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Home , Locally compact space, Pontryagin duality

Perspective

The operator approach to quantum groups

Johan Kustermans* and Stefaan Vaes†‡§

*Department of , University College Cork, Western Road, Cork, Ireland; †Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium; and the ‡Fund for Scientific Research, Flanders (Belgium) (F.W.O.) A relatively simple definition of a locally compact properties that justified calling it a quantum group, but never- in the C*-algebra setting will be explained as it was recently theless it did not fit in the framework of Kac . Woronow- obtained by the authors. At the same time, we put this definition icz also built up a whole theory of so-called compact quantum in the historical and mathematical context of locally compact groups in refs. 7 and 8, and most of them were not Kac algebras. groups, compact quantum groups, Kac algebras, multiplicative So it became a challenge to define an even larger unitaries, and theory. including both the Kac algebras and the compact quantum groups. One started looking for the definition of a locally he nowadays popular topic of quantum groups can be ap- compact quantum group. Tproached from two essentially different directions. The first and A first attempt was made by Masuda and Nakagami in ref. 9, most widespread approach is algebraic in nature. The first successes who formulated a definition of locally compact quantum groups of this approach date back to Drinfel’d (see ref. 1) and Jimbo (see (they called them Woronowicz algebras) in the framework of von ref. 2), who defined one-parameter deformations of universal Neumann algebras. A C*-algebraic version of this definition was enveloping algebras of semisimple complex Lie algebras in 1985. presented in lectures by Masuda, Nakagami, and Woronowicz, Many other classes of Hopf algebras have been studied since 1985 but was never published. The main drawback to their approach, and many received the label ‘‘quantum group.’’ The second ap- however, was the complexity of their axioms. In fact, many nice proach is analytic in nature: the basic motivation in the early features of a locally compact quantum group, which one would development of the theory was the generalization of Pontryagin like to prove from more elementary axioms, are presupposed by duality for abelian locally compact groups. Because the dual of a definition. Nevertheless, they are able to give a duality inside nonabelian group can no longer be a group, one looked for a larger their category, and their theory indeed includes compact quan- category that was self-dual. These generalized objects would be tum groups and Kac algebras, hence also locally compact groups. called quantum groups again. We also mention here the fundamental work of Baaj and Skandalis (see ref. 10) and of Woronowicz (see ref. 11) on This paper will deal with the second analytic approach. A multiplicative unitaries, about which we will tell more later. major role will be played throughout by proper generalizations Recently, we have given a much simpler definition for locally of the Haar of a locally , which is one of compact quantum groups in refs. 12 and 13, and we are able to the points where the theory differs thoroughly from the algebraic prove (approximately) all the axioms considered by Masuda, approach. Most of the Hopf algebras that are called quantum Nakagami, and Woronowicz. The major remaining problem is groups do not possess a proper generalization of a . that in our definition—just as in the definition of Masuda, Between these two approaches is the theory of multiplier Hopf Nakagami, Woronowicz or of Kac algebras—we still assume the *-algebras with , which is studied in ref. 3 by Van Daele. existence of the Haar measure. It would, of course, be more This theory is purely algebraic in nature, but nevertheless one elegant to give a definition for locally compact quantum groups has the analogue of the Haar measure. This framework of without assuming this and with the existence of the Haar multiplier Hopf *-algebras has the advantage of being easier to measure as a theorem. Taking into account all the efforts that understand than the often very technical have been made to prove the existence of the Haar measure in approach, but it is not general enough to include all locally such a general situation, it seems that this is far out of reach. compact quantum groups. On the level of formal manipulations, This paper is organized as follows. In the first section, we explain all technicalities of course disappear, and some of the formulas how C*-algebras turn up when one wants to define locally compact appearing in the operator algebra approach are very similar to quantum groups. Then we will explain the well-developed theory of

those considered by Van Daele. compact quantum groups and go on with our definition of a locally PERSPECTIVE This paper is the second in a series of papers, the first being compact quantum group, whose major properties, such as the ref. 4. Both papers can be read independently, but the remarks existence of the antipode and the uniqueness of the Haar measure, above make it clear that reading ref. 4 first will yield a better will be subsequently described. We also construct the multiplicative insight into the motivation underlying this paper. unitary and develop the duality theory. The approach to quantum groups through operator algebras dates back to the 1970s. After the pioneering work of Takesaki, Quantizing Locally Compact Groups Tannaka, Krein, and many others, the problem of finding a Let us start with the easiest step of the quantization procedure self-dual category containing the locally compact groups was that we will consider in this paper. Suppose X is a locally compact completely solved independently by Kac and Vainerman and by space. We will shift our attention to the space C0(X) of contin- Enock and Schwartz. The object they defined is called a Kac uous complex valued functions on X, vanishing at . With algebra; see ref. 5 for an overview. A Kac algebra is a von pointwise operations and the supremum norm, this is a com- Neumann algebra with much extra structure on it, the two basic mutative C*-algebra. An important theorem of Gelfand and examples being the essentially bounded measurable functions on Neumark in their paper (14) says that every commutative a and the group . A new phase in the development of the theory began with Woronowicz’ construction of the quantum SU(2) group in an Abbreviations: KMS, Kubo–Martin–Schwinger; GNS, Gelfand–Neumark–Segal. operator algebraic framework (see ref. 6). This object had all the §To whom reprint requests should be addressed. E-mail: [email protected].

PNAS ͉ January 18, 2000 ͉ vol. 97 ͉ no. 2 ͉ 547–552 Downloaded by guest on October 1, 2021 C*-algebra is of this form (Stone proved a real version of this by (m(h))(p) ϭ h(p, p). This map m is called the multiplication theorem four years earlier; see ref. 15). This is why one could map, because m(f R g) ϭ fg. speak about a C*-algebra as a locally compact quantum space or Now, trying to continue the quantization procedure, one as the functions on a quantum space, vanishing at infinity. All would like to write down the formulas above as axioms for the this was realized very early in the development of the theory of counit and antipode on an arbitrary C*-algebra with comulti- C*-algebras (1950s). plication, but then we run into all kinds of trouble. First of all, It is well known that locally compact spaces form a category examples show that it is too restrictive to assume that ⑀ is a with continuous maps as morphisms. How can this be quantized? *-homomorphism that is defined everywhere. In general, ⑀ could Let X and Y be locally compact spaces and ␪ : X 3 Y a be unbounded, and then it is not so clear how we can give a ␳ 3 ␫ R ⑀ ⑀ R ␫ continuous map. Then we can define : C0(Y) Cb(X), the meaning to and on the completed tensor product C*-algebra of bounded continuous functions on X, by putting A R A. Further, examples also show that the antipode S can be ␳(f) ϭ f ؠ ␪. Next, we will explain a C*-algebraic procedure to unbounded and need not be a *-map. This gives the same kind R ␫ ␫ R obtain C (X) out of the C*-algebra C (X). Let A be a C*- of problems for S and S. Finally, the multiplication map b 0 R algebra. Then we define M(A) as the set of all linear maps T from m, in general, also cannot be defined on the whole of A A or R A to A, which have an adjoint, i.e., for which there exists a linear M(A A). That this can be done is typical of the commutative map T* from A to A, satisfying b*T(a) ϭ (T*(b))*a for all a, b situation. ⑀ A. When T ⑀ M(A), one can prove that T is bounded as an Therefore, it will be necessary to replace both axioms by other operator from A to A and, using the operator norm of T, the axioms in order to define locally compact quantum groups. This necessarily unique adjoint T* defined above and the composition will be done further in the paper. of mappings as multiplication, we get a C*-algebra called the Compact Quantum Groups multiplier algebra of A. We can embed x ⑀ A into M(A) by the formulas T(a) ϭ xa and T*(a) ϭ x*a for all a ⑀ A. It is easy to Compact quantum groups form the best-understood part of the ϭ theory of quantum groups. Woronowicz developed this theory in verify that indeed M(C0(X)) Cb(X). If H is a and A a C*-subalgebra of B(H) that acts nondegenerately, then his fundamental papers (refs. 7 and 8). The typical example is M(A) can be identified with {x ⑀ B(H)͉xA, Ax ʕ A}. quantum SU(2), studied by Woronowicz in ref. 6. Woronowicz ϭ ϭ ␳ also introduced the differential calculus on compact quantum Now write A C0(X) and B C0(Y). We have defined : B 3 M(A). It is easy to verify that ␳ is a *-homomorphism and groups and used it to unravel the of that ␳(B)A is dense in A. This last condition expresses that ␳ is quantum SU(2). nondegenerate. Given two C*-algebras A and B, we call a map In order to motivate the definition of a compact quantum group, we go back to the quantization procedure described in the ␳ : B 3 M(A)amorphism from B to A when ␳ is a nondegenerate previous section. The compactness of a locally X *-homomorphism. The set of these morphisms is denoted by is expressed by the fact that the C*-algebra C (X) has a unit, Mor(B, A). The correspondence between ␪ and ␳ in the previous 0 therefore unital C*-algebras will be considered as compact paragraph gives a bijection between the continuous maps from quantum spaces. Now let G be a compact semigroup; then we can X to Y and the morphisms from C (Y)toC (X). Finally, if A and 0 0 define ⌬ : C(G) 3 C(G ϫ G), as before. Here C(G) denotes B are C*-algebras and ␳⑀Mor(B, A), it is possible to extend ␳ the C*-algebra of continuous functions on G, and we observe uniquely to a unital *-homomorphism from M(B)toM(A), still that C(G) ϭ C (G), because G is compact. As we explained denoted by ␳, such that ␳(m)␳(b)a ϭ ␳(mb)a for m ⑀ M(B), b 0 ⑀ ⑀ above, it seems impossible to write down easy axioms for the B, and a A. antipode and counit. Nevertheless, we can consider a weaker Now let G be a locally compact group. Inspired by the possible property of G, and that is the cancellation law: if rs ϭ quantization procedure above, we would like to translate the rt, then s ϭ t and analogously on the other side. It is not hard group structure on G to the C*-algebra C0(G). Multiplication is to prove that a compact semigroup with the cancellation law is, a continuous map from G ϫ G to G and hence can be translated ⌬ 3 ϫ in fact, a compact group, but there is more. This cancellation law to a morphism : C0(G) M(C0(G G)) by the formula can easily be translated to a property of the C*-algebra C(G), ⌬(f)(p, q) ϭ f(pq). Identifying f R g with the that sends R using the lemma of Urysohn (e.g., ref. 17). One can prove that (p, q)tof(p)g(q), we get an between C0(G) ϫ G satisfies the cancellation law if, and only if, the linear spaces C0(G), the C*-tensor product, and C0(G G). So we get a ⌬ R ⌬ R ⌬ 3 R (C(G))(C(G) 1) and (C(G))(1 C(G)) are dense in morphism : C0(G) M(C0(G) C0(G)), and associativity C G ϫ G ⌬ R ␫ ⌬ϭ ␫ R ⌬ ⌬ ( ). is translated to the formula ( ) ( ) , called This discussion makes the following definition of Woronowicz ⌬ R ␫ coassociativity. This can be given a meaning, because and more or less acceptable: ␫ R ⌬ are again morphisms and hence can be extended to the R multiplier algebra M(C0(G) C0(G)). The coassociativity is 1. DEFINITION. Consider a unital C*-algebra A together with a ⌬ R ␫ ⌬ ϭ⌬ ϭ easy to verify: (( ) (f))(p, q, r) (f)(pq, r) f((pq)r), unital *-homomorphism ⌬ : A 3 A R A such that (⌬ R ␫)⌬ϭ(␫ ␫ R ⌬ ⌬ ϭ whereas (( ) (f))(p, q, r) f(p(qr)). R ⌬)⌬ and such that the spaces ⌬(A)(A R 1) and ⌬(A)(1 R A) ⌬ ⑀ R If A is an arbitrary C*-algebra, a morphism Mor(A, A are dense in A R A. Then the pair (A, ⌬) is called a compact A), where A R A denotes the minimal C*-tensor product, is quantum group. called a comultiplication on A when it satisfies the coassociativity formula (⌬ R ␫)⌬ϭ(␫ R ⌬)⌬. Such a pair (A, ⌬) is sometimes The main reason for the success of compact quantum groups called a locally compact quantum semigroup. lies in the fact that this rather elegant definition of a compact Now one can go further and try to translate also the unit and quantum group (A, ⌬) allowed Woronowicz to generalize the the inverse to the C*-algebra level. The former gives rise to a whole theory of compact groups to the quantum group setting. ␧ 3 ރ ⑀ ϭ *-homomorphism : C0(G) given by (f) f(e) and the The pivotal result is the existence of a unique state ␸ on A, that ϭ Ϫ1 latter to a *-automorphism S of C0(G) given by (Sf)(p) f(p ). is left and right invariant, i.e., (␸ R ␫)⌬(a) ϭ (␫ R ␸)⌬(a) ϭ The properties of the unit and the inverse can be expressed by ␸(a) 1 for all a ⑀ A. The state ␸ is called the Haar state of the the formulas (␫ R ⑀)⌬ϭ(⑀ R ␫)⌬ϭ␫ for the counit ⑀ and compact quantum group (A, ⌬). m(␫ R S)⌬(f) ϭ ⑀(f)1 ϭ m(S R ␫)⌬(f) for the antipode S. Here The most important consequence of the definition is the we again used the necessary extensions to the multiplier algebra, quantum version of the classical Peter–Weyl theorem. For this, R and m is the map from M(C0(G) C0(G)) to M(C0(G)) given we need the notion of a unitary corepresentation of a compact

548 ͉ www.pnas.org Kustermans and Vaes Downloaded by guest on October 1, 2021 quantum group as a generalization of a strongly continuous weights we consider will be lower semicontinuous and densely ϭ unitary of a compact group. Let H be a defined. When A C0(X) is a commutative C*-algebra, there ␸ Hilbert space and denote the compact operators on H by B0(H). is a bijective correspondence between such weights on A and R ⌬ R ␫ ϭ ␮ ␸ ϭ͐ ␮ A unitary element U in M(A B0(H)) such that ( )(U) regular Borel measures on X given by (f) f(x)d (x) for ⌬ U13U23 is called a unitary corepresentation of (A, )onH. Here all positive functions f in C0(X). we used the so-called leg-numbering notation U13 and U23. Both A truly noncommutative phenomenon is the Kubo–Martin– R R ϭ R are elements of M(A A B0(H)) and U23 1 U, whereas Schwinger (in short, KMS) property for weights. Although the ϭ ␹ R ␫ ␹ R U13 ( )(U23), where denotes the flip map on A A, C*-algebra may be noncommutative, the KMS condition gives sending a R b to b R a. We say that U is finite dimensional if H some control over the noncommutativity under the weight. In is finite dimensional. The unitary corepresentation U is called order to make this more precise, we need the notion of a irreducible if the commutant {(␻ R ␫)(U)͉␻⑀A*}Ј in B(H)is one-parameter group and its analytic extension. ␣ ޒ 3 ␣ ␣ ϭ ␣ equal to ރ1. As in the classical case, every irreducible unitary Let : Aut(A) be a mapping such that: (i) s t sϩt ⑀ ޒ ޒ 3 x ␣ corepresentation of (A, ⌬) is finite dimensional. for all s, t , and (ii) the function A : t t(a) is norm Let Ꮽ be the of A defined by continuous for all a ⑀ A, where Aut(A) denotes the set of all *-automorphisms of the C*-algebra A. Then we call ␣ a norm- Ꮽ ϭ ͕͑␫  ␻͒͑U͉͒U a finite dimensional unitary continuous one-parameter group on A. An element a ⑀ A is called analytic with respect to ␣ if the function ޒ 3 A : t x ␣ (a) can ␻⑀ ͑ ͒ ͖ t corepresentation on a Hilbert space H and B H * . be extended to an analytic function f : ރ 3 A. In that case, the ␣ Ꮽ element z(a) is defined as f(z). Woronowicz proved that is a dense *-subalgebra of A and that ␸ Ꮽ ⌬ Ꮽ Now consider a weight on A. It is called a KMS weight if a together with the restriction of to forms a Hopf *-algebra norm-continuous one-parameter group ␴ on A exists such that: with positive integrals (see ref. 3 for definitions). The presence (i) ␸ is invariant under ␴, i.e., ␸␴ ϭ ␸ for every t ⑀ ޒ, and (ii) of the Hopf *-algebra structure on Ꮽ indicates that compact t for every a ⑀ ᏹ␸ and b ⑀ A, with b analytic with respect to ␴, one quantum groups can be studied within an algebraic framework. ␴ ⑀ ᏹ ␸ ϭ ␸ ␴ has ab, b Ϫi(a) ␸ and (ab) (b Ϫi(a)). This formula also It should be pointed out, however, that the existence of the Haar appears in the algebraic approach to quantum groups, as ex- state can be established only within the C*-algebra framework, plained in theorem 8 of ref. 4. If ␸ is a KMS weight (this is not unless the axioms are considerably strengthened. automatically true for every lower semicontinuous faithful Let (U␭)␭⑀⌳ be a complete set of mutually inequivalent ␴ ⌬ weight), such a one-parameter group is called a modular group irreducible unitary corepresentations of (A, ) on finite dimen- of ␸.If␸ is faithful, ␴ is uniquely determined by the properties sional Hilbert spaces (H␭)␭⑀⌳. Let ␭⑀⌳ and fix an orthonormal ␭ ␭ ⑀ above. The KMS condition is really the key result that allows one basis (e1( ),...,en␭( )) of H␭. For i, j {1, . . . , n␭}, we define ␻ ⑀ ␻ ϭ͗ ␭ ␭ ͘ ⑀ to develop a generalized noncommutative measure theory that i,j B(H␭)* by i,j(x) xei( ), ej( ) for all x B(H␭). ␭ ϭ ␫ R ␻ parallels classical measure theory (for instance, the Radon– Moreover, we put Uij( ) ( i,j)(U␭). The quantum version ␭ ͉␭⑀⌳ Nikodym Theorem has a generalization to weights in the von of the Peter–Weyl theorem says that the family {Uij( ) , i, ϭ Ꮽ Neumann algebra framework). j 1,...,n␭} is a Hamel basis of the and that We have now gathered enough material to formulate the quantized orthogonality relations hold between these elements. definition of a locally compact quantum group. Locally Compact Quantum Groups 2. DEFINITION. Consider a C*-algebra A and a nondegenerate Before stating the definition of a locally compact quantum group *-homomorphism ⌬ : A 3 M(A R A) such that:(i)(⌬ R ␫)⌬ϭ as it was given in refs. 12 and 13, we need some extra terminology (␫ R ⌬)⌬ and (ii) the linear spaces ⌬(A)(1 R A) and ⌬(A)(A R concerning weights on C*-algebras. The most important objects 1) are dense in A R A. associated with a locally compact group are its Haar measures, Assume, moreover, the existence of: (i) a faithful KMS weight ␸ so it is no big surprise that in the quantum group setting equally on (A, ⌬) such that ␸((␻ R ␫)⌬(x)) ϭ ␸(x)␻(1) for ␻⑀A*ϩ and ϩ fundamental roles are also played by the proper generalizations x ⑀ ᏹ␸ and (ii) a KMS weight ␺ on (A, ⌬) such that ␺((␫ R ϩ of these measures. Their importance in the more general setting ␻)⌬(x)) ϭ ␺(x)␻(1) for ␻⑀A*ϩ and x ⑀ ᏹ␺ . Then we call is even more pronounced, because—to the present—their exis- (A, ⌬) a locally compact quantum group. tence is an axiom in the definition of a quantum group. It turns out that most properties of a locally compact quantum group can The equality in condition (i) of this definition is called the left be deduced from the existence of generalized Haar measures. invariance of the weight ␸. An important property of locally The usual way to generalize measures (or rather their inte- compact quantum groups is the uniqueness of left invariant

grals) on locally compact spaces is to use weights on von weights: any lower semicontinuous left invariant weight ⌽ on PERSPECTIVE Neumann algebras or, more generally, on C*-algebras. The (A, ⌬) is proportional to ␸. It should be noted that it is possible formal definition of a weight is as follows: consider a C*-algebra to relax the KMS condition somewhat and still get an equivalent ϩ A and a function ␸ : A 3 [0, ϱ] such that: (i) ␸(x ϩ y) ϭ ␸(x) ϩ definition. Similar remarks apply to the right invariant weights. ϩ ϩ ␸(y) for all x, y ⑀ A , and (ii) ␸(rx) ϭ r␸(x) for all x ⑀ A and Also the density conditions in the definition can be slightly r ⑀ [0, ϱ[. We call ␸ a weight on A. The weight ␸ is called faithful weakened. if ␸(x) ϭ 0 N x ϭ 0 for all x ⑀ Aϩ. Denote the set of positive ϩ As already mentioned, the main drawback to this definition is integrable elements of ␸ by ᏹ␸ , and the set of all integrable the assumption of the existence of the left and right invariant ϩ ϩ elements by ᏹ␸. More precisely, ᏹ␸ ϭ {x ⑀ A ͉␸(x) Ͻϱ}, and ϩ weights (including their KMS properties), which is in sharp ᏹ␸ is the linear span of ᏹ␸ . There exists a unique linear contrast with the compact and discrete cases. So far no one has functional on ᏹ␸ which extends ␸, and this will still be denoted been able to formulate a general definition of a locally compact by ␸. We say that ␸ is densely defined when ᏹ␸ is dense in A. quantum group without assuming the existence of invariant In order to render weights useful, we have to impose a weights. continuity condition on them. The relevant continuity condition Following our paper (13), we should call the object in our is the usual lower semicontinuity as a function from Aϩ to [0, ϱ]. definition a reduced locally compact quantum group, because we Loosely speaking, this boils down to requiring the weight to require the left invariant weight to be faithful. However, given satisfy the lemma of Fatou (lower semicontinuity also implies any ‘‘locally compact quantum group,’’ one can associate with it some monotone convergence properties). From now on, all a reduced locally compact quantum group that is essentially

Kustermans and Vaes PNAS ͉ January 18, 2000 ͉ vol. 97 ͉ no. 2 ͉ 549 Downloaded by guest on October 1, 2021 equivalent to the original locally compact quantum group. G, so it should be no surprise that also in the development of the Therefore, the faithfulness of the Haar weight is not a major theory of locally compact quantum groups, such a unitary plays topic, and we will leave out the prefix ‘‘reduced.’’ an important role. From these axioms, one can construct (but this is highly non We already explained that the L2-space of the Haar measure trivial) the antipode S, which is a closed generally unbounded is replaced by the GNS construction of the Haar weight ␸ of our operator that is only densely defined. The unboundedness is locally compact quantum group (A, ⌬). Then we can define a controlled by the existence of a unique bounded *-antiautomor- unitary W on H R H such that W*(⌳(a) R ⌳(b)) ϭ (⌳ R phism R on A and a unique norm-continuous one-parameter ⌳)(⌬(b)(a R 1)) for all a, b ⑀ ᏺ␸. It is easy to check that W*is ␶ 2 ϭ ␫ ␶ group on A, such that (i) R ,(ii) R and commute, and an isometry by using the left invariance of the weight ␸. The ϭ ␶ ϭ ␶ (iii) S R Ϫi/2. Observe that the equation S R Ϫi/2 has to be coassociativity property of the comultiplication is encoded in the understood in the following sense: when a ⑀ A is analytic with ϭ formula W12W13W23 W23W12, called the Pentagon equation. respect to ␶, then a belongs to the domain of S and S(a) ϭ ␶ One can verify this equation immediately for the Kac–Takesaki R( Ϫi/2(a)). Moreover, these analytic elements give a core (or operator of a locally compact group G. essential domain) for the unbounded linear map S. The pair Still more information on (A, ⌬) is hidden in W. Because the R ␶ polar decomposition S ( , ) is called the of . The *-antiauto- weight ␸ is faithful, the representation ␲ will be faithful as well, morphism R is called the unitary antipode of (A, ⌬), and the and we can identify A with ␲(A) through ␲. Then A is the closure one-parameter group ␶ is called the scaling group of (A, ⌬). Now and the comultiplication is ,{ء(of the set {(␫ R ␻)(W)͉␻⑀B(H a Kac algebra will be precisely a von Neumann algebraic version given by ⌬(x) ϭ W*(1 R x)W for all x ⑀ A. There is also an of a locally compact quantum group satisfying the extra condi- ␶ ϭ ␫ ⑀ ޒ ϭ ␴ ϭ ␴ important link between the antipode S and the multiplicative tions t for all t (or equivalently S R) and R t ϪtR ␻⑀ ␫ R ␻ (one proves that ( )(W ,ء(where ␴ denotes the modular group of the left unitary. For every B(H ,ޒ ⑀ for all t ␸ belongs to the domain of S and S((␫ R ␻)(W)) ϭ (␫ R ␻)(W*). invariant weight . ␫ R ␻ One of the axioms of Kac algebras is the strong left invariance. Moreover, the elements ( )(W) form a core for the This gives a relation between the left Haar weight and the antipode. antipode. Now, because the antipode is constructed in our Before the theory of locally compact quantum groups as theory, the strong left invariance will be a theorem: for all a, b described above, a systematic study of such unitaries W was made ⑀ ᏺ␸, one can prove that x :ϭ (␫ R ␸)(⌬(a*)(1 R b)) belongs by Baaj and Skandalis (see ref. 10). The starting is a Hilbert R to the domain of S and S(x) ϭ (␫ R ␸)((1 R a*)⌬(b)). Here ᏺ␸ space H and a unitary W on H H, satisfying the Pentagon denotes the set of all square-integrable elements of ␸ in A, ᏺ␸ ϭ equation. This is called a multiplicative unitary. In their paper, {x ⑀ A͉␸(x*x) Ͻϱ}, and ␫ R ␸ is the slice map. This map can Baaj and Skandalis introduced two extra axioms, called regularity be characterized as follows. If x ⑀ M(A R A)ϩ, we say that x ⑀ and irreducibility, which made it possible to prove that the closure ϩ ϩ is a C*-algebra, and that the {ء(ᏹ␫R␸ when y ⑀ M(A) exists, such that ␻(y) ϭ ␸((␻ R ␫)(x)) for of the set {(␫ R ␻)(W)͉␻⑀B(H all ␻⑀A*ϩ. Then y is unique and is denoted by (␫ R ␸)(x). formula ⌬(x) ϭ W*(1 R x)W defines a comultiplication on this ϩ We denote by ᏹ␫R␸ the linear span of ᏹ␫R␸ and extend ␫ R ␸ C*-algebra. The theory they develop is very elegant and beau- to a linear map from ᏹ␫R␸ to M(A). Now one can prove that tiful: they obtain many quantum group-like features for this both ⌬(a*)(1 R b) and (1 R a*)⌬(b) belong to ᏹ␫R␸, so that C*-algebra with comultiplication, such as a generalization of the the formulas above make sense. In the commutative case Takesaki–Takai duality theorem for crossed products with abe- R ϭ ϫ ␸ M(A A) Cb(X X), corresponds to a regular Borel lian locally compact groups. Later, Baaj proved that the quantum measure on X, and ␫ R ␸ integrates out the second variable. E(2) group, as constructed by Woronowicz, did not satisfy the ⌬ ␹ R ⌬ϭ The unitary antipode anticommutes with , i.e., (R R) axiom of regularity, so that one should weaken this axiom a bit ⌬ ␹ R R, where denotes the flip-automorphism extended to M(A in order to include all locally compact quantum groups. ␸ A). This means, in particular, that R is a faithful right invariant Woronowicz also has studied multiplicative unitaries (see ref. KMS weight on A. It should be pointed out that, nevertheless, ␺ ␶ ⌬␴ ϭ 11). He replaced the axioms of regularity and irreducibility by a is needed in the construction of S, R, and . Also, t ␶ R ␴ ⌬ ⌬␶ ϭ ␶ R ␶ ⌬ ⑀ ޒ ␴ completely different, and stronger, axiom called manageability. ( t t) , and t ( t t) for all t , where denotes Woronowicz then associates with every manageable multiplica- again the modular group of the left Haar weight ␸. Further, there ␯ ␸␶ ϭ ␯Ϫt␸ ⑀ ޒ tive unitary a C*-algebra with comultiplication by the same exists a positive number such that t for all t .We formulas as above. He is also able to define an antipode S call ␯ the scaling constant. In all the known examples, one has ␯ ϭ satisfying the polar decomposition S ϭ R␶Ϫ . Now it is possible 1, but we do not know whether this holds in general. i/2 to prove that the multiplicative unitary associated with a locally The role of the L2-space of the Haar measure is played by the compact quantum group as above is always manageable. More- Gelfand–Neumark–Segal (in short, GNS) representation asso- over, the antipode with polar decomposition as obtained in ciated with the weight ␸. This is a triple (H, ␲, ⌳), where: (i) H Woronowicz’ theory is the same as the one obtained in the is a Hilbert space; (ii) ⌳ is a linear map from ᏺ␸ into H such that theory of locally compact quantum groups. ⌳(ᏺ␸) is dense in H, ͗⌳(a), ⌳(b)͘ϭ␸(b* a) for every a, b ⑀ ᏺ␸; and (iii) ␲ is a *-representation of A on H such that ␲(a)⌳(b) ϭ It is an open problem whether, conversely, every manageable ⌳(ab) for every a ⑀ A and b ⑀ ᏺ␸. Here ͗⅐, ⅐͘ denotes the inner multiplicative unitary admits a left invariant weight and hence product on H. gives rise to a locally compact quantum group. The Multiplicative Unitary Duality One of the motivations for the development of the theory of Kac As a motivating example, we return to a locally compact group algebras was the pioneering work of Takesaki (see ref. 18) on G. Another way of associating a quantum group with G is via the generalizing the theorem (e.g., ref. 17) to group C*-algebra construction, which is more involved than the nonabelian locally compact groups. The main tool in the work of above construction of C0(G). One starts by fixing a left Haar Takesaki is the so-called Kac–Takesaki operator. Let G be a measure ␮ on G and considers the normed space L1(G)of locally compact group and fix a left Haar measure on it. Then we functions on G, integrable with respect to ␮, where the norm is can define a unitary on the Hilbert space L2(G) R L2(G) given the ordinary L1-norm. Next, it is customary to turn L1(G) into and the ء by (W␰)(p, q) ϭ ␰(p, pϪ1q). The important point is that this a *-algebra by introducing the product unitary encodes all the information of the locally compact group appropriate *-operation ° on L1(G):

550 ͉ www.pnas.org Kustermans and Vaes Downloaded by guest on October 1, 2021 Ϫ g)(t) ϭ͐f(s)g(s 1t)d␮(s) for all f, g ⑀ L1(G) and almost 3. DEFINITION. We define: (i) the set Aˆ s the norm closure of {(␻ ء Y (f and (ii) the injective linear map ⌬ˆ : Aˆ 3 B(H ;{ء(all t ⑀ G, R ␫)(W)͉␻⑀B(H R H), such that ⌬ˆ (x) ϭ⌺W(x R 1)W* ⌺ for all x ⑀ A.ˆ Y fЊ(t) ϭ ␦(t)Ϫ1f(tϪ1) for all f ⑀ L1(G) and almost all t ⑀ G, We use the flip map ⌺ on H R H to guarantee that the dual where ␦ denotes the modular function of the locally compact weight constructed from ␸ will again be left invariant rather than group G, which connects the left and the right Haar measure on right invariant. Thanks to the results in ref. 11, the manageability G. It should be stressed that L1(G) is not a C*-algebra, only a of W implies that the set Aˆ is a nondegenerate C*-subalgebra of ⌬ˆ Banach *-algebra. B(H), and the mapping is a nondegenerate *-homomorphism ˆ ˆ R ˆ ⌬ˆ R ␫ ⌬ϭˆ ␫ R ⌬ˆ ⌬ˆ A possible way of obtaining a C*-algebra is by using the left from A into M(A A) such that: (i)( ) ( ) ; and ⌬ˆ ˆ ˆ R ⌬ˆ ˆ R ˆ ˆR ˆ of G. The left regular representation s x (ii) (A)(A 1) and (A)(1 A) are dense of A A. ␭ We will now introduce a notation that strengthens the analogy s of G is a of G acting on the space of 1 square integrable functions L2(G), and is defined by the formula with the classical group case. We define L (A) to be the closed Ϫ linear span of {a␸b*͉a, b ⑀ ᏺ␸}inA*. We use the notation (␭ g)(t) ϭ g(s 1t) for all g ⑀ L2(G) and s, t ⑀ G. This s (a␸b*)(x) ϭ ␸(b*xa). Now denote by A˜ the von Neumann representation gives rise to the left regular *-representation ␭ of algebra in B(H) generated by ␲(A). Then for every ␻⑀L1(A), L1(G)onL2(G), defined by ␭(f) ϭ͐f(s)␭ d␮(s) for all f ⑀ ␻␲ ϭ ␻ ء ˜ ⑀ s ␻ 1 there is a unique ˜ A , such that ˜ , and hence we can L (G), where the is formed in the strong of define the injective contractive linear mapping ␲ˆ : L1(A) 3 Aˆ ␭ 1 3 2 B(H). One can prove that : L (G) B(L (G)) is a faithful such that ␲ˆ(␻) ϭ (␻˜ R ␫)(W). In the group case, this is the left ␭ 1 *-representation. Define C*r(G) to be the closure of (L (G)) in regular representation of L1(G)onL2(G) mentioned above. We B(H). The C*-algebra C*r(G) is referred to as the reduced dual also mention that the expression ␻␮ ϭ (␻ R ␮)⌬ turns L1(A) into ␭ of G. Then the unitaries s belong to the multiplier algebra a , and that ␲ˆ becomes multiplicative this way. In ⑀ M(C*r(G)) for all s G. It is possible to prove the existence of the classical case, this comes down to the usual convolution ⌬ 3 1 a unique nondegenerate *-homomorphism : C*r(G) product on L (G) described above. R ⌬ ␭ ϭ ␭ R ␭ ⑀ M(C*r(G) C*r(G)) such that ( s) s s for all s G, and Now we want to define a left invariant weight on (Aˆ, ⌬ˆ ) using ⌬ Ᏽ 1 it turns out that the pair (C*r(G), ) is a locally compact quantum definition 2.1.6 of ref. 5. Define the subset of L (A) as follows: group. Ᏽ ϭ {␻⑀L1(A)͉ there exists a number M Ն 0 such that ͉␻(x*)͉ Let us now look at the case where G is abelian. Classical group Յ Mʈ⌳(x)ʈ for all x ⑀ ᏺ␸}. It is clear that Ᏽ is a subspace of theory tells us how to construct the dual group Gˆ . As a set, Gˆ L1(A). By Riesz’ theorem for Hilbert spaces, there exists for ␻⑀Ᏽ ␷ ␻ ⑀ ␻ ϭ͗␷ ␻ is the set of all continuous group characters on G taking values every a unique element ( ) H such that (x*) ( ), ⌳ ͘ ⑀ ᏺ in the unit circle. The group multiplication on Gˆ is just the (x) for x ␸. It can be shown that there exists a unique closed ⌳ˆ ⌳ˆ ʕ ˆ pointwise multiplication of two characters. The topology of Gˆ is densely-defined linear map from D( ) A into H, such that ˆ ␲ˆ(Ᏽ) is a core for ⌳ˆ and ⌳ˆ(␲ˆ(␻)) ϭ ␷(␻) for all ␻⑀Ᏽ. Finally, the compact-open topology. In this way, G is endowed with the ␸ ˆ structure of a commutative locally compact group. The cele- there exists a unique faithful KMS weight ˆonA, such that (H, ␫, ⌳ˆ ) is a GNS construction for ␸ˆ. brated Pontryagin duality theorem says that the mapping ␪ : G The weight ␸ˆ is a left invariant weight on (Aˆ, ⌬ˆ ). In order to 3 G^^ defined by ␪(s)(␻) ϭ ␻(s) for all ␻⑀Gˆ , and s ⑀ G,is get hold of the right invariant weight on (Aˆ, ⌬ˆ ), we produce the a group isomorphism and a . unitary antipode on (Aˆ, ⌬ˆ ). There exists a unique *-antiauto- * ˆ It is possible to identify C r(G) with C0(G) through a *-iso- morphism Rˆ on Aˆ, such that Rˆ(␲ˆ(␻)) ϭ ␲ˆ(␻R) for all ␻⑀L1(A). ␲ 3 ˆ ␲ ␭ ␻ ϭ morphism : C*r(G) C0(G) defined such that ( (f))( ) This *-antiautomorphism satisfies the relation ␹(Rˆ R Rˆ)⌬ϭˆ ⌬ˆ Rˆ. ͐ ␻ ␮ ⑀ 1 ␻⑀ ˆ f(s) (s)d (s) for all f L (G) and G. It turns out that this Thus, ␸ˆRˆ is a right invariant KMS weight on (Aˆ, ⌬ˆ ). Hence, we ␲ ␲ R ␲ ⌬ϭ map is an isomorphism of quantum groups, i.e., ( ) are led to the following conclusion: ⌬␲ and S␲ ϭ ␲S. This discussion holds for abelian groups but fails for nonabe- 4. THEOREM. The pair (Aˆ, ⌬ˆ ) is a locally compact quantum lian ones. It is impossible to define, by a general construction, an group. appropriate dual locally compact group that encodes essentially all the information about the original locally compact group. The Pontryagin duality theorem for abelian locally compact However, the reduced dual C*-algebra C*r(G) encodes, as a groups also has its generalization to the quantum group setting. quantum group all information about G. But if G is not abelian, In the same way as we constructed the dual (Aˆ, ⌬ˆ )of(A, ⌬), we ^^ ⌬^^ ˆ ⌬ˆ this C*-algebra C*r(G) is noncommutative and cannot arise as can again construct the dual (A , )of(A, )asa (C (H), ⌬) for some locally compact group H. C*-algebra of bounded operators on H (with respect to the GNS PERSPECTIVE 0 ␫ ⌳ ␸ As we explained in the Introduction, here lies the motivation construction (H, , ˆ ) for ˆ). The generalized Pontryagin duality ␲ 3 for defining Kac algebras: one was looking for a larger category, theorem states that the natural : A B(H)isa ^^ ␲ R ␲ ⌬ϭ⌬^^␲ allowing duality, and containing the locally compact groups. The *-isomorphism from A to A such that ( ) , and final definition of Kac algebras was given independently by hence an isomorphism of the locally compact quantum groups (A, ⌬) and (A^^, ⌬^^). Enock and Schwartz and Kac and Vainerman, solving the problem of duality of locally compact groups. The Universal Setting The above discussed construction of the reduced dual of a Let us consider once more a locally compact group G. In the locally compact group can be generalized to the quantum group beginning of the previous section, we constructed the reduced setting. Let us therefore return to our general quantum group 1 group C*-algebra C*r(G) by putting a certain C*-norm on L (G) (A, ⌬) with its left Haar weight ␸. 1 and completing L (G)toC*r(G) with respect to this norm, but The construction of the dual of a Kac algebra can be found in in general there does not exist a unique C*-norm on L1(G). ʈ ʈ chapter 3 of ref. 5. Essentially the same construction can be used Another possible C*-norm is the universal C*-norm . u on 1 ⑀ 1 ʈ ʈ to define the dual of a locally compact quantum group, although L (G): for every x L (G) we understand by definition that x u some simpler proofs can be given because of the stronger general is the supremum of {ʈ␪(x)ʈ͉␪ a *-representation of L1(G)ona theory at hand. Recall that above we defined the GNS construc- Hilbert space}. The universal group C*-algebra C*(G) is by ␲ ⌳ ␸ 1 ʈ ʈ tion (H, , ) for and the multiplicative unitary W. definition the completion of L (G) with respect to the norm . u.

Kustermans and Vaes PNAS ͉ January 18, 2000 ͉ vol. 97 ͉ no. 2 ͉ 551 Downloaded by guest on October 1, 2021 A classical result says that there is a one-to-one correspondence major motivation for its development. Further, the list of axioms between *-representations of the C*-algebra C*(G) and strongly is not too long, and important features such as the uniqueness of continuous unitary representations of the group G. the Haar weights, the existence of the antipode and its polar This situation can be completely generalized to the quantum decomposition, a nice duality theory, the modular element, and group setting. Given a reduced locally compact quantum group the manageability of the multiplicative unitary are established as ⌬ (A, ), one can construct the ‘‘universal’’ C*-algebra Au, such theorems. that there is a one-to-one correspondence between *-represen- The most important drawback to our definition is, of course, ⌬ tations of Au and unitary corepresentations of (Aˆ, ˆ ). the assumption of the existence of the Haar weight, so the major ⌬ Further, it is possible to construct a comultiplication u on Au, challenge is the formulation of an alternative and more elemen- ⌬ and one can show that the pair (Au, u) satisfies the same tary axiom that allows the existence of the Haar weight as a interesting properties as the reduced companion (A, ⌬), except theorem. This is really an enormous challenge, and not all the for the faithfulness of the Haar weights. So we see that given a experts believe it can be realized. reduced locally compact quantum group (A, ⌬), there are at least Another problem has existed since the development of Kac two other interesting operator algebraic companions associated algebras: it is very hard to construct nontrivial examples of ⌬ with it: the universal version (Au, u) and the von Neumann locally compact quantum groups. The most important and well algebraic version (A˜, ⌬˜), which is obtained by extending ⌬ to the known examples have been discovered by Woronowicz, e.g., the von Neumann algebra A˜ generated by ␲(A). Although these quantum E(2), quantum Lorentz, or quantum ax ϩ b-groups, three algebras A, Au, and A˜ have significantly different proper- and each single example is a subtle and beautiful piece of ties as operator algebras, they are three different realizations of mathematics. Although there exist some general construction the same underlying ‘‘quantum group’’ (each of them can be procedures—we think of the quantum double construction, the canonically recovered from the other one). Rather than seeing crossed and bicrossed products, or the locally compact quantum this as something awkward, it should be seen as an advantage, groups arising from multiplier Hopf *-algebras with positive because a certain realization might allow certain manipulations integrals—we do not obtain the most interesting examples in this that are not possible in another realization. way. It is another major challenge to develop a construction procedure that is both easy enough to be feasible and subtle Concluding Remarks enough to produce nice and nontrivial examples. Before going on with the statement of perspectives and chal- Finally, when one compares the locally compact groups with lenges of the theory, we would like to summarize what we have the locally compact quantum groups, the second theory is done so far. Roughly speaking, we proposed a definition for a developed only up to its fundamentals: given a definition, one locally compact quantum group and discussed its properties. This should now prove theorems or study special classes of locally theory includes the locally compact groups, the compact and compact quantum groups. In our opinion, a serious amount of discrete quantum groups, and the Kac algebras, which was a work can be done in this direction.

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