The Heisenberg commutation relations, commuting squares and the Haar measure on locally compact quantum groups ( + )

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1 The Heisenberg commutation relations, commuting squares and the Haar measure on locally compact quantum groups ( + ) Operator algebras and mathematical physics: conference Proceedings, Constanta (Romania), July 2 7, Editors J.-M. Combes, J. Cuntz, G.A. Elliott, G. Nenciu, H. Siedentop and S. Stratila. Theta Foundation, Bucarest (2003), pp Abstract Stefaan Vaes ( ) & Alfons Van Daele ( ) Consider a locally compact group G and the Hilbert space L 2 (G) for G with the left Haar measure. Let M denote the von Neumann algebra on L 2 (G) that consists of multiplication operators with functions in L (G). Take also the left regular representation λ given by (λ(s)ξ)(t) = ξ(s 1 t) whenever s, t G and ξ L 2 (G). Let ˆM denote the von Neumann algebra generated by {λ(s) s G}. The commutation rules between operators of M and operators of ˆM can be thought of as a generalization of the Heisenberg commutation relations (we will explain this in the paper). Denote by h the operator on L 2 (G) given by multiplying with the modular function δ of G. It can be shown that, for sufficiently many elements x M and y ˆM, the operator hyx is a trace class operator and that tr(hyx) = ˆϕ(y)ϕ(x) where ϕ is the weight on M obtained by integration with respect to the left Haar measure on G and where ˆϕ is a similar weight on the von Neumann algebra ˆM (to be explained in the paper). In the finite-dimensional case (i.e. when G is finite), we have a commuting square here (as considered in the theory of subfactors). In this case, the conditional expectations on M and ˆM are given by E(yx) = ˆϕ(y)x and Ê(yx) = ϕ(x)y resp. where x M and y ˆM. In the general case, these maps will be (unbounded) operator valued weights (in the sense of Haagerup). In this paper, we will generalize these results to the general case of locally compact quantum groups and clarify further the relation between the three topics mentioned in the title of the paper. The general case will be treated in the second part of the paper. The first part will be expository. There, we will first describe the various concepts and formulas in the finite-dimensional context. This will allow the reader to get familiar with the purely algebraic aspects of this theory before going into the more difficult (and more technical) operator algebra framework. ( + ) Talk given (by the second author) at the OAMP conference in Constantza (Romania), June ( ) Address: Institut de mathématiques de Jussieu, Algèbres d opérateurs et représentations, Plateau 7E, 175, rue du Chevaleret, F Paris (France). ( ) Address: Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B-3001 Heverlee (Belgium). address : Vaes@math.jussieu.fr, Alfons.VanDaele@wis.kuleuven.ac.be 1

2 0. Introduction As the title of this paper suggests, we want to relate several topics with each other: the Heisenberg commutation relations, commuting squares (a notion coming from subfactor theory), and the Haar measure on locally compact quantum groups. We will start in this introduction by formulating various forms of the Heisenberg commutation relations, in increasing order of generality. Originally, we have the Hilbert space L 2 (R) with the two self-adjoint operators p and q defined on it, roughly speaking, by the formulas (pf)(x) = if (x) and (qf)(x) = xf(x) whenever x R. They satisfy the formal commutation rule [p, q] = i1. We have taken ħ = 1. We will rather look at the Weyl form of this commutation rule. It says e itq e isp = e ist e isp e itq for all s, t R. Observe that these unitary operators satisfy (e itq f)(x) = e itx f(x) (e isp f)(x) = f(x s) whenever f L 2 (R) and s, t, x R. So in this form, the Heisenberg commutation rules tell us how to commute multiplication and translation operators. There is a first obvious generalization of these commutation relations. Take an abelian locally compact group G and let Ĝ be its dual group. Use t, γ to denote the value of γ, an element in Ĝ, in the element t of G. Consider L2 (G) for G with its Haar measure and define unitary representations λ of G and π of Ĝ on L2 (G) by (λ(t)ξ)(x) = ξ(t 1 x) (π(γ)ξ)(x) = x, γ ξ(x) whenever x, t G, γ Ĝ and ξ L2 (G). A simple calculation gives π(γ)λ(t) = t, γ λ(t)π(γ). To go further to the case of a general, possibly non-abelian locally compact group G, we must replace the multiplication operators with the elements γ Ĝ by general multiplication operators. We arrive at the following situation. Let G be any locally compact group. Consider G with the left Haar measure and the corresponding space L 2 (G). We still have the left regular representation λ of G on L 2 (G) as defined above. Now we consider the C -algebra C 0 (G) of complex continuous functions on G tending to 0 at infinity. This C -algebra acts by multiplication operators on L 2 (G). When we write (π(f)ξ)(x) = f(x)ξ(x) 2

3 whenever x G, ξ L 2 (G) and f C 0 (G), the commutation rules become for all t G and f C 0 (G). π(f)λ(t) = λ(t)π(f(t )) When we specialize to a finite group, we can describe the relation with a commuting square. We will discuss this in detail in example 1.2 of section 1 where we also recall the exact definition of such a commuting square (definition 1.1). Roughly speaking, we have the following. So now, let G be a finite group. Then L 2 (G) is just identified with C n where n = #G. The operators π(f) (where now f is any complex function on G) generate one subalgebra A of M n (the algebra of all n n matrices over C). The operators λ(s) with s G generate another one B. One can easily check that A B = C1 and that the linear span of operators of the form λ(s)π(f) is all of M n. The algebra M n is now considered with its usual trace and then A and B have a very special position as subalgebras of the bigger one M n. This is the commuting square (the terminology will become clear when we give a more precise definition in the next section). Also the relation with the Haar measure pops up, but in this case, it is rather trivial. When we restrict, in the above situation, the trace of M n to the subalgebra A, we get the linear functional ψ given by ψ(f) = s G f(s). This is the integral with respect to the Haar measure on G. The next generalization is obtained in the framework of finite-dimensional Hopf algebras (over C). Also this is described in the next section. Here we just observe that the algebras A and B that we had above, which are essentially the algebra of functions on G (with point-wise operations) and the convolution algebra of functions on G (the group algebra CG), have natural Hopf algebra structures and they are dual to each other (in the sense of proposition 1.4 in the next section). This observation takes us to the following step. Now, A is any finite-dimensional Hopf algebra (see definition 1.3) and B is the dual (see proposition 1.4). These two algebras act on the space A and the commutation rules of these two algebras, as acting on A are the Heisenberg commutation rules. One should think of A as the algebra of multiplication operators and of B as the algebra of translation operators (which they are in the group case!). Again only the scalar multiples of 1 are both in A and in B. And the span of products of operators in B with operators in A gives all linear maps. So at least, also here we have four algebras, the basic ingredient for having a commuting square. Unfortunately, we do not have a commuting square in the strict sense of the definition. We have to modify the notion and work with another functional f on the algebra of all linear maps (and it is in general not the trace). But still, when restricting this functional to the subalgebras, we do get the Haar measures. In this case, these are invariant functionals on the Hopf algebras (see proposition 1.6 for a precise definition). At the end of section 1, we will look at another functional and we will leave the nice framework of commuting squares. Nevertheless, this new functional g has a lot of similarities 3

4 with the other one f. From a certain point of view, it is even nicer than the first one and we will explain why. It is related with a construction method of the Haar measure on a locally compact quantum group. When A is a finite-dimensional Hopf -algebra, with a nice underlying -algebra structure (i.e. an operator algebra), then f = g and it is a trace. So we get a genuine commuting square. However, this situation is too trivial to be a good model for the general case that we want to consider in section 2. That is the reason we had to leave the -algebraic context. But in section 2, we will return to the (nicer) -algebra setting when we consider the general locally compact quantum groups. We will not say very much about section 2 in this introduction. In this section, we generalize all of this to the case of general locally compact quantum groups. We will work in the von Neumann algebra setting. And just as before, we will get two von Neumann subalgebras M and ˆM of B(H) for some Hilbert space H. Also here, we will have that the intersection M ˆM equals C1 and that the linear span ˆMM of products of elements in ˆM with elements in M is σ-weakly dense in B(H). We will construct the equivalent of the functionals f and g. Now they will be faithful normal semi-finite weights on the algebra B(H). The first functional f will again have properties that take us to an unbounded version of the commuting square (as we had it for dual pairs of Hopf algebras). The other one g will not be so nice from this point of view, but both functionals, when restricted to the subalgebras M and ˆM in the appropriate sense, give Haar weights. And in the case of g, this formula will be such that it yields a construction procedure for the Haar measures on examples of locally compact quantum groups. The two sections of this paper deal with the same material but are of a very different nature. In the first one, we only look at the algebraic aspects and in order to do that, we consider the finite-dimensional case. We also have to leave the -algebra context so that still enough complexity is retained. The results in this section are not new but they should be understood before starting with the general case. So section 1 is mainly expository. In section 2, we obtain these results for the general locally compact quantum groups. Also here, part of the results have been obtained earlier (and in an even more general context - see [V1]). Other results have been announced in [VD5] but where not yet proven. In any case, we bring together these different parts of the theory in this note and we have tried to make it self-contained in a way convenient for the reader. Moreover, it is really nice to learn about these different topics: the Heisenberg commutation relation, commuting squares and the Haar measure on locally compact quantum groups; and the relations among them. We finish this section by giving some standard references and by agreeing on some basic notational conventions in this paper. We will only work with vector spaces and algebras over C (although some of the results in section 1 are certainly also valid for other fields). We will also be using tensor products all the time. In section 1, these will be algebraic tensor products while in section 2 they will be mostly completed tensor products. We will say clearly which ones we mean. We 4

5 will use slice maps. In the algebraic setting, we have e.g. the tensor product of two vector spaces A and B and a linear functional f : A C. The slice map f ι is the linear map from A B to B given by (f ι)(a b) = f(a)b whenever a A and b B. We are using ι to denote the identity map (we will use 1 to denote the identity in an algebra). Slice maps will also be considered in the case of completed tensor products, but then one has to be much more careful. In section 1 we will work with Hopf algebras. The basic notions about Hopf algebras can be found in the standard works [A] and [S]. We will also be using the Sweedler notation from [S]. For dual pairs of Hopf algebras we refer to [VD1]. In [VD2] one can find an exposition about the Haar measure on finite-dimensional Hopf algebras and an example. Acknowledgements The first author wants to thank the whole team Algèbres d Opérateurs et Représentations of the Institute of Mathematics of the Universities Paris VI-VII for their warm hospitality. The second author likes to thank his colleagues S. Stratila and F. Boca for their hospitality in Bucharest and for giving the opportunity to present this work at the OAMP conference in Constantza in June Part of this work was done while the second author was visiting the University of Illinois at Urbana-Champaign and he would like to express his gratitude towards Prof. Z.-J. Ruan for his hospitality during this visit. 1. Commuting squares and finite quantum groups Let us start by recalling the notion of a commuting square (as it can e.g. be found in the book of Jones and Sunder [J-S], Definition 5.1.7). 1.1 Definition Let C be a finite von Neumann algebra with a faithful normal tracial state tr. Given are also von Neumann subalgebras A, B and D of C such that D A B. It is also assumed that the identity of C is in D (and so it is the identity of all these four von Neumann algebras). Such a set of von Neumann algebras is called a commuting square if E A E B = E B E A = E D where E A, E B and E D are respectively the canonical conditional expectations of C onto A, B and D. Recall that the map E A : C A is defined by tr(ae A (x)) = tr(ax) for all a A and x C. It follows from the faithfulness of tr that this map is well-defined. It also follows immediately from this formula that E A (ax) = ae A (x) for all a A and x C but one needs the trace property of tr to have also that E A (xa) = E A (x)a for all a A 5

6 and x C. Furthermore, the map E A is unital and positive. The first property follows again immediately from the definition, but the trace property is needed to prove positivity. Similarly for the two other maps E B and E D. Later we will need to look at generalizations of this concept in the case where we do not have a trace. Therefore is it important to see here when the trace property had to be used. But first, let us consider what can be called the motivating example of a commuting square. 1.2 Example Let G be a finite group. Take for the underlying Hilbert space H the space l 2 (G). The algebra C is the algebra B(H) of all linear operators on H (remember that H is finite-dimensional). For tr we take the usual trace, normalized such that tr(1) = 1. For A we take the algebra of multiplication operators. We will not distinguish between the operator and the function and so whenever we have a complex function a on G, we also have the operator a on H given by the formula (aξ)(p) = a(p)ξ(p) for all ξ l 2 (G) and p G. For B we take the algebra of convolution operators. Again we will not distinguish the notation for the function and for the operator (this will not lead to confusion when we say clearly in which of the two algebras we are working). So, an element b in B is given by a function, also denoted by b on G, such that (bξ)(p) = q G b(q)ξ(q 1 p) for ξ l 2 (G) and p G. It is well-known (and in fact not so difficult to prove) that these two algebras are -subalgebras of the bigger one C and that A B = C1. A simple calculation gives us that tr(ba) = ˆϕ(b)ϕ(a) for all a A and b B where ϕ(a) = 1 n p a(p) and ˆϕ(b) = b(e) (with n = #G and e the identity in G). Indeed, when we use δ p to denote the function on G given by δ p (p) = 1 and δ p (q) = 0 if q p, we get (baδ p )(q) = a(p) b(r)δ p (r 1 q) = a(p)b(qp 1 ) r so that tr(ba) = 1 (baδ p )(p) = n p ( 1 n It follows immediately from the above formula that ) a(p) b(e). E A (ba) = ˆϕ(b)a and E B (ba) = ϕ(a)b for all a A and b B. Of course E D (ba) = tr(ba)1 so that clearly E A E B (x) = E B E A (x) = E D (x) when x = ba. It suffices to observe that the linear span of operators ba is precisely all of C (again this is a result that is well-known and easy to show). Hence, we do have a commuting square in the sense of definition p

7 Later, it will become clear why have denoted these two functionals on A and B by ϕ and ˆϕ respectively. Other examples can be found in chapter 5 of [J-S]. We will also describe a generalization of the above example, later in this section. Roughly speaking, we have a commuting square if, for a given von Neumann algebra with a faithful normal tracial state, we have subalgebras A and B that lie in a certain position w.r.t. each other. For the results that we want to prove in section 2 of this paper, the notion of a commuting square however is too restrictive. In this section, which is mainly expository and serves the purpose of illustrating the results of the next section, but in a more simple algebraic context, we will have to leave the von Neumann algebra setting for a while in order to be able to give a relatively simple, but non-trivial example which is still finite-dimensional. Later, in section 2, we will move again to the von Neumann algebras. The example is build from a finite-dimensional Hopf algebra (and thus is a generalization of the group example above). Let us first recall the notion of a Hopf algebra (see e.g. [A] or [S]). 1.3 Definition Let A be an (associative) algebra over C with identity 1. A comultiplication (or coproduct) is a unital homomorphism : A A A satisfying coassociativity ( ι) = (ι ) (where ι denotes the identity map). The pair (A, ) is a Hopf algebra if there exist linear maps ε : A C and S : A A satisfying (ε ι) (a) = (ι ε) (a) = a m(s ι) (a) = m(ι S) (a) = ε(a)1 for all a A. Here, m denotes the multiplication in A, seen as a linear map from A A to A. One can show that ε is a homomorphism and that S is an anti-homomorphism. They are uniquely determined. The first map ε is called the counit and the second one S is called the antipode (or the coinverse). For a finite-dimensional Hopf algebra A the dual space A can be made again into a Hopf algebra: 1.4 Proposition Let A be a finite-dimensional Hopf algebra. The dual space A of all linear functionals on A becomes an algebra when the product fg of elements f, g A is defined by (fg)(a) = (f g) (a). It becomes a Hopf algebra when the coproduct on A is defined by (f)(a b) = f(ab). 7

8 Of course, the unit in A is ε and the counit and antipode on A are given by ε(f) = f(1) and S(f)(a) = f(s(a)). Observe that this cannot be done when A is no longer finitedimensional. In that case, the candidate for (f) will in general no longer belong to the (algebraic) tensor product A A. In what follows, we will use B to denote this dual algebra but considered with the opposite comultiplication. We will use a, b for the evaluation of the element b in the element a. By definition we will have a, bb = (a) a (1), b a (2), b aa, b = (b) a, b (2) a, b (1). We use the Sweedler notation (v) = (v) v (1) v (2) ; see [S]. We get the last equality because we have considered the opposite comultiplication on the dual space. This also means that the antipode has to be inverted, giving the formula a, Sb = S 1 a, b. We use this different (and in some sense unusual) convention in order to get similar formulas as in section 2 where all of this is considered in the operator algebra context where this other convention is very common. Sometimes, in the infinite-dimensional case, it is possible to pass to a certain subalgebra B of the dual space A, having the property that (B) B B and that (B, ) is a Hopf algebra. In this case, we have a so-called dual pair of Hopf algebras (see e.g. [VD1]). Essentially all what we will do in the remaining of this section can also be done for such more general dual pairs or even for dual pairs of multiplier Hopf algebras ([D-VD]), when integrals exist. Integrals automatically exist in the finite-dimensional case (see also further), but not in general. For simplicity, and because the aim of this section is mainly expository, we will stick here to the finite-dimensional case. The more general case (for algebraic quantum groups) is treated in [K-VD]. So, in what follows, let A be a finite-dimensional Hopf algebra and let B denote the dual algebra, with the opposite comultiplication as explained before. We will let A act on itself by left multiplication. We also will let B act on A by bv = (v) v (1), S(b) v (2) whenever b B and v A. For the simplicity of the notations, we would like to identify the algebras A and B with their images in the algebra L(A) of linear maps from A to itself. Therefore, we will be using V for A when we look at it as the underlying space. Hence from now on, our two algebras A and B are subalgebras of L(V ). We will also systematically use the letters v, v to denote elements in V, the letters a, a for elements in A and b, b for elements in B. With these conventions, the following result makes sense. 8

9 1.5 Proposition For all a in A and b in B we have ab = a (1), b (1) b (2) a (2). (a)(b) Moreover, the linear span C of elements of the form ba with a A and b B consists of all linear maps L(V ) of V into itself. Also A B = C1. We refer to [D-VD-Z] for a proof of this result (in a slightly different form), see also [K-VD]. The relations in this proposition are the Heisenberg commutation relations for a (twisted) dual pair of Hopf algebras. They also make sense for a dual pair of multiplier Hopf algebras (as introduced in [D-VD]), see [K-VD]. In the general case, the algebra C, which is BA, is called the Heisenberg algebra. It should be emphasized that the Heisenberg algebra is thought of as an algebra with two distinguished subalgebras, rather than as an algebra itself (as often, it is a rather trivial algebra, containing little information). So, in the finite-dimensional case as above, we have some of the ingredients of a commuting square. We have four algebras, the bigger one C containing A and B as subalgebras and the smaller one C1, contained in both A and B. We also have the trace on C, but this will not give us the necessary commutativity of the associated conditional expectations. We need to replace the trace by some other functional. This functional is defined using the left integral on A and the left integral on B. Let us therefore recall the notion of an integral on a Hopf algebra. We do this in the following proposition. 1.6 Proposition Let (H, ) be a finite-dimensional Hopf algebra. Then there exist unique non-zero linear functionals ϕ and ψ on H satisfying for all h H. (ι ϕ) (h) = ϕ(h)1 (ψ ι) (h) = ψ(h)1 The first functional ϕ is called the left integral (because it is left invariant). The second one ψ is called the right integral. These functionals are automatically faithful. This means e.g. that h = 0 if ϕ(hh ) = 0 for all h or if ϕ(h h) = 0 for all h. They are also unique, up to a scalar. We will use in what follows ϕ and ψ to denote a left and a right integral on A and we will use ˆϕ and ˆψ to denote a left and a right integral on B. It is possible to normalize these four integrals in a more or less standard way. We will discuss this further in this section, as soon as it is necessary. Now we have the following result. 1.7 Proposition Define linear maps E A : C A and E B : C B by E A (ba) = ˆϕ(b)a E B (ba) = ϕ(a)b. 9

10 They are conditional expectations, i.e. E A (xa) = E A (x)a and E A (ax) = ae A (x) for all x C and a A. Similarly for E B. Moreover, they commute: for all a A and b B we have E A E B (ba) = E B E A (ba) = ˆϕ(b)ϕ(a)1. Proof: There is not so much to prove. It is immediate from the definition that E A (xa) = E A (x)a for all x C and a A. To prove the other equality, take a, a A and b B. Using the Heisenberg commutation relations of proposition 1.5 and the left invariance of ˆϕ, we find E A (a ba) = a (1), b (1) E A (b (2) a (2) a) = a (1), b (1) ˆϕ(b (2) )a (2) a = a (1), 1 ˆϕ(b)a (2) a = ˆϕ(b)a a = a E A (ba). Similarly, using the left invariance of ϕ on A, we will be able to show that E B (bab ) = E B (ba)b for all a A and b, b B. Again the other formula, E B (bx) = be B (x) is trivial. Also the last statement is a triviality and follows immediately from the definitions. In general we will not have that ϕ(1) = 1 and ˆϕ(1) = 1. This is not a matter of normalization as it can happen that ϕ(1) = 0 and ˆϕ(1) = 0 (see e.g. an example in [VD2]). So it will be possible not to have that E A (1) = 1 or E B (1) = 1. It is possible to require extra conditions so that we will have ϕ(1) = 1 and ˆϕ(1) = 1. Then we do have E A (1) = 1 and E B (1) = 1. And if we consider the linear functional f on C given by f(ba) = ˆϕ(b)ϕ(a), we will also have f(xa) = f(e A (x)a) f(bx) = f(be B (x)) whenever a A, b B and x C. Then, because f is faithful (see proposition 1.8 below), not only on A and on B but also on C, these formulas characterize E A and E B. With these extra conditions however, we will loose too much of our generality (because we are working in the finite-dimensional case) and we will no longer get a good model for the general case that we will treat in the next section. Moreover, as there we will be working with unbounded maps, unitality will make no sense anyway. Observe that in general, we will have here that E A E B = E B E A = E D where E D (x) = f(x)1 with f defined as above. Now we will try to understand the underlying reason for still having these commuting conditional expectations although we do not have a trace. We will also discover what is the role of the functional f in this case. 10

11 First recall the following general results (see e.g. [VD4]). There exists automorphisms σ and σ of A satisfying (and defined by) ϕ(aa ) = ϕ(a σ(a)) ψ(aa ) = ψ(a σ (a)) for all a, a A. These automorphisms will be called the modular automorphisms associated with ϕ and ψ respectively. There is also an invertible element δ A such that (ϕ ι) (a) = ϕ(a)δ (ι ψ) (a) = ψ(a)δ 1. for all a A. Finally, there is a non zero scalar µ C such that for all a A. ϕ(s 2 (a)) = µϕ(a) There are many formulas relating these various objects (again see [VD4]). We will need to use several of them. We can now make an important remark. We saw that we needed to use the trace property to show that the maps E A and E B are conditional expectations (see the remark after definition 1.1). So one may aks why it still works here although we do not have a trace? The reason can be found in the following result. 1.8 Proposition There is an automorphism γ on C that leaves A and B globally invariant and such that f(xy) = f(yγ(x)) for all x, y C. This modular automorphism γ of f coincides on A with the modular automorphism σ of ϕ and on B with the modular automorphism ˆσ of ˆϕ. Proof: Take a, a A and b B. We have Hence f(aba ) = a (1), b (1) f(b (2) a (2) a ) = a (1), b (1) ˆϕ(b (2) )ϕ(a (2) a ) = a (1), 1 ˆϕ(b)ϕ(a (2) a ) = ˆϕ(b)ϕ(aa ). f(abσ(a )) = ˆϕ(b)ϕ(aσ(a )) = ˆϕ(b)ϕ(a a) = f(a ab) for all a, a A and b B. So we see that we must have γ(a) = σ(a). A similar calculation will give that we must have γ(b) = ˆσ(b) for b B. It follows from this that 11

12 there is a modular automorphism for f given by these two modular automorphisms on A and on B. In fact, one can verify that they respect the Heisenberg commutation rules. So we find that the modular automorphism γ associated with f on C coincides with the modular automorphisms of the components ϕ and ˆϕ on the subalgebras. This is the main reason why the maps E A and E B are still conditional expectations, even though we do not have traces. We will elaborate more on this property in the next section. Now we are ready to formulate and prove the following proposition. 1.9 Proposition Define h C by hv = σ(v)δ 1 = δ 1 σ (v) when v V. Then, with f defined as before on C by f(ba) = ˆϕ(b)ϕ(a), we have for all x C. f(x) = tr(hx) Of course, given the normalization of the usual trace on C (= L(V )), we need to take a related normalization of the pair (ϕ, ˆϕ). This can be done in a standard way, but then we have to allow a scalar in the above formula. We refer to [K-VD] for a more precise discussion about this normalization. In section 2, where we can require positivity, we will be more careful and select a suitable normalisation and combine it with a proper scalar. Proof: Because the Heisenberg algebra C is all of L(V ), we just need to show that hxh 1 = γ(x) for all x C. And as C is spanned by the elements ba with a A and b B, it will be sufficient to prove this formula for x A or x B. Using the equation hv = σ(v)δ 1 we see easily that ha = σ(a)h. For b B we get, with v V, ˆσ(b)hv = (σ(v)) (1) δ 1, S(ˆσ(b)) (σ(v)) (2) δ 1 = S 2 (v (1) )δ 1, S(ˆσ(b)) σ(v (2) )δ 1 = S 2 (v (1) )δ 1, ˆσ 1 (S(b)) σ(v (2) )δ 1 = v (1), S(b) σ(v (2) )δ 1 = hbv When (A, ) is a finite-dimensional Hopf -algebra (with a good underlying -algebra structure), most of the objects are trivial. We have in this case S 2 = ι, ϕ = ψ and σ = σ = ι. In particular, all the Haar measures are traces. The above properties are almost automatic. The nice thing about this of course is that we do have a commuting square in the original sense. But it is getting too trivial and it will not be a good model for the general locally compact quantum group case. Strange enough, the non- -case in 12

13 finite dimensions has more features in common with the general infinite-dimensional case where we will work in a good -algebra context. We now want to finish this section by studying another linear functional on the algebra C = BA. We use the same notations as before to formulate the following proposition Proposition Define a linear functional g on C by g(ba) = ˆψ(b)ψ(a) when a A and b B. Then g(x) = tr(kx) for all x C where now k is a scalar multiple of the operator S 2 acting on V. The modular automorphism γ of C associated with g, given by γ(x) = kxk 1 leaves A and B globally invariant and γ(a) = S 2 (a) for a A while γ(b) = S 2 (b) for b B Proof: Whenever a, a A and b B, we have g(a ba) = a (1), b (1) g(b (2) a (2) a) = a (1), b (1) ˆψ(b (2) )ψ(a (2) a) = a (1), ˆδ 1 ˆψ(b)ψ(a (2) a). One can show that, in this setting, a, ˆδ 1 = ε(σ 1 (a)). Using also that (S 2 (a)) = (σ 1 σ ) (a), we get further g(a ba) = ε(σ 1 (a (1) ) ˆψ(b)ψ(aσ (a (2) )) = ε(s 2 (a ) (1) ˆψ(b)ψ(aS 2 (a ) (2) ) = ˆψ(b)ψ(aS 2 (a (2) )) = g(bas 2 (a )). So γ(a) = S 2 (a) for a A when γ is the modular automorphism of g. Similarly for all a A and b, b B, we have g(bab ) = a (1), b (1) g(bb (2) a (2)) = a (1), b (1) ˆψ(bb (2) )ψ(a (2)) = δ 1, b (1) ˆψ(bb (2) )ψ(a) = ε(ˆσ(b (1) )) ˆψ(ˆσ 1 (b (2) )b)ψ(a) = ε(s 2 (b ) (1) ˆψ(S 2 (b ) (2) b)ψ(a) = ˆψ(S 2 (b )b)ψ(a) = g(s 2 (b )ba). 13

14 Therefore γ(b) = S 2 (b) when b B. Now, let k : V V be defined by k(v) = S 2 (v). Then it is immediate that kak 1 = S 2 (a) as elements in L(V ). Also, when v V and b B, kbk 1 v = kbs 2 (v) = k S 2 (v (1) ), S(b) S 2 (v (2) ) = v (1), S 1 (b) v (2) = (S 2 (b))v. It follows that there is a scalar α C such that g(x) = αtr(kx) for all x C. This proves the result. In [K-VD] we also give a proof of proposition 1.10, but in the more common dual pair framework. Again, we will be more careful and obtain the precise scalar, given a standard normalisation of the integrals. Also in section 2, where we have positivity and a more or less obvious normalisation of these integrals, we will obtain the precise scalar. Observe that the maps a S 2 (a) on A and b S 2 (b) on B are compatible with the commutation rules of proposition 1.5 because in this setting S 2 (a), S 2 (b) = a, b. This has to be the case for these maps to induce an automorphism on C. We see that in this case, the modular automorphism γ of g on C does not coincide on A with the modular automorphism of ψ (which is σ ) and it does not coincide on B with the modular automorphism of ˆψ (which is ˆσ ). We still can define the associated maps E A and E B in the usual way by E A (ba) = ˆψ(b)a E B (ba) = ψ(a)b when a A and b B but they will not be conditional expectations. We will obviously have ψ(e A (x)) = g(x) and ˆψ(E B )(x) = g(x) for all x C. It will follow immediately from the definition that E A (xa) = E A (x)a (and E B (bx) = be B (x)) whenever a A, b B and x C. With the elements a and b on the other side, these formulas will no longer be valid. In fact, we have ψ(e A (ax)a ) = ψ(e A (axa )) = g(axa ) = g(xa S 2 (a)) = ψ(e A (x)a S 2 (a)) = ψ(σ 1 (S 2 (a))e A (x)a ) whenever a, a A and x C. Hence E A (ax) = σ 1 (S 2 (a))e A (x). Similarly E B (xb) = E B (x)ˆσ (S 2 (b)). It would be possible to modify the notion of a commuting square further so as to include the above situation, but we will not do this. On the other hand, we want to make the following important observation. To know f one must have the modular automorphism σ of ϕ on A and ˆσ of ˆϕ on B. So we need some information about the left integral on A and the left integral on B. This is different with the other functional g. This is given in terms of the square of the antipode. So, 14

15 we know the functional g without any information on the integrals. Then, we can obtain the integrals from the functional g by restricting it to the subalgebras A and B. In other words, we can use the formula of proposition 1.10 to prove the existence of the integrals on a finite quantum group. So it is no surprise that this yields a construction procedure to find the Haar measures on locally compact quantum groups that has been proven to be useful in many examples (see e.g. [VD5]). We will come back to this at the end of the next section. 2. The case of a general locally compact quantum group Now, we will try to generalize the results of the previous section to the case of a locally compact quantum group. We recall some of the basic results on locally compact quantum groups. We begin with the definition (cf. definition 1.1 in [K-V4]). 2.1 Definition Let M be a von Neumann algebra. A comultiplication on M is a unital normal -homomorphism : M M M such that ( ι) = (ι ). Here we use the von Neumann algebraic tensor product M M of M with itself and ι is the identity map from M to M. The maps ι and ι are the normal -homomorphisms on M M obtained by extending the obvious maps from the algebraic tensor product to the von Neumann tensor product. 2.2 Definition A pair (M, ) of a von Neumann algebra M and a comultiplication on M is called a locally compact quantum group (in the von Neumann algebraic setting) if there exist normal semi-finite faithful weights ϕ and ψ such that ϕ is left invariant and ψ is right invariant. Recall that the left invariance of ϕ means that for all ω M + defined. ϕ((ω ι) (x)) = ϕ(x) ω(1) and for all x M + such that ϕ(x) <. Similarly, right invariance is Observe that it has been shown in [K-V4] that the C -algebraic formulation of locally compact quantum groups, as developed in [K-V1] and [K-V2] is equivalent with the von Neumann algebraic formulation above. The key role in showing this, is the associated multiplicative unitary. This is in fact the left regular representation, which we recall now. We also remark that the existence of the Haar weights ϕ and ψ is assumed, but their uniqueness, up to a positive scalar, is proven in [K-V2]. We fix a left invariant weight ϕ. We will use H to denote the G.N.S.-Hilbert space associated with the left invariant weight ϕ. We will also assume that M acts already on H. 15

16 The canonical map from N ϕ (= {x M ϕ(x x) < }) to H will be denoted by Λ. So, we have that Λ(x), Λ(y) = ϕ(y x) whenever x, y N ϕ and xλ(y) = Λ(xy) when y N ϕ and x M. Using these conventions, we get the following result (cf. theorem 1.2 in [K-V4]). 2.3 Proposition There is a unitary operator W on H H defined by whenever x, y N ϕ. W (Λ(x) Λ(y)) = (Λ Λ)( (y)(x 1)) It is explained in [K-V4] how the right hand side of the above equation makes sense. One shows that W satisfies the Pentagon equation W 12 W 13 W 23 = W 23 W 12 where we use the well-known leg-numbering notation. It is also true that the left leg of W sits in M. In fact {(ι ω)(w ) ω B(H) } is a σ-strongly dense subspace of M. Considering the other leg of W, we get the underlying von Neumann algebra dual quantum group. So, ˆM is the σ-strong closure of ˆM of the {(ω ι)(w ) ω B(H) }. This is indeed a von Neumann algebra on H (not obvious!). It carries a comultiplication ˆ, defined by ˆ (x) = ΣW (x 1)W Σ where x ˆM and Σ denotes the flip map on H H. Observe that (x) = W (1 x)w for x M. The appearance of Σ in ˆ precisely means that the opposite comultiplication on ˆM is taken. This convention is common in the setting of multiplicative unitaries. It is the reason why, also in the previous section, we have taken the opposite comultiplication on the dual space (see the remarks after proposition 1.4 in section 1). It is proven in [K-V4] that ( ˆM, ˆ ) is again a locally compact quantum group, with some natural left invariant weight ˆϕ, with a natural G.N.S.-map ˆΛ : N ˆϕ H. This G.N.S.-map ˆΛ is determined by the formulas ˆΛ((ω ι)(w )) = ξ(ω) and ξ(ω), Λ(a) = ω(a ). 16

17 Next, we introduce the antipode on (M, ). It follows from [K-V2] that there exists a unique σ-strongly closed linear map S on M, such that S((ι ω)(w )) = (ι ω)(w ) for all ω B(H) and such that the elements (ι ω)(w ) form a core for S. The map S, has a uniquely determined polar decomposition S = Rτ i/2, where R is an anti-automorphism of M, (τ t ) is a one-parameter group of automorphisms of M and Rτ t = τ t R. We use the notation τ i/2 to denote the analytic continuation of (τ t ) to the point i/2 in the complex plane. We call R the unitary antipode and (τ t ) the scaling group. One shows that R = χ(r R) where χ denotes the flip map on M M. Hence, ϕr is a right invariant weight on M, and we make the particular choice ψ = ϕr. We use (σ t ) to denote the modular automorphism group of ϕ. Then, one proves the existence of a positive real number ν > 0, called the scaling constant, such that ψσ t = ν t ψ. Observe that the scalar µ, introduced in section 1, equals ν i here. Further, the modular automorphism σ in section 1 agrees with σ i in this section. An application of the Radon- Nikodym theorem (see [V2], proposition 5.5) yields the existence of a non-singular, positive, self-adjoint operator δ, affiliated with M, such that ψ = ϕ δ, which formally means that ψ( ) = ϕ(δ 1/2 δ 1/2 ). This element δ satisfies (ϕ ι) (a) = ϕ(a) δ, (ι ψ) (a) = ψ(a) δ 1, which can be given a proper meaning in the operator algebra setting. Moreover, we get a canonical G.N.S.-map Γ for ψ, which is formally given by Γ(x) = Λ(xδ 1/2 ). The element δ should be thought of as the quantum analogue of the modular function of a locally compact group and is therefore called the modular element. As in the classical case, it is a character: (δ) = δ δ. On the dual locally compact quantum group ( ˆM, ˆ ), we analogously have ˆδ, the right invariant weight ˆψ and the corresponding G.N.S.-map ˆΓ. Finally, the map Λ(x) Λ(x ) for x N ϕ N ϕ, can be closed and its closure has the polar decomposition J 1/2, where J is an anti-unitary on H and is a non-singular positive self-adjoint operator on H. Analogously, we have Ĵ and ˆ related to the dual. We use the notation (σ t ) to denote the modular automorphism group of ϕ. Analogously, we have (ˆσ t ) on ˆM. We also mention that ϕτ t = ν t ϕ, which allows to define a final non-singular positive self-adjoint operator P on H given by P it Λ(x) = ν t/2 Λ(τ t (x)) for x N ϕ. The operator P has the interesting feature to be self-dual, by which we mean that P it ˆΛ(x) = ν t/2 ˆΛ(ˆτt (x)) 17

18 for x N ˆϕ. This last formula also indicates that the scaling constant of ( ˆM, ˆ ) equals ν 1. The different operators on H introduced so far are related in all possible ways. In fact, all can be written in terms of δ, ˆδ and the anti-unitaries J and Ĵ. These formulas were predicted by G. Skandalis in [Sk], although the appearance of a scaling constant was not foreseen. One should consider δ and ˆδ as the basic unbounded operators, being characters of (M, ) and ( ˆM, ˆ ), respectively. 2.4 Proposition The following formulas hold for t R: 2it = ˆδ it δ it Jδ it J Ĵ ˆδ it Ĵ, ˆ 2it = ν 2it2 ˆδ it δ it Jδ it J Ĵ ˆδ it Ĵ, P 2it = ˆδ it δ it Jδ it J Ĵ ˆδ it Ĵ. Commutation relations between the different operators are governed by the following rules: Ĵδ it Ĵ = δ it, J ˆδ it J = ˆδ it, δ isˆδit = ν istˆδit δ is, ĴJ = ν i/4 JĴ. Finally, the scaling group and unitary antipode are nicely implemented: τ t (x) = P it xp it = ˆ it x ˆ it, x M, ˆτ t (x) = P it xp it = it x it, x ˆM, R(x) = Ĵx Ĵ, x M, ˆR(x) = Jx J, x ˆM. Proof: Almost everything is contained in [K-V4], propositions 2.1, 2.12 and 2.13, which we use freely. The commutation δ isˆδit = ν istˆδit δ is follows by combining proposition 2.13 of [K-V4] and the formula ˆ it = Jδ it JP it contained in proposition 8.9 of [K-V2]. In proposition 2.13 of [K-V4], the supplementary notation is used to denote the modular operator of the weight ψ (with its G.N.S.-map Γ described above). Combining with [V2], proposition 2.5, we get Ĵ it Ĵ = δ it Jδ it J it. As above, we know that ˆ it = Jδ it JP it and by duality we get it = Ĵ ˆδ it ĴP it. Using twice this formula, we get δ it Jδ it J it = Ĵ it Ĵ = P itˆδit = Ĵ ˆδ it Ĵ itˆδit = Ĵ ˆδ it Ĵ ˆδ it it. Using the commutation relations at our disposal, we arrive at the formula for 2it. The formula for P 2it follows from the equality it = Ĵ ˆδ it ĴP it, and the formula for ˆ it follows next from the equality ˆ it = Jδ it JP it. 18

19 It is easy to prove the following result. 2.5 Proposition The linear space ˆMM is σ-strongly dense in B(H). Proof: First observe that it follows from the Pentagon equation that M ˆM is contained in the σ-strong closure of ˆMM, and hence the latter is a von Neumann algebra. Its commutant is precisely ˆM M. When we apply JĴ ĴJ to the latter, we arrive at ˆM M, by combining the previous proposition with the well known result from modular theory: JMJ = M and Ĵ ˆMĴ = ˆM. Because ˆM M = C1, we are done. This gives us the complete picture of the Heisenberg algebra. One should think of it as B(H) with two specific subalgebras M and ˆM. And we are just in the situation of proposition 1.5; we only have to consider the closure of ˆMM. It should be observed that on the von Neumann algebraic level, the Heisenberg algebra is very simple: it is just B(H). However, on the C -algebraic level, things are much more subtle. Recall that our locally compact quantum group M and its dual ˆM also have C -algebraic counterparts A and Â. These C -algebras are obtained by taking the norm closure of the sets of elements (ι ω)(w ) and (ω ι)(w ), respectively. Next, one considers the norm closure of ÂA, which will be a C -algebra. In their fundamental paper [B-S], Baaj and Skandalis assumed that this C -algebra is precisely the C -algebra of compact operators on H, and they called this property regularity. Unfortunately, the regularity condition is not satisfied by all known examples. The best known counterexample is the quantum E(2)-group. In [B1] and [B2], Baaj introduced a weaker form of regularity, supposing that the closure of ÂA contains all compact operators. He called this property semi-regularity. Recently, in [B-S-V], Baaj, Skandalis and the first author discovered examples of locally compact quantum groups which are not semi-regular. In these examples, the closure of ÂA is a highly non-trivial C -algebra which is not of type I. Let us return to the von Neumann algebra setting. These are the four algebras of the commuting square in this setting: B(H), M, ˆM and C1. Next, we need the analogue of the linear functional f (cf. proposition 1.8), which was the trace in the classical case (cf. definition 1.1 and example 1.2). Here, we need a weight on B(H). Such a weight is also given by a non-singular, positive, self-adjoint operator. And from the formula in proposition 1.8 we may guess what this operator has to be. We must, however, be careful since we are working with unbounded operators. 2.6 Definition Define a non-singular positive self-adjoint operator h on H by h it = ν it2 2 it Jδ it J. Define f to be the n.s.f. weight on B(H) with density h with respect to the trace. One can easily verify that the right hand side in this equation is a one-parameter group of unitaries. The main property that one has to use is the commutation rule is δ it = ν ist δ it is 19

20 (i.e. σ s (δ) = ν s δ, cf. proposition 2.13 in [K-V4]) and the fact that J and it commute. We know from the previous section that the main property of h that we are looking for is that it implements the modular automorphisms on the subalgebras M and ˆM. The precise result is as follows. 2.7 Proposition For all t R, x M and y ˆM, we have h it xh it = σ t (x) and h it yh it = ˆσ t (y). Proof: The first result is obvious as Jδ it J commutes with M and it implements σ t. To prove the other result, it will suffice to show that also h it = ν it2 2 ˆ it Ĵ ˆδ it Ĵ (compare with the two formulas for h in proposition 1.8). This follows immediately from proposition 2.4. We can now apply a result of Haagerup on operator valued weights. 2.8 Proposition There exist unique normal semi-finite faithful operator valued weights E : B(H) M and Ê : B(H) ˆM such that ϕ E = ˆϕ Ê = f. Proof: This follows immediately from a result of Haagerup (cf. [H], see also theorem in [St]). Indeed, the modular automorphism group for f on B(H) coincides with the modular automorphism group for ϕ on M and with the the modular automorphism group for ˆϕ on ˆM; this is precisely what we proved in proposition 2.7. We are almost done. We still would like to have formulas like in proposition 1.7. Loosely speaking, they should follow from the above result and the fact that M ˆM = C1. But, as we are working with weights on von Neumann algebras, we have to be much more careful. Fortunately, here it is possible to give an explicit formula for these operator valued weights E and Ê. 2.9 Proposition Let V and ˆV be defined as follows. V = (Ĵ Ĵ)ΣW Σ(Ĵ Ĵ), ˆV = (J J)W (J J), then E(z) = (ι ˆϕ)( ˆV (z 1) ˆV ), Ê(z) = (ι ϕ)(v (z 1)V ), whenever z B(H) and z 0. 20

21 We first make some comments. Because W M ˆM, we get that V ˆM M and ˆV M ˆM. Because ι ϕ and ι ˆϕ are well-defined operator valued weights, both right hand sides of the formulas for E(z) and Ê(z) make sense in the extended positive part of B(H). Observe also that defines an action of (M, ) on M in the sense of [V1], definition 1.1, and the crossed product is given by ( (M) ˆM 1). This crossed product is isomorphic to B(H) through the map Then, it is easy to verify that the map π : z V (z 1)V. z (ι ˆϕ)( ˆV (z 1) ˆV ) coincides with the natural faithful semi-finite normal operator valued weight from the crossed product B(H) to the algebra M on which was acted, as defined in [V1], section 3. Analogously, ˆ can be considered as an action of ( ˆM, ˆ ) on ˆM. Again, B(H) is the crossed product and z (ι ϕ)(v (z 1)V ) defines the natural faithful semi-finite normal operator valued weight from B(H) to ˆM. On a more informal level, we can say the following: because ˆV M ˆM, and because ˆV implements ˆ on ˆM, we get, for a M and x ˆM, (ι ˆϕ)( ˆV (ax 1) ˆV ) = (ι ˆϕ)((a 1) ˆ (x)) = a ˆϕ(x). This makes clear that we will indeed prove an operator algebra version of proposition 1.7. Proof (of proposition 2.9): As we described above, the formula Ẽ(z) = (ι ˆϕ)( ˆV (z 1) ˆV ) defines an n.s.f. operator valued weight from B(H) to M. Put µ = ϕ Ẽ. Then, µ is an n.s.f. weight on B(H). It is sufficient to prove that µ = f. Observe that ˆV = (u 1)W (u 1) where u = JĴ. Take b N ϕ, nice enough. By this we mean that for all z C, the operator xδ z is bounded and that its closure belongs to the Tomita algebra of ϕ. When ξ H, we introduce the notation θ ξ for the operator from C to H given by θ ξ (λ) = λξ. Then we get, for η H, Ẽ(θ u Λ(b)θ u Λ(b) )η, η = ˆϕ((ω Λ(b),uη ι)(w ) (ω uη,λ(b) ι)(w )). From [V1], proposition 7.1, we get that the last expression is equal to Jσ i/2 (b)juη 2 = Ĵσ i/2(b) σ i/2 (b)ĵη, η. 21

22 So, we arrive at Ẽ(θ u Λ(b)θ u Λ(b) ) = Ĵσ i/2(b) σ i/2 (b)ĵ. Using a bar to denote the closure of an operator, we get µ(θ u Λ(b)θ u Λ(b) ) = ψ(σ i/2(b) σ i/2 (b)) = Λ(σ i/2 (b)δ 1/2 ) 2 = h 1/2 Λ(b) 2 = h 1/2 u Λ(b) 2. Defining z := θ u Λ(b)θu Λ(b), we conclude that µ(z) = f(z) <. Remember that we want to prove that µ = f. Using standard techniques, it is now sufficient to prove that span {u Λ(b) b N ϕ, b as above } is invariant under h it and µ Ad h it = µ, for all t R. The first result follows immediately from the formula h it u Λ(b) = u h it Λ(b) = ν ( t it2 )/2 u Λ(σ t (b)δ it ). To prove the final formula, take x, y N ϕ. Using the formula τ t = (σ t σ t), from [K-V2], proposition 6.8, where (σ t) is the modular automorphism group of ψ, we get W ( it δ it P it )(Λ(x) Λ(y)) = ν t/2 (Λ Λ)( (δ it τ t (y))(σ t (x) 1)) = ν t/2 (Λ Λ)((δ it δ it )(σ t σ t) (y)(σ t (x) 1)) = ν t/2 (Λ Λ)((δ it 1)(σ t σ t )( (y)(x 1))(1 δ it )) = (δ it it Jδ it J it )W (Λ(x) Λ(y)). Taking the adjoint, and replacing t by t, we get W ( it δ it it Jδ it J) = ( it P it δ it )W. Applying (J J) (J J) to this equation, and using the formula ˆ it = JP it δ it J, we get ν it2 2 ˆV (h it it δ it ) = ( it ˆ it ) ˆV. Because ˆϕ is invariant under (ˆσ t ), which is implemented by ˆ it, we get Ẽ(h it zh it ) = σ t (Ẽ(z)) for all positive z B(H). Hence, µ(h it zh it ) = µ(z) for all positive z B(H). 22

23 Combining the previous proposition with the remarks following the statement of this proposition, we observe that E is the natural operator valued weight from the crossed product B(H) to M. Because f = ϕ E, we see that f is precisely the dual weight of ϕ on the crossed product B(H). From Definition 3.4 in [V1], we then get a natural G.N.S.-construction for f, on the Hilbert space H H. The G.N.S.-representation is the map π given by π(z) = V (z 1)V and the G.N.S.-map is denoted by Λ f and determined by Λ f (xa) = ˆΛ(x) Λ(a) whenever x N ˆϕ, a N ϕ and such that these elements xa span a core for Λ f σ-strong topology on B(H) and the norm topology on H H. in the Next, we want to define a second weight g on the Heisenberg algebra B(H), as defined in proposition 1.10 in the algebraic case. In order to do so, we define a non-singular positive self-adjoint operator ρ on H by the formula ρ it = ν it2 2 ˆδit δ it. It is easy to verify that h is and ρ it commute. Hence, we can consider a new n.s.f. weight g on B(H) given by g := f ρ. By this, we mean that the Radon-Nikodym derivative of g with respect to f is ρ, and this makes sense because ρ is invariant under the modular automorphisms of f. It is clear that h it ρ it = P it. Hence, g is the n.s.f. weight on B(H) with density P 1 with respect to the trace. Because P it implements τ t on M and ˆτ t on ˆM, the weight g is precisely the analogue of the functional g described in proposition We now state the following precise result, but comment immediately on its more informal meaning Proposition If a D(S), S(a) N ϕ and x N ˆψ, then If x D(Ŝ), Ŝ(x) N ˆϕ and a N ψ, then g(a x xa) = ˆψ(x x) ϕ(s(a) S(a)). g(x a ax) = ψ(a a) ˆϕ(Ŝ(x) Ŝ(x)). Observe that we can make the following informal computation now. Let x ˆM and a M be sufficiently regular. g(x xaa ) = g(τ i (a )x xa) = ˆψ(x x) ϕ(s(τ i (a)) S(a)) = ˆψ(x x) ϕ(r(τ i/2 (a)) R(τ i/2 (a))) = ˆψ(x x) ψ(τ i/2 (aa )) = ν i/2 ˆψ(x x) ψ(aa ). 23