence of a modular automorphism that controls the deviation of the total integral from being a. Using integration, we finally construct the dual˚-algebra of a multiplier pB, Γq-Hopf˚-algebroid ( §1.7).

Preliminaries on non-unital algebras
To handle non-unital algebras, we use extra non-degeneracy assumptions and multiplier algebras [28, appendix] which are recalled below.
Let R be an algebra, not necessarily unital. Given a left R-module M, we say that R has local units for M if for each finite subset F Ď M, there exists some r P R such that rm " m for all m P F [30]. The corresponding notion for right R-modules is defined similarly. We say that R has local units if it has local units for R, regarded as a left and as a right R-module.
Let R and S be algebras with local units, let N be an R-S-bimodule and assume that R and S have local units for N. A multiplier of N is a pair T " pT R , T S q, where T R : R Ñ N is a left Rmodule map and T S : S Ñ N a right S-module map satisfying T R prqs " rT S psq for all r P R, s P S. Given such a multiplier, we write rT :" T R prq and T s :" T S psq for all r P R, s P S. We denote the set of all multipliers of N by MpNq. Clearly, N embeds into MpNq and MpNq carries a natural structure of an R-S-bimodule that is compatible with this embedding.
Regarding R as an R-R-bimodule, MpRq becomes an algebra via rpT T 1 q :" prT qT 1 and pT T 1 qr :" T pT 1 rq, and R embeds into MpRq as an essential ideal. If R is a˚-algebra, then so is MpRq, where rT˚r 1 " pr 1˚T r˚q˚for all r, r 1 P R, T P MpRq.
The bimodule N becomes an MpRq-MpSq-bimodule via r 1 prnsqs 1 :" pr 1 rqnpss 1 q for all r 1 P MpRq, r P R, n P N, s P S, s 1 P MpSq, and MpNq is an MpRq-MpSq-bimodule via rpr 1 T s 1 q :" pprr 1 qT qs 1 and pr 1 T s 1 qs " r 1 pT ps 1 sqq for all r P R, r 1 P MpRq, T P MpNq, s P S, s 1 P MpSq.

The category of pB, Γq ev -algebras
Let B be a commutative˚-algebra with local units, let Γ be a group that acts on B on the left, and let e P Γ be the unit.
A pB, Γq-module is a Γ-graded B-bimodule V " À γPΓ V γ for which B has local units, where each V γ is a B-bimodule and vb " γpbqv for all v P V γ , b P B, γ P Γ. A morphism of pB, Γq-modules V and W is a morphism of Γ-graded B-bimodules.
A pB, Γq-algebra is a Γ-graded˚-algebra A " À γPΓ A γ which has local units in A e and is equipped with a˚-homomorphism B Ñ MpAq that turns A into a pB, Γq-module. Such a pB, Γqalgebra is proper if B maps into A.
Given a pB, Γq-algebra A and γ P Γ, we denote by MpAq γ Ď MpAq the space of all multipliers T P MpAq satisfying TA γ 1 Ď A γγ 1 and A γ 1 T Ď A γ 1 γ for all γ 1 P Γ.
A morphism of pB, Γq-algebras A and C is a non-degenerate, B-linear˚-homomorphism π : A Ñ MpCq satisfying πpA γ q Ď MpCq γ for all γ P Γ. Such a morphism is proper if it maps A into C.
Using the extension of non-degenerate homomorphisms to multipliers, one defines the composition of morphisms and checks that pB, Γq-algebras form a category.
The tensor product B b B is a˚-algebra with local units and a natural action of ΓˆΓ. Replacing pB, Γq by pB, Γq ev :" pB b B, ΓˆΓq in the definition above, we obtain the category of all pB, Γq ev -algebras.
Let A be a pB, Γq ev -algebra. We call an element x P A homogeneous and write B x " γ,B x " γ 1 if x P A γ,γ 1 for some γ, γ 1 P Γ. Thus, B x B y " B xy ,B xBy "B xy and B x˚" B´1 x ,B x˚"B´1 x for all homogeneous x, y P A. Define r " r A : B Ñ MpAq and s " s A : B Ñ MpAq by rpbqa " pb b 1qa and spbqa " p1 b bqa for all a P A, b P B. We write r A, A r , s A, A s if we consider A as a B-module via left or right multiplication via r or s, respectively.
Clearly, B is a pB, Γq-algebra and B b B is a pB, Γq ev -algebra with respect to the trivial gradings. Every pB, Γq-algebra A can be regarded as a pB, Γq ev -algebra, where A pγ,γq " A γ and A pγ,γ 1 q " 0 whenever γ ‰ γ 1 , and pb b b 1 qa " bb 1 a for all b, b 1 P B, a P A. Conversely, every pB, Γq ev -algebra A can be considered as a pB, Γq-algebra via r : B Ñ MpAq and the grading given by A γ :" À γ 1 A γ,γ 1 , or via s : B Ñ MpAq and the grading given by A γ 1 :" À γ A γ,γ 1 . We write pA, rq and pA, sq, respectively, to denote the resulting pB, Γq-algebras.
Denote by B¸Γ the crossed product for the action of Γ on B, that is, the universal algebra containing B and Γ such that e " 1 B and bγ¨b 1 γ 1 " bγpb 1 qγγ 1 for all b, b 1 P B, γ, γ 1 P Γ. This is a pB, Γq-algebra with respect to the natural inclusion B Ñ B¸Γ and the involution and grading given by pbγq˚" γ´1b˚and pB¸Γq γ " Bγ for all b P B, γ P Γ.
The fiber product of pB, Γq ev -algebras A and C is defined as follows. The subalgebra is a pB, Γq ev -algebra, where B abc " B a ,B abc "B c for all a P A, c P C and prˆsqpb b b 1 q " r A pbq b Cq be the ideal generated by ts A pbqb1´1br C pbq : b P Bu.
Then the quotient Cqq is a pB, Γq ev -algebra again, called the fiber product of A and C. Write abc for the image of an element a b c in AbC.
The assignment pA,Cq Þ Ñ AbC is functorial, associative and unital. Indeed, for all morphisms of pB, Γq ev -algebras π 1 : A 1 Ñ C 1 , π 2 : A 2 Ñ C 2 , there exists a morphism π 1b π 2 : A 1b A 2 Ñ C 1b C 2 , a 1b a 2 Þ Ñ π 1 pa 1 qbπ 2 pa 2 q; (1) for all pB, Γq ev -algebras A,C, D, there exists an isomorphism and for each pB, Γq ev -algebra A, there exist isomorphisms These isomorphisms are compatible in a natural sense and endow the category of pB, Γq evalgebras with a monoidal structure. From now on, we shall use them without further notice. The category of pB, Γq ev -algebras carries automorphisms p´q op and p´q co such that for each pB, Γq-algebra A and each morphism φ : A Ñ C, we have A co " A as an algebra, A op is the opposite˚-algebra of A, that is, the same vector space with the same involution and reversed multiplication, and pA op q γ,γ 1 " A γ´1,γ 1´1 for all γ, γ 1 P Γ, pA co q γ,γ 1 " A γ 1 ,γ for all γ, γ 1 P Γ, These automorphisms are involutive and commute, that is, Furthermore, they are compatible with the monoidal structure as follows. Given pB, Γq-algebras A,C, there exist isomorphisms pAbCq op Ñ A opb C op and pAbCq co Ñ C cob A co given by abc Þ Ñ abc and abc Þ Ñ cba, respectively. Moreover, pB¸Γq co " B¸Γ, there exists an isomorphism S B¸Γ : B¸Γ Ñ pB¸Γq op , bγ Þ Ñ γ´1b, and all of these isomorphisms and the isomorphisms in (2) and (3) are compatible in a natural sense.

Multiplier pB, Γq-Hopf˚-algebroids
We shall work with variants of the h-Hopf algebroids and pB, Γq-Hopf˚-algebroids considered in [6,9] and [20], respectively, where the basis need no longer be unital. These variants consist of a pB, Γq ev -algebra and a comultiplication, counit and antipode, which will be introduced one after the other. To quickly proceed to the main part of this article, we postulate all the usual properties of these maps as axioms and leave a study of the axiomatics for later.
Given a pB, Γq ev -algebra A, we denote byMpAbAq Ď MpAbAq the set of all T P MpAbAq for which all products of the form T pxb1 MpAq q, pxb1 MpAq qT, T p1 MpAqb yq, p1 MpAqb yqT where x P A γ,e , y P A e,γ , γ P Γ, lie in AbA. Evidently,MpAbAq is a˚-subalgebra of MpAbAq.

Definition.
A comultiplication on a pB, Γq ev -algebra A is a morphism ∆ from A to AbA satisfying ∆pAq ĎMpAbAq and p∆b idq˝∆ " pidb∆q˝∆. A (proper) multiplier pB, Γq-˚-bialgebroid is a (proper) pB, Γq ev -algebra with a comultiplication. A morphism of multiplier pB, Γq--bialgebroids pA, ∆ A q, pB, ∆ B q is a morphism φ from A to B satisfying ∆ B˝φ " pφbφq˝∆ A .
Let pA, ∆q be a multiplier pB, Γq-˚-bialgebroid. We adopt the Sweedler notation and write ∆paq " ÿ a p1qb a p2q , p∆b idqp∆paqq " ÿ a p1qb a p2qb a p3q " pidb∆qp∆paqq and so on for each a P A. In general, a p1q and a p2q do not stand for elements of A because ∆paq need not lie in AbA, but only inMpAbAq. Therefore, this notation requires extra care; see [28,30] for a detailed explanation in the context of multiplier Hopf algebras. We shall need to form products of the form ∆pxqp1 b yq or py b 1q∆pxq when B y ‰ e orB y ‰ e, respectively, which are defined as follows. The multiplication on A b A induces a canonical AbA-A b A-bimodule structure on s A b x p2q y, Similarly, one can define the maps 1q.
iii) The condition pεb idq˝∆ " id A " pidbεq˝∆ is equivalent to the relations ÿ rpε 7 px p1q qqx p2q y " xy " ÿ xy p1q spε 5 py p2q qq for all x, y P A, and hence to commutativity of the diagrams Furthermore, this condition is equivalent to the relations ÿ xy p2q rpε 5 py p1q qq " xy " ÿ spε 7 px p2q qqx p1q y for all x, y P A.
The definition of the antipode involves the isomorphism σ A,A : pAbAq co,op Ñ A co,opb A co,op , xby Þ Ñ ybx.
Given an antipode S on a multiplier pB, Γq-˚-bialgebroid pA, ∆q and an element a P A, we shall henceforth always regard Spaq as an element of A and not of A co,op .
1.3.6 Remarks. Let pA, ∆, ε, Sq be a multiplier pB, Γq-Hopf˚-algebroid. i) In Sweedler notation, commutativity of the diagrams in Definition 1.3.4 amounts to ÿ Spx p1q qx p2q y " spε 5 pxqqy, ÿ xy p1q Spy p2q q " xrpε 7 pyqq for all x, y P A, ii) If S 1 is an antipode as well, then S 1 " S because for all x, y, z P A, For every multiplier pB, Γq-Hopf˚-algebroid, the maps T 1 and T 2 defined above are bijections.
1.3.7 Proposition. Let pA, ∆q be a multiplier pB, Γq-˚-bialgebroid. If pA, ∆q has a counit ε and an antipode S, then the maps T 1 , T 2 , T 3 , T 4 are bijective and for all x, y P A, Proof. We only prove the assertion concerning T 1 . One first checks that the formula given for T´1 1 yields a well-defined map T 1 s A, and then that for all x, y P A and u, v P A e,e , As in the case of multiplier bialgebras or Hopf algebroids, this result should have a converse.

Bi-measured multiplier pB, Γq-˚-bialgebroids
We now introduce the main objects of this article -multiplier pB, Γq-Hopf˚-algebroids equipped with certain integrals. In §2, we shall construct completions of such objects in the form of measured quantum groupoids.
As on a groupoid, integration on a multiplier pB, Γq-˚-bialgebroid pA, ∆q proceeds in stages. First, one needs partial integrals φ, ψ : A Ñ B with suitable left or right invariance properties, and second a suitable weight µ : B Ñ C that is compatible with the action of Γ. The results in [9] suggest that dynamical quantum groups that are compact in a suitable sense even possess a bi-invariant integral h : A Ñ B b B that can be obtained from a Peter-Weyl decomposition of A.
We first focus on the weight µ and the bi-integral h, and discuss left and right integrals in the next subsection.
Let us briefly recall some terminology. Let C be a˚-algebra with local units. A linear map µ : C Ñ C is faithful if µpCcq " 0 implies c " 0, and positive if µpc˚cq ě 0 for all c P C. Assume that µ is positive. Then µ is˚-linear, because positivity of φppb`cq˚pb`cqq and φppb`icq˚pb`icqq implies µpb˚cq " φpc˚bq for all b, c P C, and faithful as soon as µpc˚cq ‰ 0 whenever c ‰ 0.

Definition.
A weight for pB, Γq is a faithful, positive linear map µ : B Ñ C that is quasiinvariant with respect to Γ in the sense that for each γ P Γ, there exists some D γ P MpBq such that µpγpbD γ qq " µpbq for all b P B.
Let pA, ∆q be a multiplier pB, Γq-˚-bialgebroid. The following definition is inspired by the notion of a Haar functional introduced in [9].

Definition.
A bi-integral on pA, ∆q is a morphism of pB, Γq ev -modules h : A Ñ B b B satisfying ∆pker hqp1bA e,e q Ď ker hbA and ∆pker hqpA e,eb 1q Ď Ab ker h. If pA, ∆q is proper and hprpbqspb 1 qq " b b b 1 for all b, b 1 P B, we call such a bi-integral normalized.

Lemma.
Let pA, ∆q be proper and let h be a normalized bi-integral on pA, ∆q.
Proof. i) We only prove the first equation. Let ω : pA, rq Ñ B be a morphism of pB, Γq-modules sending I :" ker h to 0. Then Iq " 0 and hence pidbωqp∆pIqq " 0. Moreover, if b, b 1 , b 2 P B and u P A e,e , then For ω " m B˝h , these calculations imply for all a P I and b, b 1 P B pidbm B˝h qp∆paqq " 0 " hpaq, pidbm B˝h qp∆prpbqspb 1 qqq " rpbqspb 1 q " hprpbqspb 1 qq.
Since A " I`rpBqspBq, we can conclude pidbm B˝h q˝∆ " h. ii) Let x P ker h and choose u, Then Thus, ker h Ď ker h 1 . Since h and h 1 are normalized and iii) One easily verifies that σ B˝h˝S is a normalized bi-integral. By ii), it equals h.

Definition.
A proper multiplier pB, Γq-˚-bialgebroid pA, ∆q is bi-measured if it is equipped with a normalized bi-integral h : A Ñ B b B and a weight µ for pB, Γq such that ν :" pµ b µq˝h is faithful and positive.
1.4.6 Remark. Given a bi-measured proper multiplier pB, Γq-Hopf˚-algebroid as above, h is evidently faithful, and also˚-linear. To see this, note that

Left and right integrals
For large parts of this article, the multiplier pB, Γq-Hopf˚-algebroids under consideration need not be equipped with a bi-integral, but only with left and right integrals φ, ψ. The definition of these integrals involves slice maps of the following form. Let pA, ∆q be a multiplier pB, Γq-˚-bialgebroid and let φ : pA, rq Ñ B be a morphism of pB, Γqmodules. Then there exists a unique linear map idbφ :MpAbAq Ñ MpAq such that Likewise, every morphism ψ : pA, sq Ñ B yields a slice map ψb id :MpAbAq Ñ MpAq.
iii) If φ is a left integral, then also φp´spbqq is left integral for each b P B. Likewise, if ψ is a right integral, then also ψp´rpbqq is a right integral for each b P B.
We shall frequently use the following strong invariance relations: φqp∆pzqp1bxqqq for every left integral φ and all x, z P A; idqppxb1q∆pzqq for every right integral ψ and all x, z P A.
Proof. Using Sweedler notation, we calculate Normalized bi-integrals yield left and right integrals as follows: 1.5.4 Lemma. Assume that pA, ∆q is proper, h is a normalized bi-integral on pA, ∆q, and µ : B Ñ C is linear. Then φ :" pid bµq˝h and ψ :" pµ b idq˝h are a left and a right integral, respectively, and φ˝S˘1 " ψ.
Proof. Repeating the proof of Lemma 1.4.4 i) with ω :" φ " pid bµq˝h, we find for all a P ker h and b, b 1 P B. Since A " pker hq`rpBqspBq, we can conclude pidbφq˝∆ " r˝φ. The assertion on ψ follows similarly, and the last equation follows from Lemma 1.4.4 iii).

Measured multiplier pB, Γq-˚-bialgebroids
Much of the ensuing material applies not only to bi-measured proper multiplier pB, Γq-Hopf -algebroids but also to the following class of objects.
1.6.2 Remarks. i) Given a measured multiplier pB, Γq-Hopf˚-algebroid as above, the maps φ and ψ are˚-linear. This can be seen from a similar argument as in Remark 1.4.6.
iii) One could probably drop the assumption ν " ν´1 and assume the existence of an invertible multiplier δ such that ν´1paq " νpaδq for all a P A. In the applications we have in mind, in particular, in the bi-measured case, the stricter assumption above is satisfied.
Proof. The maps D andD are bijective because D γ is invertible for each γ P Γ. The remaining assertions follow from straightforward calculations, for example, iii) ωpDpaqa 1 q " ωpaD´1pa 1 qq and ωpDpaqa 1 q " ωpaD´1pa 1 qq for all a, a 1 P A.
Proof. i) For ω " ν, the assertion follows from the relation ker φ`ker ψ Ď ker ν. To obtain the assertion for ω " φ, ψ, use the fact that µ is faithful.
iii) This follows immediately equation (9) and i).
We shall now show that ν " µ˝φ has a modular automorphism and thus satisfies an algebraic variant of the KMS-condition. Let us briefly recall this concept.

Theorem.
i) There exists a modular automorphism θ for ν.
ii) θ is a pB, Γq ev -module automorphism of A.
Proof. i) The proof repeatedly uses strong invariance of φ and ψ, and closely follows [29], where the corresponding result was obtained for multiplier Hopf algebras. We proceed in three steps.
ii) We first show that θ respects the grading. Let c, d P A be homogeneous. Then the element a in (13) is homogeneous as well, with grading given by B a " B c andB a "B d because ψpdSpc p2q qq " 0 unlessB d " B c p2q "B c p1q , and similarly a 1 in (13) is homogeneous with the same degree like a. To see that θ is B b B-linear, use the relation νpyθprpbqspb 1 qxqq " νprpbqspb 1 qxyq " νpxyrpbqspb 1 qq " νpyrpbqspb 1 qθpxqq, where x, y P A and b, b 1 P B, and faithfulness of ν.
ii) ψ˝θ " ψ and ψpxyq "B x pψpyθDpxqqq for all x, y P A.
iii) h˝θ " h and hpxyq " pB x bB x qphpyθ D,D pxqqq for all x, y P A if h is a bi-invariant integral and ν " pµ b µq˝h.
Proof. Assertion i) follows from the fact that µ is faithful and that for all x, y P A, b P B, µpbφpθpxqqq " νprpbqθpxqq " νpθprpbqxqq " νprpbqxq " µpbφpxqq, Assertions ii) and iii) follow similarly.
1.6.7 Proposition. Assume that A s is a flat B-module. Then ∆˝θ D " pS 2b θ D q˝∆.
Proof. Let x, y P A. Using Sweedler notation, we calculate ÿ SpspB´1 x pφpxy p2q qqqy p1q q (Proposition 1.6.6) Since A s is a flat B-module and maps of the form a Þ Ñ φpyaq, where y P A, separate the points of A, we can conclude ř θ D pxq p1qb θ D pxq p2q " ř S 2 px p1q qbθ D px p2q q.

The dual˚-algebra
Let pA, ∆, ε, S, φ, ψ, µq be a measured multiplier pB, Γq-Hopf˚-algebroid. Denote by MpAq 1 the dual vector space of MpAq and letÂ :" tνpx´q : x P Au Ď MpAq 1 ThenÂ " tνp´xq : x P Au by Theorem 1.6.5 and for each ω PÂ, there exist unique B-module maps r ω : r MpAq Ñ B, ω r : MpAq r Ñ B, s ω : s MpAq Ñ B, ω s : MpAq s Ñ B whose compositions with µ are equal to ω, because ν " µ˝φ " µ˝ψ and µ is faithful. Using either of these B-module maps, one can equipÂ with the structure of a˚-algebra. We shall choose an approach that fits well with the duality on the operator-algebraic level in the next section. First, we define an abstract Fourier transform A ÑÂ, x Þ Ñx :" νpSpxq´q.
Evidently,x s " ψpSpxq´q andx r " φpSpxq´q, and by Proposition 1.6.6, sx " ψp´θpSpxqqq and rx " φp´θpSpxqqq. For all x, a P A, we define a right convolution a˚x :" ÿ a p2q rpx s pa p1q qq " ÿ a p2q rpψpSpxqa p1q qq P A.
1.7.1 Remark. One could also work with the transform A ÑÂ, x Þ Ñx :" νp´Spxqq, and the left convolution defined by If φ˝S " ψ, for example, if we are in the bi-measured case (see Remark 1.6.2 ii)), then } Spxq˚Spaq " ÿ spφpSpaq p2q S 2 pxqqqSpaq p1q " Spa p2q rpψpSpxqa p1q qqq " Spa˚xq for all a, x P A.

1.7.3
Proposition.Â has a structure of a pB, Γq ev -matrix-algebra, where for all x, y P A, b P B, rpbqx " z xrpbq,xrpbq " z xspbq,ŝpbqx " z rpbqx,xŝpbq " z spbqx, Proof. The multiplication is associative and turnsÂ into an algebra by (19). This algebra is non-degenerate because A˚Â spans A by surjectivity of T 2 . The˚-operation is involutive because˚˝S is involutive, and anti-multiplicative because
We shall construct operator-algebraic completions of this algebraic object in the form of a Hopf C˚-bimodule, Hopf-von Neumann bimodule and a measured quantum groupoid. Along the way, we shall impose further assumptions on B, Γ, µ, A which were mentioned already in the introduction, most notably properness of A.
The basic idea is to use the GNS-representations for the weight µ on the basis B and the functional ν on the total algebra A, respectively. Naturally, some restrictions have to be made on B, Γ, µ. To show that ν admits a bounded GNS-representation and to lift the comultiplication to the level of operator algebras, we use a fundamental unitary. To take full advantage of this unitary, we describe its domain and range as relative tensor products, and show that it is a pseudomultiplicative unitary in the sense of [23] and [25]. The necessary modules are introduced in §2.2, and the unitary itself is constructed in §2.3. This part uses Connes' spatial theory [19], and the relative tensor product of Hilbert spaces over C˚-algebras which was introduced in [21]. The fundamental unitary then gives rise to completions of A andÂ in the form of Hopf C˚-bimodules and two Hopf-von Neumann bimodules; see §2.4- §2.6.To obtain the full structure of a measured quantum groupoid, we finally extend the integrals φ, ψ to the level of von Neumann algebras and show that these extensions are left or right invariant again in §2.7.
Before we turn to details, let us briefly sketch the construction of the fundamental unitary, which we denote by W . Its domain and range can be described as separated completions of the relative tensor products s A b Note that positivity of these forms is not evident because φ is not assumed to be completely positive in any sense. Given that positivity, the map extends to a unitary on the respective completions because it is surjective by Proposition 1.3.7 and isometric as shown by the following calculation: The adjoint of this extension is the fundamental unitary W .
Similarly, one can construct and employ another unitary V which is an extension of the map yq. We shall focus on W because this unitary is given preference in the theory of locally compact quantum groups and measured quantum groupoids.

Preparations concerning the base
We define an inner product on B by xb|b 1 y :" µpb˚b 1 q for all b, b 1 P B, and denote by K the Hilbert space obtained by completion, and by Λ µ : B Ñ K the canonical inclusion. To proceed, we have to impose the following assumption: (A2) For each b P B, the following equivalent conditions hold: i) there exists a K ě 0 such that µpc˚b˚bcq ď Kµpc˚cq for all c P B; ii) there exists an operator π µ pbq P LpKq such that π µ pbqΛ µ pcq " Λ µ pbcq for all c P B.

Remark.
To apply the constructions below, it may be useful to first perform a base change, similarly as described in [20, §2], to replace B by an algebra of the form C c pΩq, where Ω is a locally compact space with an action of Γ. Then condition (A2) is automatically satisfied. For example, one can take Ω to be the set of all˚-homomorphisms χ : B Ñ C, equipped with the weakest topology that makes the function Ω Ñ C, χ Þ Ñ χpbq, continuous for each b P B, and perform a base change along the canonical map B Ñ MpC c pΩqq. Note, however, that such a base change can not simply be applied to left and right integrals, but only to bi-integrals.
Assumption (A2) immediately implies the existence of a˚-homomorphism π µ : B Ñ LpKq which can be regarded as a GNS-representation for µ.
Recall that a Hilbert algebra is a˚-algebra with an inner product such that left multiplication by each element is bounded, the resulting˚-representation is non-degenerate, and the involution is pre-closed with respect to the norm induced by the inner product. Since B is commutative, the map Λ µ pBq Ñ Λ µ pBq given by Λ µ pbq Þ Ñ Λ µ pb˚q extends to an anti-unitary operator J µ on K, and hence Λ µ pBq Ď K together with the˚-algebra structure inherited from B is a Hilbert algebra. We thus obtain • a von Neumann algebra N :" π µ pBq 2 Ď LpKq, • a n.s.f. weightμ on N such thatμpπ µ pb˚bqq " xΛ µ pbq|Λ µ pbqy " µpb˚bq for all b P B, • a left ideal Nμ :" tx P N :μpx˚xq ă 8u Ď N of square-integrable elements, • a closed map Λμ : Nμ Ñ K such that pK, Λμ, id N q is a GNS-representation forμ; this is the closure of the map π µ pBq Ñ K given by π µ pbq Ñ Λ µ pbq.

Various module structures
We define an inner product on A by xa|a 1 y :" νpa˚a 1 q for all a, a 1 P A, denote by H the Hilbert space obtained by completion, and by Λ ν : A Ñ H the canonical inclusion.
The maps introduced above yield various module structures on H as follows. Let We shall use the following concepts introduced in [21,23].

Lemma.
The Hilbert space H is a C˚-pb, bq-module with respect to either two of the spaces p βpπ µ pbqqΛ ν paq " Λ ν parpbqq, p αpπ µ pbqqΛ ν paq " Λ ν paspbqq for all b P B, a P A.
Proof. Let E, F be any two of the spaces listed above. Then rEHs " H and rEπ µ pBqs " E because xrpBqspBqArpBqspBqy " A, and rE˚Es " rπ µ pBqs because φpAq " B " ψpAq. Thus, pH, Eq is a C˚-b-module. The formulas for the associated representations are easily verified. Using these formulas and the relation xrpBqspBqArpBqspBqy " A, one easily checks that rρ E pπ µ pBqqFs " F and rρ F pπ µ pBqqEs " E.

The fundamental unitary
To define the domain and the range of the fundamental unitary, we use Connes' relative tensor product of Hilbert modules and the module structures introduced above. Connes' original manuscript on the construction remained unpublished; we therefore refer to [19] and [22] for details. The relative tensor product H β b µ α H is the separated completion of the algebraic tensor product DpH β ,μq b K b DpH α ,μq with respect to the sesquilinear form given by This Hilbert space can naturally be identified with the separated completions of the algebraic tensor products DpH β ,μq b H and H b DpH α ,μq with respect to the sesquilinear forms given by xξ b η|ξ 1 b η 1 y " xη|αpxξ|ξ 1 y β,μ qη 1 y and xξ b η|ξ 1 b η 1 y " xξ|βpxη|η 1 y α,μ qξ 1 y, (27) respectively, via and we shall use these identifications without further notice. Replacing the representations β, α by α, p β or p α, β, respectively, one obtains the relative tensor products H α b Clearly, this condition implies the existence of a unitary representation U : Γ Ñ LpKq such that U γ Λ µ pcq " Λ µ pγpcD 1 2 γ qq, U γ π µ pbqUγ " π µ pγpbqq for all b, c P B, γ P Γ. (29) Similarly as in (9), we define linear maps D for all a P A. These maps share all the properties of the maps D,D listed in Lemma 1.6.3. Short calculations show that for all x, y P A, Indeed, for all x, y P A and b P B, Proof. The maps Λ, Λ 1 are surjective because Λ ν pAq Ď H is dense, and they are well-defined and isometric because (27), (25) and (30) imply for all x, y P A xΛpx b yq|Λpx 1 b y 1 qy " νpx˚spφpy˚y 1 qqx 1 q, xΛ 1 px b yq|Λ 1 px 1 b y 1 qy " νpx˚rpB y pφpy˚y 1 qqqx 1 q.

Proposition.
There exists a unitary W : Proof. Calculation (22) and Lemma 2.3.1 imply that the map Λ ν pxq b µ α H such that for all x, y P A, Proof. The formula above defines an isometry V because (27), (25) and (31) imply for all x, x 1 , y, y 1 P A, and by right-invariance of ψ (see Remark 1.5.2 i)), the expressions above coincide. Bijectivity of V and the inversion formula follow from Proposition 1.3.7.

Boundedness of the canonical representations
The first application of the fundamental unitary W is to show that left multiplication on A and right convolution byÂ extend to representations on the Hilbert space H.
For later use, we calculate the slices of V , which are defined similarly as those of W˚.

The Hopf-von Neumann bimodules
We next show that the fundamental unitary W is pseudo-multiplicative in the sense of [25] and therefore yields two Hopf-von Neumann bimodules, which are completions of A andÂ, respectively. First, we need further preliminaries. The relative tensor product is functorial so that there exist bounded linear operators S b In particular, the commuting representations α, β, p α, p β yield six representations αb µ id, p αb µ id, Proof. This follows immediately from the fact that π µ pBq Ď N is weakly dense, the definition of W , and the formulas for α, β, p α, p β given in Lemma 2.2.2.
The relative tensor product is associative in a natural sense. The intertwining relations for W obtained above imply that all operators in the diagram below are well-defined, Proof. A short calculation shows that the adjoints of both compositions are given by Proof. We calculate 2.5.6 Remark. Under the identification (28), for all a, x, y P A and ζ P K

The Hopf C˚-bimodules
The fundamental unitary W is regular C˚-pseudo-multiplicative unitaries in the sense of [23], and therefore yields Hopf C˚-bimodules which are completions of A andÂ. To prove this, we again need some preliminaries concerning the relative tensor product in the setting of C˚algebras; for details, see [21] and [23]. The construction is parallel to the von Neumannalgebraic setting and differs mainly in notation. As before, let b " pK, rπ µ pBqs, rπ µ pBqsq. The relative tensor product H E : H of the C˚-bmodules pH, E : ψ q and pH, E : φ q is the separated completion of the algebraic tensor product E : φ with respect to the sesquilinear form given by It can be regarded as a twofold internal tensor product of Hilbert C˚-modules and identified with certain separated completions E : ψ α H and H β E : φ of the algebraic tensor products E : Comparing the sesquilinear forms (26) with (36) and using (25), one finds that there exists an isomorphism For each ξ P E : ψ and η P E : φ , there exist bounded linear operators We denote their adjoints by xξ| 1 and xη| 2 , respectively, and write |E : ψ y 1 " t|ξy 1 : ξ P E : ψ u, |E : φ y 2 " t|ηy 2 : η P E : φ u et cetera. Comparing with (33), we see that under the identification (38), λ p α,β Λ ν pxq " |Λ : ψ pxqy 1 and ρ p α,β Λ ν pyq " |Λ : φ pyqy 2 for all x, y P A. Replacing E : ψ and E : φ by E : φ and E φ , respectively, one similarly defines the relative tensor product H E : p β H, and bounded linear operators |ξy 1 , |ηy 2 : H Ñ H E : φ and η P E φ . Thus, W can be regarded as a unitary H E : To show that it is a C˚pseudo-multiplicative unitary in the sense of [23], we only need to prove: 2.6.1 Proposition. The following equations for subspaces of LpH,H E : Hq hold: The proof uses the following straightforward result: 2.6.2 Lemma. For all x, x 1 , y, y 1 P A and γ P tα, β, p βu, γ 1 P tα, p α, βu, Proof of Proposition 2.6.1. We only prove the first equation; the others follow similarly: Since the maps θ, D´1 2 , S and T 3 are bijections, we can conclude rtΛ : φ pyqΛ : φ pxq˚: x, y P Aus " rtxΛ : φ pxq| 2 W˚|Λ : φ pyqy 1 : x, y P Aus.
The assertion on V follows from a similar calculation.
Recall from [23] that a Hopf C˚-bimodule over b consists of a C˚-pb, bq-module pL, E, Fq, a non-degenerate C˚-algebra C Ď LpLq satisfying ρ E pπ µ pBqq Ď MpCq and ρ F pπ µ pBqq Ď MpCq, and a non-degenerate˚-homomorphism ∆ C : C Ñ C Fb E C that is co-associative and compatible with E and F in a suitable sense, where C Fb E C " tT P LpL F b b E Lq : T |Fy 1`T˚| Fy 1 Ď r|Fy 1 Cs and T |Ey 2`T˚| Ey 2 Ď r|Ey 2 Csu is the fiber product of C with itself relative to F and E.

The measured quantum groupoid
To obtain a measured quantum groupoid, we finally extend ν, φ, ψ to normal, semi-finite, faithful weights on the level of von Neumann algebras. We impose the following simplifying assumptions: (A4) pA, ∆q is proper in the sense that rpBqspBq Ď A.
(A5) There exists a net pu i q i in B such that pπ µ pu i qq i is a net of positive elements in the unit ball of π µ pBq that converges in Mprπ µ pBqsq strictly to 1 and such that pπ µ pu 2 i qq i is increasing.
Note that a net pu i q i as in (A5) exists always if we drop the condition that pπ µ pu 2 i qq i should be increasing.
Let us also note that in the bi-measured case where φ, ψ and ν arise from a bi-integral h on pA, ∆q, the extensions of φ, ψ, ν and the invariance of these extensions can be proved quite easily, see Remark 2.7.5 and 2.7.11.
For the extension of ν, we do not need the assumptions (A4) and (A5), but use the modular automorphism θ for ν obtained in Theorem 1.6.5, the theory of Hilbert algebras [19], and results of Kustermans and van Daele [10].
2.7.1 Lemma. Λ ν pAq Ď H is a Hilbert algebra with respect to the˚-algebra structure inherited from A.
The modular automorphism θ is related to the modular automorphism group σν as follows: 2.7.2 Proposition. π ν pAq Ď Tν and σν ni pπ ν paqq " π ν pθ´npaqq for all a P A, n P Z.
Let A θ :" ta P A : θpaq " au Ď A. Note that this space is a˚-subalgebra and, by (A4), contains rpBqspBq.
Proof. i) The first assertion follows from the fact that σν t pxq " ∆ it ν x∆´i t ν and ∆´1 ν x∆ν " x for each x P πνpA θ q by Proposition 2.7.2, and the second assertion follows from the fact that σν t is normal for all t P R and acts trivially on π ν prpBqspBqq.
ii) Combine i) and Lemma 2.2.2.

Proposition.
There exist unique n.s.f. weights T L from M to αpNq and T R from M to βpNq such thatμ˝α´1˝T L "ν "μ˝β´1˝T R .
As usual, let N T L :" tx P M : T L px˚xq P Nu and similarly define N T R . for all x P N T R , η P DpH α ,μq.
To prove Theorem 2.7.6, we construct increasing approximations of the weightsμ,ν,φ,ψ by bounded positive maps, using an approximate unit pu i q i in B with the properties assumed in (A5). Let u i, j :" rpu i qspu j q P A, and define for all i, j bounded, normal, positive, linear maps µ i : N Ñ C, x Þ Ñ xΛ µ pu i q|xΛ µ pu i qy, ν i, j : M Ñ C, x Þ Ñ xΛ ν pu i, j q|xΛ ν pu i, j qy, φ i, j : M Ñ N, x Þ Ñ Λ φ pu i, j q˚xΛ φ pu i, j q, ψ i, j : M Ñ N, x Þ Ñ Λ ψ pu i, j q˚xΛ ψ pu i, j q.
Given a net pλ κ q κ of real numbers, we write pλ κ q κ Õ λ if it is increasing and converges to λ. Likewise, given a von Neumann algebra C with a net pω κ q κ in C`and a n.s.f. weight ω, we write pω κ q κ Õ ω if ω κ px˚xq Õ ωpx˚xq for all x P C.

Proposition.
The following relations hold: pµ i q i Õμ, pν i, j q i, j Õν and pυ˝φ i, j q i, j Õ υ˝φ, pυ˝ψ i, j q i, j Õ υ˝ψ for all υ P N`.
The proof requires some preparations. We shall focus on the weightsν andφ; the caseμ is quite simple and the caseψ is similar to the caseφ. Recall that an element ξ P H is rightbounded with respect to the Hilbert algebra Λ ν pAq if there exists an operator R ξ P LpHq such that π ν paqξ " R ξ Λ ν paq for all a P A. Note that then R ξ P A 1 . Let us call ξ P H right-contractive if ξ is right-bounded and }R ξ } ď 1. Thenν is given bỹ for all x P M.
iii) If a P A and ξ P K is right-bounded with respect to Λ µ pBq, then Λ φ paqξ " p βpR ξ qΛ ν paq.
iii) If a P A and ξ " Λ µ pbq for some b P B, then R ξ " π µ pbq and Λ φ paqξ " Λ ν parpbqq " p βpπ µ pbqqΛ ν paq. Now, the assertion follows for all right-bounded ξ because Λ µ pBq is a core for Λμ and the right-bounded elements coincide with ΛμpNμq.
Proof of Proposition 2.7.8. We only prove the assertions concerning pν i, j q i, j and pφ i, j q i, j ; the others follow similarly. Let ξ i, j :" Λ ν pu i, j q and R i, j :" R ξ i, j " J ν π ν pu i, j qJ ν for all i, j. By Lemma 2.7.9, each ξ i, j is right-contractive and hence ν i, j px˚xq " }xΛ ν pu i, j q} 2 ďνpx˚xq for all i, j and all x P M. The net pν i, j q i, j in M`is increasing because (i) pRi , j R i, j q i, j is increasing by assumption on pu i q i , (ii) ν i, j pπ ν pa˚aqq " }R ξ i, j Λ ν paq} 2 for all a P A, and (iii) π ν pAq Ď M is weakly dense. For each right-contractive ξ P H and each x P M, }xξ} 2 " lim i, j }xπ ν pu i, j qξ} 2 " lim i, j }xR ξ Λ ν pu i, j q} 2 ď lim i, j }xΛ ν pu i, j q} 2 " lim i, j ν i, j px˚xq because R ξ P M 1 and Rξ R ξ ď 1. Therefore,νpx˚xq ď lim i, j ν i, j px˚xq.
A similar argument as above and Lemma 2.7.9 ii) show that for each υ P N`, the net pυφ i, j q i, j is increasing. Taking pointwise limits, we obtain a normal semi-finite weight ω from M to N such that for each y P M, the element ωpy˚yq in the extended positive partN`is defined by υpωpyqq " sup i, j υpφ i, j py˚yqq for all υ P N`. Then for all y P M, µpωpy˚yqq Ô i, j,k }yΛ φ pu i, j qΛ µ pu k q} 2 " }y p βpπ µ pu k qqξ i, j } 2 kÑ8 Ý ÝÝ Ñ }yξ i, j } 2 " ν i, j py˚yq Õ i, jν py˚yq and henceμ˝ω "ν. By [19,Theorem 4.18], ω "φ.
Proof of Theorem 2.7.6. To prove the assertion concerningφ and T L , we show that