A NEW QUANTUM DEFORMATION OF ax + b GROUP.

Transcription

1 A NEW QUANTUM DEFORMATION OF ax + b GROUP. W. PUSZ AND S. L. WORONOWICZ Abstract. The paper is devoted to locally compact quantum groups that are related to classical ax + b group. We discuss in detail the quantization of the deformation parameter assumed with no justification in the previous paper. Next we construct on the C -level a larger family of quantum deformations of ax + b group corresponding to the deformation parameter q 2 running over an interval in unit circle. To this end, beside the reflection operator β known from the previous paper we use a new unitary generator w. It commutes with a, b and βwβ = s sgn b w, where s S 1 is a new deformation parameter related to q 2. At the end we discuss the groups at roots of unity. 0. Introduction In last years a lot of effort was devoted to constructing explicit examples of non-compact locally compact quantum groups. The present paper inscribes into this line of research. It is devoted to quantum deformations of the group ax + b of affine transformations of real line. Such quantum ax + b groups were presented first in 19. We shall use the adjective old to distinguish them from the new ax + b groups constructed in section 4 of present paper. We go back to the subject for the following reasons. At first the quantizations of the deformation parameter introduced in the previous paper was not discussed in detail. Now we give strong arguments that the values of = π 2k+3 are the only ones allowed within the setting considered in 19. Secondly one of the important formulae in 19 was not proven. We fill this gap. Third, the old quantum ax + b admits a large set of automorphisms. In 19 we identified only four of them. Now we show that the group of automorphisms is as large as S 1. These automorphisms play an important role in constructing the new quantum ax + b groups. The new groups do exist for running over an interval in R no more quantization of deformation parameter. The new quantum groups constructed in the present paper seem to be very important. They will serve as building blocks in construction of quantum SL2, R group. This is our next target. Let G be the ax + b group. On the classical level G consists of all transformations of the form 0.1 R x ax + b R, where a and b are real parameters labeling the elements of the group. We shall assume that a > 0. Assigning to each element of the group the values of the parameters we define two unbounded continuous real functions on G. To denote the functions we shall use the same letters: a, b CG. Then the C -algebra C G of all continuous functions vanishing at infinity on G is generated by log a and b: { } uniformly closed linear envelope C G = flog agb : f, g C R. Functions a and b may be considered as elements affiliated with C G. Composing two transformations of the form 0.1 with parameters a 1, b 1 and a 2, b 2 one obtains the transformation with parameters a 1 a 2, a 1 b 2 + b 1. This result leads to the following formulae describing the comultiplication: 0.2 a = a a, b = a b + b I. At the moment the elements of G are considered as affine transformations of R. However one may realize them as unitary operators acting on a Hilbert space. To this end, to any transformation of the form 0.1 we assign unitary operator V a,b BL 2 R introduced by the formula: Va,b f x = a 1/2 f a 1 x b for any f L 2 R. Then G may be identified with the set of unitary operators: 0.3 G = { V a,b : a, b R; a > 0 }. 1

2 2 W. PUSZ AND S. L. WORONOWICZ This identification preserves the group structure and the topology. More precisely V 1,0 = I and 0.4 V a1,b 1V a2,b 2 = V a1a 2,a 1b 2+b 1 for any a 1, a 2 0, and b 1, b 2 R. Moreover a sequence V an,b n converges to V a,b in strong topology if and only if a n a > 0 and b n b. In particular 0.3 with the strong operator topology is a locally compact space. One can also show that 0.3 is a closed subset of BL 2 R in strong operator topology. For any Hilbert space H we denote by KH the C -algebra of all compact operators acting on H. According to the general theory 13 the strongly continuous family of unitaries 0.3 is described by a single unitary V MKL 2 R C G. The C -algebra C G is generated in the sense of 13 by V. Formula 0.4 means that id V = V 12 V 13. This way we arrive to the notion of a quantum group of unitary operators. Let H be a Hilbert space. We shall consider pairs A, V, where A is a C -algebra and V is a unitary element of the multiplier algebra MKH A. If A is generated by V MKH A then A, V is called a quantum family of unitary operators. We say that the family is closed with respect to operator multiplication if there exists a morphism MorA, A A such that 0.5 id V = V 12 V 13. Then is unique because A is generated by V. Finally A, V is said to be a quantum group of unitary operators, if it is closed with respect to the operator multiplication and if A, is a locally quantum group in the sense of Kustermans and Vaes 3. It should be possible to formulate the last condition directly in terms of A, V. However this is not the subject of the present paper. Let us go back to the ax + b group. In the quantum setting functions a and b are replaced by selfadjoint elements a = a > 0 and b = b that no longer commute. Instead they satisfy the relation 0.6 ab = q 2 ba, where the deformation parameter q 2 is a number of modulus 1. Unfortunately in our case elements a and b are represented by unbounded operators and the products ab and ba may not be well defined because of the domain problem. For this reason we replace 0.6 by the so called Zakrzewski relation. It says that for any τ R: a iτ ba iτ = e τ b. In this formula is a real constant such that q 2 = e i. For technical reasons we shall assume that 0 < < π 2. The reader should notice that for τ = i the above relation reduces to 0.6. The second problem is related to the comultiplication. We would like to keep formulae 0.2. However in general a b + b I is not selfadjoint and in the best case we may expect that b is a selfadjoint extension of a b + b I: a b + b I b. To choose the extension in a well defined way we have to use additional operators independent of a and b. For old quantum ax + b groups we use a selfadjoint unitary β commuting with a and anticommuting with b. For new groups the situation is even more complicated. It means that the algebra A is no longer generated by log a and b. It is not obvious, how to present quantum ax+b group as a quantum group of unitary operators A, V. The crucial point is the formula V = V a, b,... expressing V in terms of a, b and perhaps some other elements related to A. The equation 0.5 takes the form V a a, a b + b I,... = V a I, b I,... V I a, I b,..., where a b + b I is a suitable selfadjoint extension of a b + b I. To find solutions of this equation we spent a lot of time making use of our experience in the area of quantum exponential functions and quantum groups cf. 15, 14, 11, 18, 17, 19, 7, 9. As a result we got formulae 3.8 and 4.8 that are starting points in our presentation. Let us briefly discuss the content of the paper. Sections 1 and 2 are devoted to mathematical tools used in the paper. In the first one we recall the Zakrzewski commutation relation and related quantum exponential function with a slightly modified notation. Most of the results presented in that section come from 17; the essentially new result is contained in Proposition 1.4. The second

3 A NEW QUANTUM DEFORMATION OF ax + b GROUP. 3 section deals with the notion of a C -algebra generated by affiliated elements. We prove a number of results used in the main part of the paper. Section 3 is devoted to the quantum ax + b groups introduced in 19. These groups exist only for special values of deformation parameter q 2 = e i with = π 2k+3, where k = 0, 1, 2,.... This fact was not really shown in 19. The special values of the deformation parameter were chosen to proceed with some computations. It was not clear that at the expense of some complications one is not able to construct quantum ax + b group for larger set of values of the deformation parameter. Now, presenting the ax + b group as a quantum group of unitary operators we obtain the quantization of the deformation parameter as a precise mathematical statement cf. Theorem 3.3. More precisely for q 2 = e i we shall construct a C -algebra A with distinguished selfadjoint elements a, b and iβb affiliated with it the so called reflection operator β is a unitary involution which is not affiliated with A. These elements satisfy in a well defined sense the relations ab = q 2 ba, aβ = βa and bβ = βb. The algebra A is generated by a unitary element V MKL 2 R A. The pair A, V is defined for all 0 < < π/2. However, the existence of satisfying the condition 0.5 selects much smaller subset of admissible s. We shall prove that exists if and only if is of the form indicated above. Next we derive formulae showing how acts on generators of A. In particular we prove an elegant formula describing the action of the comultiplication on the reflection operator. This formula appeared with no proof in the previous paper cf. formula 4.16 of 19. At the end of section 3 we find an interesting action of S 1 on the algebra A: for any s S 1 we have an automorphism φ s of A and φ ss = φ s o φs. If exists i.e. if = π 2k+3, where k = 0, 1, 2,... then o φs = φ s φ s o. New quantum groups related to classical ax + b group are constructed in section 4. Using one of the automorphism φ s described at the end of section 3 we consider the corresponding crossed product. In other words we extend the algebra A by adding a new unitary generator w implementing φ s. This enlargement of the algebra opens new possibilities. In particular we obtain a new admissible values of the deformation parameter. Now = π p, where p is a number larger than 2 such that e iπp = s. The latter relation distinguishes a discrete set of possible p. However the quantization of disappears because changing s we may cover the whole interval 0, π 2. For s = 1 we obtain p = 2k + 3, where k = 0, 1, 2,.... In this case the new quantum ax + b group reduces to a semidirect product of the old one by S 1. For s 1 we get essentially new examples of locally compact quantum groups. In the next section section 5 we investigate the multiplicative unitaries for the quantum groups constructed in section 4. We prove their modularity and find the unitary antipode and scaling group. In particular the objects constructed in section 4 satisfy all the axioms of Kustermans and Vaes 3 and the ones of Masuda, Nakagami and Woronowicz 4. At the end of the section we briefly discuss the duals of the new quantum ax + b groups. The last section is devoted to new quantum ax + b groups with q 2 being a root of unity. In this case we may pass to groups with smaller size. To this end we have to assume that the unitary generator w satisfies the additional relation of the form w N = I. The word size used in the previous paragraph has a precise meaning. It is based on the Stone - von Neumann theorem. Let A, be one of the quantum ax + b group considered in this paper and π be a representation of A acting on a Hilbert space H π such that kerb = {0}. Then operators logπa and log πb satisfy the same commutation relations as position x QM and momentum p QM in quantum mechanics. By the Stone - von Neumann theorem the pair logπa, log πb is unitarily equivalent to a direct sum of k copies of x QM, p QM. The number k will be called the multiplicity of π. We say that the size of the group A, is equal k if there exists a faithful representation of A with multiplicity k and if k is the smallest number with this property. In the classical situation the algebra of functions A is generated by log a and b. This is not the case when we consider quantum ax + b groups: we use additional generators such as β and w. It means that together with the quantum deformation we pass to a sort of extension of the group. The size says us, how large the extension is. The old quantum ax+b groups are of size 2. This is the minimal value. One can show that the old quantum ax + b groups are the only ones with size 2. The new groups introduced in section 4 are of infinite size. On the other hand the groups at roots of unity considered in the last section are of size 2N, where N is the number appearing in the relation w N = I.

4 4 W. PUSZ AND S. L. WORONOWICZ Our approach extensively uses the C -algebra language and the theory of selfadjoint operators on Hilbert space. For the basic facts concerning the general C -algebra theory we refer to 1, 6. The notation used in the paper follows the one explained in 13, 12. In particular MA is the multiplier algebra of a C -algebra A. The affiliation relation in the sense of C -algebra theory is denoted by η and A η is the set of all affiliated elements unbounded multipliers. It is known that MA A η. A morphism from A to a C -algebra B is by definition any -homomorphism π : A MB such that πab is dense in B. Let us recall that any such π has the unique extension to a unital -homomorphism π : MA MB and to -preserving map π : A η B η respectively both denoted by the same symbol. The set of all morphisms from A to B is denoted by MorA, B. With some abuse of notation, the symbol RepA will stay for the set of all non-degenerate representations of a C -algebra A. For any π RepA, we denote by H π the carrier Hilbert space of π. Then π MorA, KH π. In the paper we mostly deal with concrete C -algebras. By definition they are norm closed -subalgebras of the algebra BH of all bounded operators acting on some separable Hilbert space H. As a rule, C -algebras we deal with are separable. Non separable ones will appear only as a multiplier algebras. In particular BH = MKH. We shall denote by C H the set of all non-degenerate separable C -algebras of operators acting on a Hilbert space H. We recall that an algebra A BH is non-degenerate if AH is dense in H. We shall use functional calculus for strongly commuting selfadjoint operators. If T and β are selfadjoint operators acting on a Hilbert space H and T and β strongly commute then T = Λ r der, ϱ, β = Λ ϱ der, ϱ, where der, ϱ is the common spectral measure supported by the joint spectrum Λ R 2 of T, β. Moreover for any measurable complex valued function on Λ we have ft, β = fr, ϱder, ϱ. Λ In this context the characteristic function χ will appear quite often. By definition for any sentence R, we have { 0 if R is false, χr = 1 if R is true. Typically R is a formula involving inequality sign. For example χr 0 is equal 0 for positive r and 1 for r = 0 or negative. Consequently χt 0 is the spectral projection assigned to the negative part of the spectrum of a selfadjoint operator T. The corresponding spectral subspace will be denoted by HT 0: HT 0 = χt 0H. Similarly χt = λ is the orthogonal projection on the eigenspace HT = λ of T corresponding to the eigenvalue λ R. We refer to 17 for more detailed explanation of this notation. Let Npr = rχr < 0. Then for any selfadjoint T, 0.7 NpT = T χt < 0 is a selfadjoint operator acting on H. This is the negative part of the operator T. Another function frequently used in the paper is the one that returns the sign of the argument: sgn r = χr > 0 χr < 0. Then sgn T is the partial isometry that appears in the polar decomposition: T = sgn T T. 1. A special function and selfadjoint extensions. In this section we recall in a slightly modified version with a certain loss of generality the basic definitions and statements of 17. The only essentially new result is contained in formula Later on it will help us to prove the formula announced in 19, formula We start with a modified version the quantum exponential function introduced in 17. Let R and 0 < < π 2. Instead of function F defined on the set R { 1, 1} R + {0} we shall use function G defined on R { 1, 1}. It is related to the function F by the formula 1.1 G r, ϱ = F r, ϱχr < 0

5 A NEW QUANTUM DEFORMATION OF ax + b GROUP. 5 for any r R and ϱ = ±1. Taking into account definition 17, formula 1.19 we obtain V θ log r for r > G r, ϱ = π 1 + iϱ r Vθ log r πi for r < 0, where θ = 2π and V θ is the meromorphic function on C such that { 1 } V θ x = exp log1 + t θ dt 2πi t + e x 0 for all x C such that Ix < π. In addition G 0, ±1 = 1. Then G r, ϱ is a continuous function on R { 1, 1} and 1.3 G r, ϱ = G r, ϱ ϱχr < 0 = ϱ χr < 0. The asymptotic behavior of G r, ϱ for large r is described by the formula { } log r G r, ϱ C exp, 2i where C is a phase factor depending only on sgn r and ρ and means that the difference goes to 0 when r ± see Statements 9 and 10 of 17, Theorem 1.1. It is known that the quantum exponential function assumes values of modulus 1. Therefore if T and β are operators acting on a Hilbert space H, T is selfadjoint and β is unitary selfadjoint commuting with T then G T, β is unitary. Now we recall the concept of selfadjoint extension of a symmetric operator defined by a reflection operator. Let Q be a symmetric operator acting on a Hilbert space H and ρ be a unitary selfadjoint operator ρ = ρ, ρ 2 = I anticommuting with Q. Then we denote by Q ρ the restriction of Q to the domain {x DQ : ρ Ix DQ}. It is known cf. 17, Proposition 5.1 that Q ρ is a selfadjoint extension of Q. We shall use the following simple Proposition 1.1. Let Q, X and ρ be operators acting on a Hilbert space H such that Q is symmetric, X is selfadjoint, ρ is unitary selfadjoint, ρq = Qρ and ρx = Xρ. Assume that the restrictions of Q and X to Hρ = 1 coincide: 1.5 Q Hρ= 1 = X Hρ= 1. Then X = Q ρ. Proof. Let H 1 = Hρ = 1 and H 2 = Hρ = 1. Then H = H 1 H 2 and all bounded and some unbounded operators may be represented by 2 2 matrices. In particular I, 0 ρ =. 0, I Remembering that Q and X anticommute with ρ we obtain: 0, Q 0, X Q = and X = Q +, 0 X +, 0 where Q + and X + are operators acting from H 1 to H 2 and Q and X are operators acting from H 2 to H 1. Clearly Q + Q Q is symmetric and X = X+ X is selfadjoint. Assumption 1.5 means that Q + = X +. Therefore On the other hand X = 0, Q + Q +, 0 Q = 0, Q + Q, 0.., It shows that X Q and DX = {x DQ : ρ Ix DQ}. Let R. We shall use the Zakrzewski relation o cf. 17. Let R and S be selfadjoint operators acting on a Hilbert space H with the polar decompositions R = sgn R R and S =

6 6 W. PUSZ AND S. L. WORONOWICZ sgn S S. For simplicity we shall assume that one of the operators R and S has trivial kernel. If ker S = {0}, then sgn S is unitary selfadjoint and sgn S commutes with R R o S and S iλ R S iλ = e λ R. for any λ R. If ker R = {0}, then sgn R is unitary selfadjoint and sgn R commutes with S R o S and R iλ S R iλ = e λ S. for any λ R. If ker R = ker S = {0}, then the two above conditions are equivalent. One can easily show that antiunitary operators reverse the direction of Zakrzewski relation: 1.6 R o S and J is an = JSJ o JRJ antiunitary involution Let R and S be selfadjoint operators with trivial kernels and R o S. It is known 17, Example 3.1 that in this case, the operators e i /2 S 1 R and e i /2 SR 1 are selfadjoint and sgn e i /2 S 1 R = sgn e i /2 SR 1 = sgn Rsgn S. We shall use the following result cf. 17, Theorem 5.2: Proposition 1.2. Let R, S and τ be operators acting on a Hilbert space H. Assume that R and S are selfadjoint with trivial kernels, R o S, and that τ is unitary, selfadjoint anticommuting with R and S. We set T = e i /2 S 1 R. Then T is a selfadjoint operator with trivial kernel, T commutes with τ, R + S is a closed symmetric operator and the selfadjoint extension 1.7 R + S τ = G T, τ SG T, τ = G T 1, τrg T 1, τ Remark 1.3. If τ is another unitary, selfadjoint operator anticommuting with R and S and if in addition there exists a unitary selfadjoint operator ρ that commutes with τ, τ and S and anticommutes with R then 1.8 R + S τ = R + S τ = τ = τ. Indeed if R + S τ = R + S τ, then cf. 1.7 G T, τ SG T, τ = G T, τ SG T, τ. It shows that the unitary operator U = G T, τ G T, τ commutes with S and hence with S. Clearly 1.9 G T, τ = G T, τ U. Moreover T o S due to Zakrzewski relation R o S and ρ anticommutes with T. As we know τ and τ anticommute with S, hence they commute with S. We shall use Proposition 2.4 see the next section. Setting R 1 = R 2 = T, ρ 1 = τ, ρ 2 = τ, U 1 = I, U 2 = U and replacing S by S we have all the assumptions of that proposition satisfied. Therefore 1.9 implies the equality τ NpT = τ NpT. It means that τ and τ coincide on HT < 0. Then τ and τ coincide on ρht < 0 for any operator ρ commuting with τ and τ. If ρ commutes with S and anticommutes with R then it anticommutes with T and ρht < 0 = HT > 0. In this case τ and τ coincide on HT < 0 HT > 0 = H this is because ker T is trivial. Hence τ = τ. We shall prove a result of the same flavor as 1.7: Proposition 1.4. Let R and S be strictly positive selfadjoint operators acting on a Hilbert space H such that R o S and let τ, ρ, σ and ξ be unitary selfadjoint operators commuting with R and S. Assume that τ commutes with ξ and anticommutes with ρ and σ and 1.10 ξχτ = 1 = αρσχτ = 1,

7 A NEW QUANTUM DEFORMATION OF ax + b GROUP. 7 where α = i e iπ2 2. We set: T = e i /2 S 1 R. Then T is a positive selfadjoint operator with trivial kernel, σs π + ρr π is a closed symmetric operator anticommuting with τ and the selfadjoint extension π σs + ρr π τ = G τt, ξ σs π G τt, ξ 1.11 = G τt 1, ξρr π G τt 1, ξ Proof. At first we shall prove the first equality of Inserting S 1 instead of R and R instead of S in 17, Example 3.1 we see that T is a positive selfadjoint operator with trivial kernel and 1.12 T ik = e i 2 k2 S ik R ik = e i 2 k2 R ik S ik for any k R. Denote by X the right hand side of the first equality in We know that G τt, ξ is unitary in what follows we write G τt, ξ 1 instead of G τt, ξ. Operator S π commutes with σ and τ whereas σ and τ anticommute. Therefore σs π is a selfadjoint operator anticommuting with τ. So is X. Let Q = σs π + ρr π. Clearly Q is a symmetric operator anticommuting with τ. By virtue of Proposition 1.1 it is sufficient to show that 1.13 Q Hτ=1 = X Hτ=1. Restricting G τt, ξ σs π G τt, ξ to Hτ = 1 we may replace the second τ by 1 and the first τ by 1 this is because σ maps Hτ = 1 onto Hτ = 1: X Hτ=1 = G T, ξ 1 σs π G T, ξ Hτ=1 and using 1.2 we obtain 1.14 X Hτ=1 = 1 + iξt π 1 1 π Vθ log T πi σs Vθ log T Hτ=1 Now we shall move σs π to the right end of It is known cf. 17, relation 1.30 that the function V θ x has no poles and no zeroes in strip Σ = {x C : 0 Ix π}. Therefore functions V θ x and V θ x 1 are continuous on Σ and holomorphic inside Σ. Moreover cf. 17, the asymptotic formula 1.37, V θ x 1 when Rx whereas Ix stays bounded and using formula 1.32 of 17 one can easily show that for any λ > 0, functions e λx2 V θ x and e λx2 V θ x 1 are bounded on Σ. Furthermore T is a strictly positive selfadjoint operator and T o S. Therefore T π o S π and using Statement 3 of Theorem 3.1 of 17 we obtain S π Vθ log T = V θ log T + iπ S π. Inserting this formula into 1.14 and using in the second step formula 1.28 of 17 we get: X Hτ=1 = 1 + iξt π 1 Vθ log T πi 1 V θ log T + πiσs π Hτ=1 1 = 1 + iξt π 1 + T 2π σs π Hτ=1 = 1 iξt π π S σ Hτ=1. On the other hand multiplying both sides of 1.10 by σ from the right we obtain ξσχτ = 1 = αρχτ = 1. Therefore ρ Hτ=1 = αξσ Hτ=1 and Q Hτ=1 = S π + αξr π σ Hτ=1. To end this part of the proof it is sufficient to show that 1.15 S π + αξr π π π = 1 iξt S. We shall use It shows that for any x, y H and any k R we have y S ik x i e i 2 k2 y ξr ik x = I + iξt ik y S ik x Let x D S π π π D R. If y DT then both sides of the above formula have continuous holomorphic continuation to the strip π Ik 0. Inserting k = i π we obtain π π π π y S x + α y ξr x = I + iξt y S x.

8 8 W. PUSZ AND S. L. WORONOWICZ This formula holds for any y in the domain of I + iξt π. Therefore S π x D I iξt π and S π x + αξr π x = I iξt π π S x. This way we showed that 1.16 S π π π π + αξr I iξt S. To prove the converse inclusion we use again Let x D S π π and S x D T π. Then for any y H and k R: e i 2 k2 R ik y S π ik x = y T ik S π x. If y DR π then both sides of the above formula have continuous holomorphic continuation to the strip π Ik 0. Inserting k = i π we obtain iα R π y x = y T π S π x. This formula holds for any y DR π. Therefore x DR π. This way we showed the inclusion DT π S π DR π. Consequently D I iξt π π π S D S + αξr π. Combining this result with 1.16 we get 1.15 and This way the first equality of 1.11 is shown. The second equality may be shown in the same manner. However it is simpler to use the following trick based on 1.6. Let J be an antiunitary involutive operator acting on H and R n = JSJ, S n = JRJ, τ n = JτJ, ρ n = JσJ, σ n = JρJ, ξ n = JξJ. The subscript n stands for new. One can easily show that the new operators satisfy all the assumptions of our theorem. In particular ξ n χτ n = 1 = Jξχτ = 1J = Jαρσχτ = 1J = ασ n ρ n χτ n = 1 = ασ n ρ n χτ n = 1 = αρ n σ n χτ n = 1. In the present case T n = e i /2 Sn 1 R n = Je i /2 R 1 SJ = JT 1 J and the first equality of 1.11 takes the form: σ n S π n + ρ n R π n = G JτT τn 1 J, JξJ σ n S π n G JτT 1 J, JξJ = JG τt 1, ξρr π G τt 1, ξ J. A moment of reflection shows that the left hand side of this formula equals J σs π and the second equality of 1.11 follows immediately. + ρr π J τ To end the proof we have to show that the operator Q = σs π + ρr π is closed. Operator ξt π is selfadjoint. Therefore operator I iξt π is invertible with the inverse I iξt π 1 BH. Using this fact one can easily show that the composition I iξt π π S σ is a closed operator. Restricting this operator to Hτ = 1 we obtain Q Hτ=1. Hence Q Hτ=1 is closed. Remembering that Q anticommute with τ we conclude that Q is closed. Remark 1.5. According to 1.10 operator αρσχτ = 1 is selfadjoint. remembering that ρ and σ anticommute with τ one can show that ρσ = α 2τ σρ. Using this fact and Conversely let τ, ρ, σ be unitary selfadjoint operators commuting with R, S and let τ anticommutes with ρ and σ. If the above relation is satisfied, then using 1.10 to define ξ on Hτ = 1 and extending it in an arbitrary way to a unitary selfadjoint operator defined on the whole space we obtain the quadruple ρ, σ, τ, ξ of operators satisfying the assumptions of Proposition 1.4. We end this section with the reformulation of Theorem 6.1 of 17. Theorem 1.6. Let R, S be a pair of selfadjoint operators acting on a Hilbert space H such that ker R = ker S = {0} and R o S and let ρ, σ be unitary selfadjoint operators on H. Assume that ρ commutes with R, ρ anticommutes with S, σ commutes with S and σ anticommutes with R. We set: T = e i /2 S 1 R, where α = i e iπ2 2. Then τ = αρσχs < 0 + ασρχs > 0, 1. T is selfadjoint, sgn T = sgn R sgn S, T o R and T o S 2. τ is unitary selfadjoint, τ commutes with T and τ anticommutes with R and S.

9 A NEW QUANTUM DEFORMATION OF ax + b GROUP G satisfies the following exponential function equality: 1.17 G R, ρg S, σ = G T, τ G S, σg T, τ = G R + S τ, σ, where R + S τ is the selfadjoint extension of R + S corresponding to the reflection operator τ and σ = G T, τ σg T, τ. Proof. By direct computation one can easily show that τ 2 = I, τ = τ and τχt < 0 = αρχr < 0σχS < 0 + ασχs < 0ρχR < 0. Now, our theorem follows immediately from 17, Theorem 6.1. Remark 1.7. In Theorem 1.6, operator τ may be replaced by τ = αρσχr > 0 + ασρχr < 0. Operator σ is not affected by this change. Indeed, using the formula sgn T = sgn R sgn S, one can verify that τ χt < 0 = τχt < 0. It shows that G T, τ = G T, τ. 2. The special functions and affiliation relation. In this section we shall use the concept of a C -algebra generated by a set of affiliated elements 13, Definition 4.1, page 501. Let C, A be C -algebras and V be an element affiliated with C A. We say that A is generated by an element V η C A if and only if for any π RepA and any B C H π we have: 2.1 id πv η C B = π MorA, B In general the above condition is not easy to verify. We shall use the following criterion cf. 13, Example 10, page 507: Proposition 2.1. Let C, A be C -algebras and V be a unitary element of MC A. Assume that there exists a faithful representation φ of C such that: 1. For any φ-normal linear functional ω on C we have ω idv A 2. The smallest -subalgebra of A containing {ω idv : ω is φ-normal} is dense in A. Then A is generated by V MC A. We recall that a linear functional ω on C is said to be φ-normal if there exists a trace-class operator ρ acting on H φ such that ωc = Trρφc for all c C. Let Λ be the locally compact space obtained from R { 1, 1} by gluing points r, 1 and r, 1 for all r 0. Then: } fr, 1 = fr, 1 C Λ = {f C R { 1, 1} :. for all r 0 If R, ρ are operators acting on a Hilbert space H, R is selfadjoint, ρ is unitary selfadjoint and ρ commutes with R then the mapping 2.2 C Λ f πf = fr, ρ BH is a representation of C Λ acting on H. Operators R and ρ NpR are determined by π. Indeed R = πf 1 and ρ NpR = πf 2, where f 1, f 2 are elements of C Λ η = CΛ introduced by the formulae 2.3 f 1 r, ϱ = r, f 2 r, ϱ = ϱ Npr for any r R and ϱ = ±1. Using 13, Example 2, page 497 we see that f 1, f 2 generate C Λ. Therefore for any π Rep C Λ and any B C H π we have: πf 1, πf 2 η B = π MorC Λ, B = πf η B for any f CΛ In particular for π introduced by 2.2 we obtain the following result: R, ρ NpR η B 2.4 = fr, ρ η B. f CΛ

10 10 W. PUSZ AND S. L. WORONOWICZ Our special function G is continuous and satisfies the relation G r, 1 = G r, 1 for all r 0. In other words G CΛ. For any r R, ϱ = ±1 and t > 0 we set: 2.5 F t; r, ϱ = G r, ϱg tr, ϱ. Let R + = {t R : t > 0}. Then F is a continuous function on R + Λ with values of modulus 1 and we may treat F as unitary element of M C R + C Λ. We shall prove the following Proposition 2.2. The C -algebra C Λ is generated by F M C R + C Λ. Proof. We shall use Proposition 2.1 with C = C R +, A = C Λ and V = F. Let φ be the natural representation of C R + acting on L 2 R +. For any g C R +, φg is the multiplication by g. Then φ is faithful and a linear functional ω on C R + is φ-normal if and only if it is of the form ωg = gtϕt dt, R + where ϕ L 1 R +. Applying ω id to F M C R + C Λ we obtain an element of M C Λ i.e. a bounded continuous function on Λ. Clearly for any r R and ϱ = ±1 we have ω idf r, ϱ = F t; r, ϱϕt dt R = G r, ϱ G tr, ϱϕt dt. R + Taking into account the asymptotic behavior 1.4 and using the Riemann Lebesgue lemma one can verify that the integral on right hand side tends to 0 when r ±. In other words, ω idf C Λ. Using Statement 7 of Theorem 1.1 of 17 one can easily show that 1 r 2.7 lim G tr, ϱ 1 = t 0 + t 2i sin /2 for all r R and ϱ = ±1. Let r, r R and ϱ, ϱ = ±1. Assume for the moment that ω idf r, ϱ = ω idf r, ϱ for all φ-normal functionals ω. Then G r, ϱg tr, ϱ = G r, ϱ G tr, ϱ for all t > 0. Going to the limit when t +0 we get G r, ϱ = G r, ϱ. Comparing this formula with the previous one we see that G tr, ϱ = G tr, ϱ for all t > 0. Formula 2.7 shows now that r = r and by 1.3 ϱχr < 0 = ϱ χr < 0. This way we have shown that the functions 2.6 separate points of Λ. Now, using the Stone - Weierstrass theorem applied to the one point compactification of Λ we conclude that the smallest -algebra containing all functions 2.6 is dense in C Λ. The following Proposition will be very useful in proving many technical details important in the future considerations. Proposition 2.3. Let R, ρ, U, S be operators acting on a Hilbert space H and C C H. Assume that: 1. R is selfadjoint and ρ is unitary selfadjoint commuting with R, 2. U is unitary, 3. S is positive selfadjoint, ker S = {0}, S commutes with ρ and U and R o S, 4. Operators R, ρ NpR, U and log S are affiliated with C. Then G R, ρ MC and 1. For any φ RepC and any B C H φ we have: φlog S, φ G R, ρ U φr, φρ NpR, φu = are affiliated with B are affiliated with B

11 A NEW QUANTUM DEFORMATION OF ax + b GROUP For any φ 1, φ 2 RepC such that H φ1 = H φ2 we have: φ φ 1 S = φ 2 S, 1 R = φ 2 R, φ 1 G R, ρ U = φ 2 G R, ρ U = φ 1 ρ NpR = φ 2 ρ NpR, φ 1 U = φ 2 U Proof. Relation G R, ρ MC follows immediately from 2.4. Ad 1. Let λ R. Using the commutation relations satisfied by operators R, ρ, U, S we have: S iλ G R, ρ US iλ = G tr, ρ U, where t = e λ > 0. Applying a representation φ of C to both sides of the above relation we get φs iλ φ G R, ρ U φs iλ = φ G tr, ρ U. If φlog S, φ G R, ρ U η B, then all factors on the left hand side of the above equation belong to MB and depend continuously on λ we use strict topology on MB. Therefore φ G tr, ρ U MB for any t R + and the mapping R + t φ G tr, ρ U MB is strictly continuous. Applying the hermitian conjugation and multiplying from the left by φ G R, ρ U MB we see that φ G R, ρ G tr, ρ = φ F t; R, ρ MB and the mapping 2.8 R + t φ F t; R, ρ MB is strictly continuous. In the above relations F is the function introduced by 2.5. According to the general theory 13, strictly continuous bounded mappings from R + into MB correspond to elements of MC R + B. A moment of reflection shows that the mapping 2.8 corresponds to the element id φo πf, where π is the representation of C Λ introduced by 2.2. This way we have shown that id φo πf MC R + B. Using now Proposition 2.2 we conclude that φo π MorC Λ, B. Therefore φo π maps continuous functions on Λ into elements affiliated with B. Applying this rule to functions f 1, f 2 cf. 2.3 and G we obtain: φr, φρ NpR η B and φg R, ρ MB. Comparing the last relation with the assumed one φ G R, ρ U MB we see that φu MB. Statement 1 is shown. Ad 2. Let φ = φ 1 φ 2. Then H φ = H φ1 H φ2 and φc = φ 1 c φ 2 c. In our case H φ1 = H φ2. We set: B = {m m : m KH φ1 }. Then B C H φ. One can easily verify that for any c η C we have: φc η B φ 1 c = φ 2 c. Now Statement 2 follows immediately from Statement 1. We shall use slightly different version of Statement 2 of the above proposition. Proposition 2.4. Let R 1, ρ 1, U 1, R 2, ρ 2, U 2, S be operators acting on a Hilbert space H. Assume that for each k = 1, 2 the operators R k, ρ k, U k, S satisfy the assumptions 1-3 of the previous Proposition. Then 2.9 G R 1, ρ 1 U 1 = G R 2, ρ 2 U 2 = R 1 = R 2, ρ 1 NpR 1 = ρ 2 NpR 2, U 1 = U 2. Proof. Let C = KH KH and for any m 1, m 2 KH we set φ k m 1 m 2 = m k k = 1, 2. We use Proposition 2.3 with R, ρ, U and S replaced by R 1 R 2, ρ 1 ρ 2, U 1 U 2 and S S, Now 2.9 follows immediately from Statement 2 of Proposition 2.3. Proposition 2.5. Let X and Y be selfadjoint operators acting on Hilbert spaces K and H respectively. Assume that the spectral measure of X is absolutely continuous with respect to the Lebesgue measure. Then for any A C H we have: e ix Y is affiliated = Y is affiliated with A with KK A

12 12 W. PUSZ AND S. L. WORONOWICZ Proof. For any normal linear functional ω on BK and t R we set f ω t = ω e itx. Then f ω is a continuous function on R. Remembering that the spectral measure of X is absolutely continuous with respect to the Lebesgue measure and using the Riemann-Lebesgue lemma one can easily show that f ω t 0 when t ±. Therefore f ω C R. Let t, t R, t t. Assume for the moment that f ω t = f ω t for all ω. Then e itx = e it X and e it t X 2π = I. It shows that the spectral measure of X is supported by the set t t Z, which is in contradiction with the assumption saying that the spectral measure of X is absolutely continuous with respect to the Lebesgue measure. This way we showed that functions f ω separate points of R. By the Stone Weierstrass theorem, the smallest -subalgebra of C R containing all f ω is dense in C R. By the general theory strongly continuous mappings from R into the set of unitary operators acting on K correspond to unitary multipliers of KK C R. Let X MKK C R be the unitary corresponding to the mapping R t e itx BK. Then for any normal linear functional ω on BK we have ω idx = f ω. Using Proposition 2.1 we see that C R is generated by X MKK C R. f C R we set: πf = fy. For any Then π is a representation of C R acting on the Hilbert space H π = H. A moment of reflection shows that id πx = e ix Y. If e ix Y is affiliated with KK A then π MorC R, A and π maps continuous functions on R into elements affiliated with A. Applying this rule to the coordinate function ft = t we obtain Y = πf η A. 3. Constructions related to old quantum ax + b groups. In this section we recall the main results of 19. The quantum ax+b group will be presented as a quantum group of unitary operators. We shall construct a pair A, V, where A is a C -algebra and V is a unitary element of MKK A, where K is a Hilbert space endowed with a certain structure and KK denotes the algebra of all compact operators acting on K. A, V may be treated as a quantum family of unitary operators acting on K labeled by elements of quantum space related to the C -algebra A. Our construction will depend on a real parameter. We shall assume that 0 < < π/2. Negative value of leads to the C -algebra anti-isomorphic to that with positive. On the other hand the restriction < π/2 is related to the technical assumption used in the theory of the quantum exponential function 17. The main result of this section is contained in Theorem 3.2. It states that A, V is a quantum group if and only if = with k = 0, 1, 2,.... π 2k+3 To define A we consider three operators a, b and β acting on the Hilbert space L 2 R. Operator a is strictly positive selfadjoint and such that for any τ R and any x L 2 R we have a iτ x t = e τ/2 xe τ t In other words a is the analytic generator of one-parameter group of unitaries corresponding to the homotheties of R. Operator b is the multiplication operator: bxt = txt. By definition domain Db consists of all x L 2 R such that the right hand side of the above equation is square integrable. Finally, β is the reflection: for any x L 2 R we have: βxt = x t Clearly β is unitary selfadjoint. One can easily verify that aβ = βa and bβ = βb. By the last relation ibβ is selfadjoint. Moreover 3.1 a iτ ba iτ = e τ b for any τ R. This relation means that a o b.

13 A NEW QUANTUM DEFORMATION OF ax + b GROUP. 13 Theorem 3.1. Let { 3.2 A = f 1 b + βf 2 b glog a : f 1, f 2, g C R f 2 0 = 0 Then: 1. A is a nondegenerate C -algebra of operators acting on L 2 R, 2. log a, b and ibβ are affiliated with A: log a, b, ibβ η A, 3. log a, b and ibβ generate A. } norm closed linear envelope. Proof. Ad 1. Using the relation bβ = βb one can easily show that 3.3 B = { f 1 b + βf 2 b : f 1, f 2 C R f 2 0 = 0 } norm closed linear envelope is a non-degenerate C -algebra of operators acting on L 2 R. Let C 0 R, B denote the set of all continuous mappings from R into B with compact support. Then { norm closure 3.4 A = fta it dt : f C 0 R, B}. R To prove this formula it is sufficient to notice that for ft = f 1 b + βf 2 b ϕt, where t R and ϕ C 0 R we have fta it dt = f 1 b + βf 2 b glog a, R where gλ = R ϕteiλt dt λ R and by the Riemann-Lebesque Lemma, g C R. On the other hand 3.1 shows that the unitaries a it t R implement a one parameter group of automorphisms of B. Using now the standard technique of the theory of crossed products cf. 6, Section 7.6 one can easily show that 3.4 is a non-degenerate C -algebra of operators acting on L 2 R. Statement 1 is proven. Ad 2. We recall cf. 5, 13 that a closed operator T is affiliated with a C -algebra A if the z-transform z T = T I + T T 1 2 MA and if I + T T 1 2 A is dense in A. Inspecting definition 3.2 one can easily show that z log a = log a I + log a is a right multiplier of A and that A I + log a is dense in A. Passing to adjoint operators we see that z log a = z log a is a left multiplier hence z log a MA and that I + log a A is dense in A. It shows that log a is affiliated with A. For T = b and T = iβb we have z T = bi + b and z T = iβbi + b respectively. In both cases I + T T 1 2 = I + b Taking into account definition 3.2 one can easily show that I + T T 1 2 A is dense in A and that z T is a left multiplier of A. However in both cases z T is selfadjoint. Therefore z T is also a right multiplier and z T MA. It shows that b and iβb are affiliated with A. Ad 3. We shall use Theorem 3.3 of 13. By definition 3.2, I + b 2 1 I + log a 2 1 A. To end the proof it is sufficient to show that a, b, iβb separate representations of A. If c A is of the form 3.5 c = f 1 b + βf 2 b glog a, where f 1, f 2, g C R, f 2 0 = 0 and f 2 is differentiable at point 0 R, then f 2 t = itht, where t R and h C R and 3.6 πc = f 1 πb + πiβbhπb gπlog a for any representation π of A. One can easily see that elements of the form 3.5 form a dense subset of A. Formula 3.6 shows now that π is determined uniquely by πlog a, πb and πiβb. Now we pass to the description of the Hilbert space K cf. the first paragraph of this section. The structure of K is determined by a triple of selfadjoint operators â, b, β acting on K and having the following properties: 1. â > 0, ker â = ker b = {0} and â o b,

14 14 W. PUSZ AND S. L. WORONOWICZ 2. β is a unitary involution, β commutes with â and anticommutes with b. One of the possible choices is: K = L 2 R and â, b, β = a, b, β. possibility that is even more interesting: 3.7 â, b, β = b 1, e i /2 b 1 a, αβ, However there is another where α = ±1. The reader easily verifies that these operators possess the required properties. Zakrzewski relation â o b implies that the spectral measures of â and b are absolutely continuous with respect to the Lebesgue measure. Moreover Spâ = R + and Sp b = R. The latter fact follows from the relation β b = b β. Let 3.8 V = G b b, β β e i log ba log a. This is the basic object considered in this section. We shall prove Theorem V is a unitary operator and V MKK A, 2. A is generated by V MKK A. Proof. Let R = b b, ρ = β β, U = e i log ba log a, S = â 1 I and C = KK A. Then all the assumptions of Proposition 2.3 are satisfied. Clearly V = G R, ρ U MC and Statement 1 is proved. Let π RepA and B C H π. Then id π is a representation of C acting on K H π. The reader should notice that id πs = â 1 I is affiliated with KK B. Assume that id πv MKK B. By Statement 1 of Proposition 2.3, operators: id πr = b πb, id πρ NpR and id πu = e i log ba πlog a are affiliated with KK B. Using now Proposition A.1 of 19 we see that πb is affiliated with B. One can easily verify that β I commutes with ρ and anticommutes with R. Therefore ρ NpR β Iρ NpR β I = ρnpr Np R = ρr and applying id π to both sides we get id πρ NpR β Iid πρ NpR β I = id πρr = i β b πiβb The operators β I and id πρ NpR appearing on the left hand side are affiliated with KK B. Therefore i β b πiβb η KK B and using again Proposition A.1 of 19 we see that πiβb is affiliated with B. Moreover, remembering that e i log ba πlog a η KK B and using Proposition 2.5 we see that πlog a is affiliated with B. According to Statement 3 of Theorem 3.1, b, iβb and log a generate A. Therefore π MorA, B. We showed that id πv MKK B implies π MorA, B. It means that A is generated by V MKK A. Now we are able to formulate the main result of this section: Theorem 3.3. There exists MorA, A A such that id V = V 12 V 13 Proof. Let = π, k = 0, 1, 2,.... 2k α = ie iπ2 2, T = I e i /2 b 1 a b, τ = I β β αχ b b I < 0 + αχ b b I > 0. and 3.10 W = G T, τ e i I log b log a.

15 A NEW QUANTUM DEFORMATION OF ax + b GROUP. 15 Clearly W is a unitary operator acting on K L 2 R L 2 R. We shall prove that 3.11 V 12 V 13 = W V 12 W. To make our formulae shorter we set U = e i log ba log a, Z = e i log b log a. Using the relations â o b, â β = βâ and a o b one can easily verify that 3.12 U b IU = b a, U β IU = β I, 3.13 Za IZ = a a. With the above notation V = G b b, β β U and Using 3.12 we get and 3.14 V 12 V 13 = V 12 V 13 = G b b I, β β I U 12 G b I b, β I β U 13. U 12 G b I b, β I β = G b a b, β I β U 12 G b a b, β I β G b b I, β β I U12 U 13. Let us consider the first factor in We apply Theorem 1.6 with 3.15 Then T and τ are given by 3.9 and R = b a b, ρ = β I β, S = b b I, σ = β β I, G b a b, β I β G b b I, β β I = G T, τ G b b I, β β I G T, τ. Now 3.14 takes the form 3.16 V 12 V 13 = G T, τ G b b I, β β I G T, τ U 12 U 13. We shall move G T, τ to the end of the right hand side of this formula. computations and using 3.13 we obtain: It turns out that U 12 U 13 = e i log ba loga a = Z 23 U 12 Z log â loga a commutes with T, Performing simple 3.18 log â loga a commutes with τ. Indeed Zakrzewski relation a o b implies b 1 o a. Using the both relations we see that a a commutes with e i /2 b 1 a b. Therefore loga a commutes with e i /2 b 1 a b and log â loga a commutes with T = I e i /2 b 1 a b. Relation 3.17 is shown. To prove 3.18 we use Zakrzewski relations a o b and â o b. They show that a commutes with sgn b and â commutes with sgn b. Therefore log â loga a commutes with sgn b b I = sgn b sgn b I and 3.18 follows. Taking into account 3.17 and 3.18 we see that G T, τ commutes with U 12 U 13. Now relation 3.16 takes the form: 3.19 V 12 V 13 = G T, τ G b b I, β β I Z 23 U 12 Z 23 G T, τ. Finally b I and β I commute with log b log a. Therefore G b b I, β β I commutes with Z 23. Clearly G b b I, β β I U 12 = V 12 and W = G T, τ Z 23. Now 3.11 follows immediately from By the Zakrzewski relation â iλ bâ iλ = e λ b for all λ R. Multiplication by a strictly positive number does not change the sign of an operator. Using this fact one can easily show that τ commutes with â iλ I I. Consequently τ commutes with â I I. Since T = I e i /2 b 1 a b

16 16 W. PUSZ AND S. L. WORONOWICZ and I log b log a obviously commute with â I I, we conclude that W commutes with â I I. Now we are ready to prove the main statement. =. Let MorA, A A and id V = V 12 V 13. We go back to the notation used in the proof of Theorem 3.2. In particular C = KK A. For any c C we set: φ 1 c = id c, φ 2 c = W c IW. Then φ 1 and φ 2 are representations of C acting on the same Hilbert space K L 2 R L 2 R. One can easily verify that φ 1 â I = â I I = φ 2 â I. Formula 3.11 shows that φ 1 V = φ 2 V. In our notation cf. the beginning of the proof of Theorem 3.2, â I = S and V = G R, ρ U, where in particular R = b b. Statement 2 of Theorem 2.3 shows now that φ 1 R = φ 2 R. It means that b b = W b b I W. Taking into account 3.10 and using Proposition 1.2 we get: 3.20 We recall that τ = I β β b b = G T, τ b b IG T, τ = b a b + b b I. αχ b b I < 0 + αχ b b I > 0. Inspecting last two formulae we observe that b is the only operator appearing in the first leg position. We know that b is selfadjoint. Therefore replacing in both sides of 3.20 operator b by a real number λ we obtain a formula that should hold for almost all λ Sp b. For positive λ we get 3.21 b = a b + b I where 3.22 τ + = β β On the other hand for negative λ we have 3.23 b = where 3.24 τ = β β τ + αχb I < 0 + αχb I > 0. a b + b I τ αχb I > 0 + αχb I < 0. Clearly the two expressions for b must coincide. Let us notice that the operator I β commutes with τ +, τ and b I and anticommutes with a b. Therefore τ + = τ by Remark 1.3. Comparing 3.22 and 3.24 we get α = α. Remembering that α = i e iπ2 2 and 0 < < π 2 we conclude that = k = 0, 1, 2,.... π 2k+3 =. Assume that = π 2k+3 for some k = 0, 1, 2,.... Then formula 3.10 essentially simplifies. In this case α = 1 k, τ = 1 k I β β and W = W 23 = I W, where 3.25 W = G e i /2 b 1 a b, 1 k β β e i log b log a. Formula 3.11 takes the form 3.26 V 12 V 13 = W 23 V 12 W 23. For any c A we set 3.27 c = W c IW. Then is a representation of A acting on L 2 R L 2 R. We know that V MKK A. Formula 3.26 shows that id V = V 12 V 13. Clearly V 12, V 13 MKK A A. Therefore id V = V 12 V 13 MKK A A. Remembering that A is generated by V we conclude that MorA, A A. τ

17 π A NEW QUANTUM DEFORMATION OF ax + b GROUP. 17 Let = 2k+3 k = 0, 1, 2,.... Then formula 3.27 makes it possible to calculate c for any c A. The same holds for any c affiliated with A. We shall show that 3.28 a = a a, b = a b + b I 1 k β β, ib 2k+3 β = a 2k+3 ib 2k+3 β + ib 2k+3 β I sgnb b. Formula for a follows immediately from 3.13; the reader should notice that operators e i /2 b 1 a b and β β commute with a a. The formula for b was in fact shown in the proof of Theorem 3.3; in the present case τ + = τ = 1 k β β and the second formula of 3.28 coincides with 3.21 and with 3.23 as well. It remains to prove the third formula. We know that b commutes with ib 2k+3 β. Taking into account 3.25 we obtain 3.29 ib 2k+3 β = W ib 2k+3 β IW = G e i /2 b 1 a b, 1 k β β ib 2k+3 β I G e i /2 b 1 a b, 1 k β β. To compute the right hand side we use Proposition 1.4 with R = a b, S = b I, τ = sgnb b, ξ = 1 k β β, ρ = I isgn bβ, σ = isgn bβ I. Remembering that β 2 = I and β anticommutes with b and hence commutes with b one can easily check that these operators fulfill all assumption of Proposition 1.4. In this case we have T = e i /2 b 1 a b and τt = e i /2 b 1 a b. According to our assumption π = 2k + 3 is an odd positive integer. Therefore σs π = isgn bβ I b 2k+3 I = ib 2k+3 β I, ρr π = I isgn bβ a 2k+3 b 2k+3 = a 2k+3 ib 2k+3 β and formula 1.11 takes the form ib 2k+3 β I + a 2k+3 ib 2k+3 β 3.30 sgnb b = G e i /2 b 1 a b, 1 k β β ib 2k+3 β I G e i /2 b 1 a b, 1 k β β. Comparing 3.29 with 3.30 we get the last formula of This formula appeared without proof in 19. Remark 3.4. Let s S 1 be a number of modulus 1. Replacing in the above computations σ = isgn bβ I and ρ = isgn bβ I by σ = s sgn b β I and ρ = s sgn b β I respectively, one can prove that 3.31 s sgn b b 2k+3 β = a 2k+3 s sgn b b 2k+3 β + s sgn b b 2k+3 β I. sgnb b If s = i then s sgn b = i sgn b and 3.31 reduces to the previous formula. For s = 1 we get 3.32 b 2k+3 β = a 2k+3 b 2k+3 β + b 2k+3 β I sgnb b. Assume now that K = L 2 R and that the operators â, b, β are given by 3.7. Then operator 3.8 coincides with 3.25: V = W. Relation 3.26 takes the form: W 23 W 12 = W 12 W 13 W 23. This is the famous pentagon equation of Baaj and Skandalis 2. It means that W is a multiplicative unitary. It is known that W is modular 8. This property enables us to introduce unitary antipode, scaling group and Haar weight see 8, 16, 19, 10, 20 for details. In 19 we discussed the cyclic group of four elements acting on a quantum ax + b group. In fact this action may be extended to an action of S 1. At the beginning we set no condition for R. For any s S 1 and any closed operator c we set φ s c = w s c w s,