The operator algebra approach to quantum groups

Abstract

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.

The nowadays popular topic of quantum groups can be approached from two essentially different directions. The first and most widespread approach is algebraic in nature. The first successes of this approach date back to Drinfel'd (see ref. 1) and Jimbo (see ref. 2), who defined one-parameter deformations of universal enveloping algebras of semisimple complex Lie algebras in 1985. Many other classes of Hopf algebras have been studied since 1985 and many received the label “quantum group.” The second approach is analytic in nature: the basic motivation in the early development of the theory was the generalization of Pontryagin duality for abelian locally compact groups. Because the dual of a nonabelian group can no longer be a group, one looked for a larger category that was self-dual. These generalized objects would be called quantum groups again.

This paper will deal with the second analytic approach. A major role will be played throughout by proper generalizations of the Haar measure of a locally compact group, which is one of the points where the theory differs thoroughly from the algebraic approach. Most of the Hopf algebras that are called quantum groups do not possess a proper generalization of a Haar measure.

Between these two approaches is the theory of multiplier Hopf *-algebras with integrals, which is studied in ref. 3 by Van Daele. This theory is purely algebraic in nature, but nevertheless one has the analogue of the Haar measure. This framework of multiplier Hopf *-algebras has the advantage of being easier to understand than the often very technical operator algebra approach, but it is not general enough to include all locally compact quantum groups. On the level of formal manipulations, all technicalities of course disappear, and some of the formulas appearing in the operator algebra approach are very similar to those considered by Van Daele.

This paper is the second in a series of papers, the first being ref. 4. Both papers can be read independently, but the remarks above make it clear that reading ref. 4 first will yield a better insight into the motivation underlying this paper.

The approach to quantum groups through operator algebras dates back to the 1970s. After the pioneering work of Takesaki, Tannaka, Krein, and many others, the problem of finding a self-dual category containing the locally compact groups was completely solved independently by Kac and Vainerman and by Enock and Schwartz. The object they defined is called a Kac algebra; see ref. 5 for an overview. A Kac algebra is a von Neumann algebra with much extra structure on it, the two basic examples being the essentially bounded measurable functions on a locally compact group and the group von Neumann algebra.

A new phase in the development of the theory began with Woronowicz' construction of the quantum SU(2) group in an operator algebraic framework (see ref. 6). This object had all the properties that justified calling it a quantum group, but nevertheless it did not fit in the framework of Kac algebras. Woronowicz also built up a whole theory of so-called compact quantum groups in refs. 7 and 8, and most of them were not Kac algebras. So it became a challenge to define an even larger category including both the Kac algebras and the compact quantum groups. One started looking for the definition of a locally compact quantum group.

A first attempt was made by Masuda and Nakagami in ref. 9, who formulated a definition of locally compact quantum groups (they called them Woronowicz algebras) in the framework of von Neumann algebras. A C*-algebraic version of this definition was presented in lectures by Masuda, Nakagami, and Woronowicz, but was never published. The main drawback to their approach, however, was the complexity of their axioms. In fact, many nice features of a locally compact quantum group, which one would like to prove from more elementary axioms, are presupposed by definition. Nevertheless, they are able to give a duality inside their category, and their theory indeed includes compact quantum groups and Kac algebras, hence also locally compact groups.

We also mention here the fundamental work of Baaj and Skandalis (see ref. 10) and of Woronowicz (see ref. 11) on multiplicative unitaries, about which we will tell more later.

Recently, we have given a much simpler definition for locally compact quantum groups in refs. 12 and 13, and we are able to prove (approximately) all the axioms considered by Masuda, Nakagami, and Woronowicz. The major remaining problem is that in our definition—just as in the definition of Masuda, Nakagami, Woronowicz or of Kac algebras—we still assume the existence of the Haar measure. It would, of course, be more elegant to give a definition for locally compact quantum groups without assuming this and with the existence of the Haar measure as a theorem. Taking into account all the efforts that have been made to prove the existence of the Haar measure in such a general situation, it seems that this is far out of reach.

This paper is organized as follows. In the first section, we explain how C*-algebras turn up when one wants to define locally compact quantum groups. Then we will explain the well-developed theory of compact quantum groups and go on with our definition of a locally compact quantum group, whose major properties, such as the existence of the antipode and the uniqueness of the Haar measure, will be subsequently described. We also construct the multiplicative unitary and develop the duality theory.

Quantizing Locally Compact Groups

Let us start with the easiest step of the quantization procedure that we will consider in this paper. Suppose X is a locally compact space. We will shift our attention to the space C₀(X) of continuous complex valued functions on X, vanishing at infinity. With pointwise operations and the supremum norm, this is a commutative C*-algebra. An important theorem of Gelfand and Neumark in their paper (14) says that every commutative C*-algebra is of this form (Stone proved a real version of this theorem four years earlier; see ref. 15). This is why one could speak about a C*-algebra as a locally compact quantum space or as the functions on a quantum space, vanishing at infinity. All this was realized very early in the development of the theory of C*-algebras (1950s).

It is well known that locally compact spaces form a category with continuous maps as morphisms. How can this be quantized? Let X and Y be locally compact spaces and θ : X → Y a continuous map. Then we can define ρ : C₀(Y) → C_b(X), the C*-algebra of bounded continuous functions on X, by putting ρ(f) = f ○ θ. Next, we will explain a C*-algebraic procedure to obtain C_b(X) out of the C*-algebra C₀(X). Let A be a C*-algebra. Then we define M(A) as the set of all linear maps T from A to A, which have an adjoint, i.e., for which there exists a linear map T* from A to A, satisfying b*T(a) = (T*(b))*a for all a, b ε A. When T ε M(A), one can prove that T is bounded as an operator from A to A and, using the operator norm of T, the necessarily unique adjoint T* defined above and the composition of mappings as multiplication, we get a C*-algebra called the multiplier algebra of A. We can embed x ε A into M(A) by the formulas T(a) = xa and T*(a) = x*a for all a ε A. It is easy to verify that indeed M(C₀(X)) = C_b(X). If H is a Hilbert space and A a C*-subalgebra of B(H) that acts nondegenerately, then M(A) can be identified with {x ε B(H)|xA, Ax ⊆ A}.

Now write A = C₀(X) and B = C₀(Y). We have defined ρ : B → M(A). It is easy to verify that ρ is a *-homomorphism and that ρ(B)A is dense in A. This last condition expresses that ρ is nondegenerate. Given two C*-algebras A and B, we call a map ρ : B → M(A) a morphism from B to A when ρ is a nondegenerate *-homomorphism. The set of these morphisms is denoted by Mor(B, A). The correspondence between θ and ρ in the previous paragraph gives a bijection between the continuous maps from X to Y and the morphisms from C₀(Y) to C₀(X). Finally, if A and B are C*-algebras and ρ ε Mor(B, A), it is possible to extend ρ uniquely to a unital *-homomorphism from M(B) to M(A), still denoted by ρ, such that ρ(m)ρ(b)a = ρ(mb)a for m ε M(B), b ε B, and a ε A.

Now let G be a locally compact group. Inspired by the quantization procedure above, we would like to translate the group structure on G to the C*-algebra C₀(G). Multiplication is a continuous map from G × G to G and hence can be translated to a morphism Δ : C₀(G) → M(C₀(G × G)) by the formula Δ(f)(p, q) = f(pq). Identifying f ⊗ g with the function that sends (p, q) to f(p)g(q), we get an isomorphism between C₀(G) ⊗ C₀(G), the C*-tensor product, and C₀(G × G). So we get a morphism Δ : C₀(G) → M(C₀(G) ⊗ C₀(G)), and associativity is translated to the formula (Δ ⊗ ι)Δ = (ι ⊗ Δ)Δ, called coassociativity. This can be given a meaning, because Δ ⊗ ι and ι ⊗ Δ are again morphisms and hence can be extended to the multiplier algebra M(C₀(G) ⊗ C₀(G)). The coassociativity is easy to verify: ((Δ ⊗ ι)Δ(f))(p, q, r) = Δ(f)(pq, r) = f((pq)r), whereas ((ι ⊗ Δ)Δ(f))(p, q, r) = f(p(qr)).

If A is an arbitrary C*-algebra, a morphism Δ ε Mor(A, A ⊗ A), where A ⊗ A denotes the minimal C*-tensor product, is called a comultiplication on A when it satisfies the coassociativity formula (Δ ⊗ ι)Δ = (ι ⊗ Δ)Δ. Such a pair (A, Δ) is sometimes called a locally compact quantum semigroup.

Now one can go further and try to translate also the unit and the inverse to the C*-algebra level. The former gives rise to a *-homomorphism ɛ : C₀(G) → ℂ given by ε(f) = f(e) and the latter to a *-automorphism S of C₀(G) given by (Sf)(p) = f(p⁻¹). The properties of the unit and the inverse can be expressed by the formulas (ι ⊗ ε)Δ = (ε ⊗ ι)Δ = ι for the counit ε and m(ι ⊗ S)Δ(f) = ε(f)1 = m(S ⊗ ι)Δ(f) for the antipode S. Here we again used the necessary extensions to the multiplier algebra, and m is the map from M(C₀(G) ⊗ C₀(G)) to M(C₀(G)) given by (m(h))(p) = h(p, p). This map m is called the multiplication map, because m(f ⊗ g) = fg.

Now, trying to continue the quantization procedure, one would like to write down the formulas above as axioms for the counit and antipode on an arbitrary C*-algebra with comultiplication, but then we run into all kinds of trouble. First of all, examples show that it is too restrictive to assume that ε is a *-homomorphism that is defined everywhere. In general, ε could be unbounded, and then it is not so clear how we can give a meaning to ι ⊗ ε and ε ⊗ ι on the completed tensor product A ⊗ A. Further, examples also show that the antipode S can be unbounded and need not be a *-map. This gives the same kind of problems for S ⊗ ι and ι ⊗ S. Finally, the multiplication map m, in general, also cannot be defined on the whole of A ⊗ A or M(A ⊗ A). That this can be done is typical of the commutative situation.

Therefore, it will be necessary to replace both axioms by other axioms in order to define locally compact quantum groups. This will be done further in the paper.

Compact Quantum Groups

Compact quantum groups form the best-understood part of the theory of quantum groups. Woronowicz developed this theory in his fundamental papers (refs. 7 and 8). The typical example is quantum SU(2), studied by Woronowicz in ref. 6. Woronowicz also introduced the differential calculus on compact quantum groups and used it to unravel the representation theory of quantum SU(2).

In order to motivate the definition of a compact quantum group, we go back to the quantization procedure described in the previous section. The compactness of a locally compact space X is expressed by the fact that the C*-algebra C₀(X) has a unit, therefore unital C*-algebras will be considered as compact quantum spaces. Now let G be a compact semigroup; then we can define Δ : C(G) → C(G × G), as before. Here C(G) denotes the C*-algebra of continuous functions on G, and we observe that C(G) = C₀(G), because G is compact. As we explained above, it seems impossible to write down easy axioms for the antipode and counit. Nevertheless, we can consider a weaker possible property of G, and that is the cancellation law: if rs = rt, then s = t and analogously on the other side. It is not hard to prove that a compact semigroup with the cancellation law is, in fact, a compact group, but there is more. This cancellation law can easily be translated to a property of the C*-algebra C(G), using the lemma of Urysohn (e.g., ref. 17). One can prove that G satisfies the cancellation law if, and only if, the linear spaces Δ(C(G))(C(G) ⊗ 1) and Δ(C(G))(1 ⊗ C(G)) are dense in C(G × G).

This discussion makes the following definition of Woronowicz more or less acceptable:

1. Definition. Consider a unital C*-algebra A together with a unital *-homomorphism Δ : A → A ⊗ A such that (Δ ⊗ ι)Δ = (ι ⊗ Δ)Δ and such that the spaces Δ(A)(A ⊗ 1) and Δ(A)(1 ⊗ A) are dense in A ⊗ A. Then the pair (A, Δ) is called a compact quantum group.

The main reason for the success of compact quantum groups lies in the fact that this rather elegant definition of a compact quantum group (A, Δ) allowed Woronowicz to generalize the whole theory of compact groups to the quantum group setting. The pivotal result is the existence of a unique state ϕ on A, that is left and right invariant, i.e., (ϕ ⊗ ι)Δ(a) = (ι ⊗ ϕ)Δ(a) = ϕ(a) 1 for all a ε A. The state ϕ is called the Haar state of the compact quantum group (A, Δ).

The most important consequence of the definition is the quantum version of the classical Peter–Weyl theorem. For this, we need the notion of a unitary corepresentation of a compact quantum group as a generalization of a strongly continuous unitary group representation of a compact group. Let H be a Hilbert space and denote the compact operators on H by B₀(H). A unitary element U in M(A ⊗ B₀(H)) such that (Δ ⊗ ι)(U) = U₁₃U₂₃ is called a unitary corepresentation of (A, Δ) on H. Here we used the so-called leg-numbering notation U₁₃ and U₂₃. Both are elements of M(A ⊗ A ⊗ B₀(H)) and U₂₃ = 1 ⊗ U, whereas U₁₃ = (χ ⊗ ι)(U₂₃), where χ denotes the flip map on A ⊗ A, sending a ⊗ b to b ⊗ a. We say that U is finite dimensional if H is finite dimensional. The unitary corepresentation U is called irreducible if the commutant {(ω ⊗ ι)(U)|ω ε A*}′ in B(H) is equal to ℂ1. As in the classical case, every irreducible unitary corepresentation of (A, Δ) is finite dimensional.

Let 𝒜 be the subset of A defined by

Woronowicz proved that 𝒜 is a dense *-subalgebra of A and that 𝒜 together with the restriction of Δ to 𝒜 forms a Hopf *-algebra with positive integrals (see ref. 3 for definitions). The presence of the Hopf *-algebra structure on 𝒜 indicates that compact quantum groups can be studied within an algebraic framework. It should be pointed out, however, that the existence of the Haar state can be established only within the C*-algebra framework, unless the axioms are considerably strengthened.

Let (U_λ)_λεΛ be a complete set of mutually inequivalent irreducible unitary corepresentations of (A, Δ) on finite dimensional Hilbert spaces (H_λ)_λεΛ. Let λ ε Λ and fix an orthonormal basis (e₁(λ), … , e_nλ(λ)) of H_λ. For i, j ε {1, … , n_λ}, we define ω_i,j ε B(H_λ)* by ω_i,j(x) = 〈xe_i(λ), e_j(λ)〉 for all x ε B(H_λ). Moreover, we put U_ij(λ) = (ι ⊗ ω_i,j)(U_λ). The quantum version of the Peter–Weyl theorem says that the family {U_ij(λ)|λ ε Λ, i, j = 1, … , n_λ} is a Hamel basis of the vector space 𝒜 and that quantized orthogonality relations hold between these elements.

Locally Compact Quantum Groups

Before stating the definition of a locally compact quantum group as it was given in refs. 12 and 13, we need some extra terminology concerning weights on C*-algebras. The most important objects associated with a locally compact group are its Haar measures, so it is no big surprise that in the quantum group setting equally fundamental roles are also played by the proper generalizations of these measures. Their importance in the more general setting is even more pronounced, because—to the present—their existence is an axiom in the definition of a quantum group. It turns out that most properties of a locally compact quantum group can be deduced from the existence of generalized Haar measures.

The usual way to generalize measures (or rather their integrals) on locally compact spaces is to use weights on von Neumann algebras or, more generally, on C*-algebras. The formal definition of a weight is as follows: consider a C*-algebra A and a function ϕ : A⁺ → [0, ∞] such that: (i) ϕ(x + y) = ϕ(x) + ϕ(y) for all x, y ε A⁺, and (ii) ϕ(rx) = rϕ(x) for all x ε A⁺ and r ε [0, ∞[. We call ϕ a weight on A. The weight ϕ is called faithful if ϕ(x) = 0 ⇔ x = 0 for all x ε A⁺. Denote the set of positive integrable elements of ϕ by ℳ_ϕ⁺, and the set of all integrable elements by ℳ_ϕ. More precisely, ℳ_ϕ⁺ = {x ε A⁺|ϕ(x) < ∞}, and ℳ_ϕ is the linear span of ℳ_ϕ⁺. There exists a unique linear functional on ℳ_ϕ which extends ϕ, and this will still be denoted by ϕ. We say that ϕ is densely defined when ℳ_ϕ is dense in A.

In order to render weights useful, we have to impose a continuity condition on them. The relevant continuity condition is the usual lower semicontinuity as a function from A⁺ to [0, ∞]. Loosely speaking, this boils down to requiring the weight to satisfy the lemma of Fatou (lower semicontinuity also implies some monotone convergence properties). From now on, all weights we consider will be lower semicontinuous and densely defined. When A = C₀(X) is a commutative C*-algebra, there is a bijective correspondence between such weights ϕ on A and regular Borel measures μ on X given by ϕ(f) = ∫ f(x)dμ(x) for all positive functions f in C₀(X).

A truly noncommutative phenomenon is the Kubo–Martin–Schwinger (in short, KMS) property for weights. Although the C*-algebra may be noncommutative, the KMS condition gives some control over the noncommutativity under the weight. In order to make this more precise, we need the notion of a one-parameter group and its analytic extension.

Let α : ℝ → Aut(A) be a mapping such that: (i) α_s α_t = α_s+t for all s, t ε ℝ, and (ii) the function ℝ → A : t ↛ α_t(a) is norm continuous for all a ε A, where Aut(A) denotes the set of all *-automorphisms of the C*-algebra A. Then we call α a norm-continuous one-parameter group on A. An element a ε A is called analytic with respect to α if the function ℝ → A : t ↛ α_t(a) can be extended to an analytic function f : ℂ → A. In that case, the element α_z(a) is defined as f(z).

Now consider a weight ϕ on A. It is called a KMS weight if a norm-continuous one-parameter group σ on A exists such that: (i) ϕ is invariant under σ, i.e., ϕσ_t = ϕ for every t ε ℝ, and (ii) for every a ε ℳ_ϕ and b ε A, with b analytic with respect to σ, one has ab, bσ_−i(a) ε ℳ_ϕ and ϕ(ab) = ϕ(bσ_−i(a)). This formula also appears in the algebraic approach to quantum groups, as explained in theorem 8 of ref. 4. If ϕ is a KMS weight (this is not automatically true for every lower semicontinuous faithful weight), such a one-parameter group σ is called a modular group of ϕ. If ϕ is faithful, σ is uniquely determined by the properties above. The KMS condition is really the key result that allows one to develop a generalized noncommutative measure theory that parallels classical measure theory (for instance, the Radon–Nikodym Theorem has a generalization to weights in the von Neumann algebra framework).

We have now gathered enough material to formulate the definition of a locally compact quantum group.

2. Definition. Consider a C*-algebra A and a nondegenerate *-homomorphism Δ : A → M(A ⊗ A) such that: (i) (Δ ⊗ ι)Δ = (ι ⊗ Δ)Δ and (ii) the linear spaces Δ(A)(1 ⊗ A) and Δ(A)(A ⊗ 1) are dense in A ⊗ A.

Assume, moreover, the existence of: (i) a faithful KMS weight ϕ on (A, Δ) such that ϕ((ω ⊗ ι)Δ(x)) = ϕ(x)ω(1) for ω ε A^*₊ and x ε ℳ_ϕ⁺ and (ii) a KMS weight ψ on (A, Δ) such that ψ((ι ⊗ ω)Δ(x)) = ψ(x)ω(1) for ω ε A^*₊ and x ε ℳ_ψ⁺. Then we call (A, Δ) a locally compact quantum group.

The equality in condition (i) of this definition is called the left invariance of the weight ϕ. An important property of locally compact quantum groups is the uniqueness of left invariant weights: any lower semicontinuous left invariant weight Φ on (A, Δ) is proportional to ϕ. It should be noted that it is possible to relax the KMS condition somewhat and still get an equivalent definition. Similar remarks apply to the right invariant weights. Also the density conditions in the definition can be slightly weakened.

As already mentioned, the main drawback to this definition is the assumption of the existence of the left and right invariant weights (including their KMS properties), which is in sharp contrast with the compact and discrete cases. So far no one has been able to formulate a general definition of a locally compact quantum group without assuming the existence of invariant weights.

Following our paper (13), we should call the object in our definition a reduced locally compact quantum group, because we require the left invariant weight to be faithful. However, given any “locally compact quantum group,” one can associate with it a reduced locally compact quantum group that is essentially equivalent to the original locally compact quantum group. Therefore, the faithfulness of the Haar weight is not a major topic, and we will leave out the prefix “reduced.”

From these axioms, one can construct (but this is highly non trivial) the antipode S, which is a closed generally unbounded operator that is only densely defined. The unboundedness is controlled by the existence of a unique bounded *-antiautomorphism R on A and a unique norm-continuous one-parameter group τ on A, such that (i) R² = ι, (ii) R and τ commute, and (iii) S = Rτ_−i/2. Observe that the equation S = Rτ_−i/2 has to be understood in the following sense: when a ε A is analytic with respect to τ, then a belongs to the domain of S and S(a) = R(τ_−i/2(a)). Moreover, these analytic elements give a core (or essential domain) for the unbounded linear map S. The pair (R, τ) is called the polar decomposition of S. The *-antiautomorphism R is called the unitary antipode of (A, Δ), and the one-parameter group τ is called the scaling group of (A, Δ). Now a Kac algebra will be precisely a von Neumann algebraic version of a locally compact quantum group satisfying the extra conditions τ_t = ι for all t ε ℝ (or equivalently S = R) and Rσ_t = σ_−tR for all t ε ℝ, where σ denotes the modular group of the left invariant weight ϕ.

One of the axioms of Kac algebras is the strong left invariance. This gives a relation between the left Haar weight and the antipode. Now, because the antipode is constructed in our theory, the strong left invariance will be a theorem: for all a, b ε 𝒩_ϕ, one can prove that x := (ι ⊗ ϕ)(Δ(a*)(1 ⊗ b)) belongs to the domain of S and S(x) = (ι ⊗ ϕ)((1 ⊗ a*)Δ(b)). Here 𝒩_ϕ denotes the set of all square-integrable elements of ϕ in A, 𝒩_ϕ = {x ε A|ϕ(x*x) < ∞}, and ι ⊗ ϕ is the slice map. This map can be characterized as follows. If x ε M(A ⊗ A)⁺, we say that x ε ℳ_ι⊗ϕ⁺ when y ε M(A)⁺ exists, such that ω(y) = ϕ((ω ⊗ ι)(x)) for all ω ε A^*₊. Then y is unique and is denoted by (ι ⊗ ϕ)(x). We denote by ℳ_ι⊗ϕ the linear span of ℳ_ι⊗ϕ⁺ and extend ι ⊗ ϕ to a linear map from ℳ_ι⊗ϕ to M(A). Now one can prove that both Δ(a*)(1 ⊗ b) and (1 ⊗ a*)Δ(b) belong to ℳ_ι⊗ϕ, so that the formulas above make sense. In the commutative case M(A ⊗ A) = C_b(X × X), ϕ corresponds to a regular Borel measure on X, and ι ⊗ ϕ integrates out the second variable.

The unitary antipode anticommutes with Δ, i.e., χ(R ⊗ R)Δ = ΔR, where χ denotes the flip-automorphism extended to M(A ⊗ A). This means, in particular, that ϕR is a faithful right invariant KMS weight on A. It should be pointed out that, nevertheless, ψ is needed in the construction of S, R, and τ. Also, Δσ_t = (τ_t ⊗ σ_t)Δ, and Δτ_t = (τ_t ⊗ τ_t)Δ for all t ε ℝ, where σ denotes again the modular group of the left Haar weight ϕ. Further, there exists a positive number ν such that ϕτ_t = ν^−tϕ for all t ε ℝ. We call ν the scaling constant. In all the known examples, one has ν = 1, but we do not know whether this holds in general.

The role of the L²-space of the Haar measure is played by the Gelfand–Neumark–Segal (in short, GNS) representation associated with the weight ϕ. This is a triple (H, π, Λ), where: (i) H is a Hilbert space; (ii) Λ is a linear map from 𝒩_ϕ into H such that Λ(𝒩_ϕ) is dense in H, 〈Λ(a), Λ(b)〉 = ϕ(b* a) for every a, b ε 𝒩_ϕ; and (iii) π is a *-representation of A on H such that π(a)Λ(b) = Λ(ab) for every a ε A and b ε 𝒩_ϕ. Here 〈⋅, ⋅〉 denotes the inner product on H.

The Multiplicative Unitary

One of the motivations for the development of the theory of Kac algebras was the pioneering work of Takesaki (see ref. 18) on generalizing the Pontryagin duality theorem (e.g., ref. 17) to nonabelian locally compact groups. The main tool in the work of Takesaki is the so-called Kac–Takesaki operator. Let G be a locally compact group and fix a left Haar measure on it. Then we can define a unitary on the Hilbert space L²(G) ⊗ L²(G) given by (Wξ)(p, q) = ξ(p, p⁻¹q). The important point is that this unitary encodes all the information of the locally compact group G, so it should be no surprise that also in the development of the theory of locally compact quantum groups, such a unitary plays an important role.

We already explained that the L²-space of the Haar measure is replaced by the GNS construction of the Haar weight ϕ of our locally compact quantum group (A, Δ). Then we can define a unitary W on H ⊗ H such that W*(Λ(a) ⊗ Λ(b)) = (Λ ⊗ Λ)(Δ(b)(a ⊗ 1)) for all a, b ε 𝒩_ϕ. It is easy to check that W* is an isometry by using the left invariance of the weight ϕ. The coassociativity property of the comultiplication is encoded in the formula W₁₂W₁₃W₂₃ = W₂₃W₁₂, called the Pentagon equation. One can verify this equation immediately for the Kac–Takesaki operator of a locally compact group G.

Still more information on (A, Δ) is hidden in W. Because the weight ϕ is faithful, the representation π will be faithful as well, and we can identify A with π(A) through π. Then A is the closure of the set {(ι ⊗ ω)(W)|ω ε B(H)_∗}, and the comultiplication is given by Δ(x) = W*(1 ⊗ x)W for all x ε A. There is also an important link between the antipode S and the multiplicative unitary. For every ω ε B(H)_∗, one proves that (ι ⊗ ω)(W) belongs to the domain of S and S((ι ⊗ ω)(W)) = (ι ⊗ ω)(W*). Moreover, the elements (ι ⊗ ω)(W) form a core for the antipode.

Before the theory of locally compact quantum groups as described above, a systematic study of such unitaries W was made by Baaj and Skandalis (see ref. 10). The starting point is a Hilbert space H and a unitary W on H ⊗ H, satisfying the Pentagon equation. This is called a multiplicative unitary. In their paper, Baaj and Skandalis introduced two extra axioms, called regularity and irreducibility, which made it possible to prove that the closure of the set {(ι ⊗ ω)(W)|ω ε B(H)_∗} is a C*-algebra, and that the formula Δ(x) = W*(1 ⊗ x)W defines a comultiplication on this C*-algebra. The theory they develop is very elegant and beautiful: they obtain many quantum group-like features for this C*-algebra with comultiplication, such as a generalization of the Takesaki–Takai duality theorem for crossed products with abelian locally compact groups. Later, Baaj proved that the quantum E(2) group, as constructed by Woronowicz, did not satisfy the axiom of regularity, so that one should weaken this axiom a bit in order to include all locally compact quantum groups.

Woronowicz also has studied multiplicative unitaries (see ref. 11). He replaced the axioms of regularity and irreducibility by a completely different, and stronger, axiom called manageability. Woronowicz then associates with every manageable multiplicative unitary a C*-algebra with comultiplication by the same formulas as above. He is also able to define an antipode S satisfying the polar decomposition S = Rτ_−i/2. Now it is possible to prove that the multiplicative unitary associated with a locally compact quantum group as above is always manageable. Moreover, the antipode with polar decomposition as obtained in Woronowicz' theory is the same as the one obtained in the theory of locally compact quantum groups.

It is an open problem whether, conversely, every manageable multiplicative unitary admits a left invariant weight and hence gives rise to a locally compact quantum group.

Duality

As a motivating example, we return to a locally compact group G. Another way of associating a quantum group with G is via the group C*-algebra construction, which is more involved than the above construction of C₀(G). One starts by fixing a left Haar measure μ on G and considers the normed space L¹(G) of functions on G, integrable with respect to μ, where the norm is the ordinary L¹-norm. Next, it is customary to turn L¹(G) into a *-algebra by introducing the convolution product ∗ and the appropriate *-operation ° on L¹(G):

• (f ∗ g)(t) = ∫ f(s)g(s⁻¹t)dμ(s) for all f, g ε L¹(G) and almost all t ε G,

• f°(t) = δ(t)⁻¹ for all f ε L¹(G) and almost all t ε G,

where δ denotes the modular function of the locally compact group G, which connects the left and the right Haar measure on G. It should be stressed that L¹(G) is not a C*-algebra, only a Banach *-algebra.

A possible way of obtaining a C*-algebra is by using the left regular representation of G. The left regular representation s ↛ λ_s of G is a unitary representation of G acting on the space of square integrable functions L²(G), and is defined by the formula (λ_sg)(t) = g(s⁻¹t) for all g ε L²(G) and s, t ε G. This representation gives rise to the left regular *-representation λ of L¹(G) on L²(G), defined by λ(f) = ∫ f(s)λ_sdμ(s) for all f ε L¹(G), where the integral is formed in the strong topology of B(H). One can prove that λ : L¹(G) → B(L²(G)) is a faithful *-representation. Define C^*_r(G) to be the closure of λ(L¹(G)) in B(H). The C*-algebra C^*_r(G) is referred to as the reduced dual of G. Then the unitaries λ_s belong to the multiplier algebra M(C^*_r(G)) for all s ε G. It is possible to prove the existence of a unique nondegenerate *-homomorphism Δ : C^*_r(G) → M(C^*_r(G) ⊗ C^*_r(G)) such that Δ(λ_s) = λ_s ⊗ λ_s for all s ε G, and it turns out that the pair (C^*_r(G), Δ) is a locally compact quantum group.

Let us now look at the case where G is abelian. Classical group theory tells us how to construct the dual group Ĝ. As a set, Ĝ is the set of all continuous group characters on G taking values in the unit circle. The group multiplication on Ĝ is just the pointwise multiplication of two characters. The topology of Ĝ is the compact-open topology. In this way, Ĝ is endowed with the structure of a commutative locally compact group. The celebrated Pontryagin duality theorem says that the mapping θ : G → G^{^^} defined by θ(s)(ω) = ω(s) for all ω ε Ĝ, and s ε G, is a group isomorphism and a homeomorphism.

It is possible to identify C^*_r(G) with C₀(Ĝ) through a *-isomorphism π : C^*_r(G) → C₀(Ĝ) defined such that π(λ(f))(ω) = ∫ f(s)ω(s)dμ(s) for all f ε L¹(G) and ω ε Ĝ. It turns out that this map π is an isomorphism of quantum groups, i.e., (π ⊗ π)Δ = Δπ and Sπ = πS.

This discussion holds for abelian groups but fails for nonabelian ones. It is impossible to define, by a general construction, an appropriate dual locally compact group that encodes essentially all the information about the original locally compact group. However, the reduced dual C*-algebra C^*_r(G) encodes, as a quantum group all information about G. But if G is not abelian, this C*-algebra C^*_r(G) is noncommutative and cannot arise as (C₀(H), Δ) for some locally compact group H.

As we explained in the Introduction, here lies the motivation for defining Kac algebras: one was looking for a larger category, allowing duality, and containing the locally compact groups. The final definition of Kac algebras was given independently by Enock and Schwartz and Kac and Vainerman, solving the problem of duality of locally compact groups.

The above discussed construction of the reduced dual of a locally compact group can be generalized to the quantum group setting. Let us therefore return to our general quantum group (A, Δ) with its left Haar weight ϕ.

The construction of the dual of a Kac algebra can be found in chapter 3 of ref. 5. Essentially the same construction can be used to define the dual of a locally compact quantum group, although some simpler proofs can be given because of the stronger general theory at hand. Recall that above we defined the GNS construction (H, π, Λ) for ϕ and the multiplicative unitary W.

3. Definition. We define: (i) the set  s the norm closure of {(ω ⊗ ι)(W)|ω ε B(H)_∗}; and (ii) the injective linear map Δ̂ :  → B(H ⊗ H), such that Δ̂(x) = ΣW(x ⊗ 1)W* Σ for all x ε Â.

We use the flip map Σ on H ⊗ H to guarantee that the dual weight constructed from ϕ will again be left invariant rather than right invariant. Thanks to the results in ref. 11, the manageability of W implies that the set  is a nondegenerate C*-subalgebra of B(H), and the mapping Δ̂ is a nondegenerate *-homomorphism from  into M( ⊗ Â) such that: (i) (Δ̂ ⊗ ι)Δ̂ = (ι ⊗ Δ̂)Δ̂; and (ii) Δ̂(Â)( ⊗ 1) and Δ̂(Â)(1 ⊗ Â) are dense subsets of Â⊗ Â.

We will now introduce a notation that strengthens the analogy with the classical group case. We define L¹(A) to be the closed linear span of {aϕb*|a, b ε 𝒩_ϕ} in A*. We use the notation (aϕb*)(x) = ϕ(b*xa). Now denote by à the von Neumann algebra in B(H) generated by π(A). Then for every ω ε L¹(A), there is a unique ω̃ ε Ã_∗, such that ω̃π = ω, and hence we can define the injective contractive linear mapping π̂ : L¹(A) →  such that π̂(ω) = (ω̃ ⊗ ι)(W). In the group case, this is the left regular representation of L¹(G) on L²(G) mentioned above. We also mention that the expression ωμ = (ω ⊗ μ)Δ turns L¹(A) into a Banach algebra, and that π̂ becomes multiplicative this way. In the classical case, this comes down to the usual convolution product on L¹(G) described above.

Now we want to define a left invariant weight on (Â, Δ̂) using definition 2.1.6 of ref. 5. Define the subset ℐ of L¹(A) as follows: ℐ = {ω ε L¹(A)| there exists a number M ≥ 0 such that |ω(x*)| ≤ M∥Λ(x)∥ for all x ε 𝒩_ϕ}. It is clear that ℐ is a subspace of L¹(A). By Riesz' theorem for Hilbert spaces, there exists for every ω ε ℐ a unique element υ(ω) ε H such that ω(x*) = 〈υ(ω), Λ(x)〉 for x ε 𝒩_ϕ. It can be shown that there exists a unique closed densely-defined linear map Λ̂ from D(Λ̂) ⊆ Â into H, such that π̂(ℐ) is a core for Λ̂ and Λ̂(π̂(ω)) = υ(ω) for all ω ε ℐ. Finally, there exists a unique faithful KMS weight ϕ̂ on Â, such that (H, ι, Λ̂) is a GNS construction for ϕ̂.

The weight ϕ̂ is a left invariant weight on (Â, Δ̂). In order to get hold of the right invariant weight on (Â, Δ̂), we produce the unitary antipode on (Â, Δ̂). There exists a unique *-antiautomorphism R̂ on Â, such that R̂(π̂(ω)) = π̂(ωR) for all ω ε L¹(A). This *-antiautomorphism satisfies the relation χ(R̂ ⊗ R̂)Δ̂ = Δ̂R̂. Thus, ϕ̂R̂ is a right invariant KMS weight on (Â, Δ̂). Hence, we are led to the following conclusion:

4. Theorem. The pair (Â, Δ̂) is a locally compact quantum group.

The Pontryagin duality theorem for abelian locally compact groups also has its generalization to the quantum group setting. In the same way as we constructed the dual (Â, Δ̂) of (A, Δ), we can again construct the dual (A^{^^}, Δ^{^^}) of (Â, Δ̂) as a C*-algebra of bounded operators on H (with respect to the GNS construction (H, ι, Λ̂) for ϕ̂). The generalized Pontryagin duality theorem states that the natural embedding π : A → B(H) is a *-isomorphism from A to A^{^^} such that (π ⊗ π)Δ = Δ^{^^}π, and hence an isomorphism of the locally compact quantum groups (A, Δ) and (A^{^^}, Δ^{^^}).

The Universal Setting

Let us consider once more a locally compact group G. In the beginning of the previous section, we constructed the reduced group C*-algebra C^*_r(G) by putting a certain C*-norm on L¹(G) and completing L¹(G) to C^*_r(G) with respect to this norm, but in general there does not exist a unique C*-norm on L¹(G). Another possible C*-norm is the universal C*-norm ∥.∥_u on L¹(G): for every x ε L¹(G) we understand by definition that ∥x∥_u is the supremum of {∥θ(x)∥ | θ a *-representation of L¹(G) on a Hilbert space}. The universal group C*-algebra C*(G) is by definition the completion of L¹(G) with respect to the norm ∥.∥_u. A classical result says that there is a one-to-one correspondence between *-representations of the C*-algebra C*(G) and strongly continuous unitary representations of the group G.

This situation can be completely generalized to the quantum group setting. Given a reduced locally compact quantum group (A, Δ), one can construct the “universal” C*-algebra A_u, such that there is a one-to-one correspondence between *-representations of A_u and unitary corepresentations of (Â, Δ̂).

Further, it is possible to construct a comultiplication Δ_u on A_u, and one can show that the pair (A_u, Δ_u) satisfies the same interesting properties as the reduced companion (A, Δ), except for the faithfulness of the Haar weights. So we see that given a reduced locally compact quantum group (A, Δ), there are at least two other interesting operator algebraic companions associated with it: the universal version (A_u, Δ_u) and the von Neumann algebraic version (Ã, Δ̃), which is obtained by extending Δ to the von Neumann algebra à generated by π(A). Although these three algebras A, A_u, and à have significantly different properties as operator algebras, they are three different realizations of the same underlying “quantum group” (each of them can be canonically recovered from the other one). Rather than seeing this as something awkward, it should be seen as an advantage, because a certain realization might allow certain manipulations that are not possible in another realization.

Concluding Remarks

Before going on with the statement of perspectives and challenges of the theory, we would like to summarize what we have done so far. Roughly speaking, we proposed a definition for a locally compact quantum group and discussed its properties. This theory includes the locally compact groups, the compact and discrete quantum groups, and the Kac algebras, which was a major motivation for its development. Further, the list of axioms is not too long, and important features such as the uniqueness of the Haar weights, the existence of the antipode and its polar decomposition, a nice duality theory, the modular element, and the manageability of the multiplicative unitary are established as theorems.

The most important drawback to our definition is, of course, the assumption of the existence of the Haar weight, so the major challenge is the formulation of an alternative and more elementary axiom that allows the existence of the Haar weight as a theorem. This is really an enormous challenge, and not all the experts believe it can be realized.

Another problem has existed since the development of Kac algebras: it is very hard to construct nontrivial examples of locally compact quantum groups. The most important and well known examples have been discovered by Woronowicz, e.g., the quantum E(2), quantum Lorentz, or quantum ax + b-groups, and each single example is a subtle and beautiful piece of mathematics. Although there exist some general construction procedures—we think of the quantum double construction, the crossed and bicrossed products, or the locally compact quantum groups arising from multiplier Hopf *-algebras with positive integrals—we do not obtain the most interesting examples in this way. It is another major challenge to develop a construction procedure that is both easy enough to be feasible and subtle enough to produce nice and nontrivial examples.

Finally, when one compares the locally compact groups with the locally compact quantum groups, the second theory is developed only up to its fundamentals: given a definition, one should now prove theorems or study special classes of locally compact quantum groups. In our opinion, a serious amount of work can be done in this direction.

Abbreviations

KMS

Kubo–Martin–Schwinger

GNS

Gelfand–Neumark–Segal