Modular properties of type I locally compact quantum groups
aa r X i v : . [ m a t h . QA ] A ug Modular properties of type I locally compact quantumgroups
Jacek Krajczok ∗ Institute of Mathematics, Polish Academy of Sciences
Abstract
The following paper is devoted to the study of type I locally compact quantumgroups. We show how various operators related to the modular theory of the Haarintegrals on G and b G act on the level of direct integrals. Using these results wederive a web of implications between properties such as unimodularity or tracialityof the Haar integrals. We also study in detail two examples: discrete quantum group \ SU q (2) and the quantum az + b group. A remarkable feature of the theory of compact quantum groups introduced by Woronowicz([22, 23]) is the fact that the Haar integral need not be tracial (in such case one says that acompact quantum group G is not of Kac type). Whether G is of Kac type or not, is relatedto a number of other properties. To name a few, the Haar integral of G is tracial if, andonly if its scaling group is trivial and this happens if, and only if the dual discrete quantumgroup b G is unimodular (equivalently has tracial integrals). In fact, behind all these objectsand properties stands a family ( ρ α ) α ∈ Irr( G ) of positive invertible operators (see [13]) and G is of Kac type if, and only if ρ α = H α for all α ∈ Irr( G ).A theory of locally compact quantum groups was proposed by Kustermans and Vaes ([11]).As general quantum group can be non-unimodular, each quantum group G has two Haarintegrals: left ϕ and right ψ . It is still possible that these are non-tracial, however nowthe situation is more complicated and the above simple equivalences from the world ofcompact quantum groups are no longer valid.An intermediate step between the theory of compact and general locally compact quantumgroups is formed by the so called type I locally compact quantum groups. Roughly speak-ing, similarly to the classical (i.e. not quantum) setting, these are quantum groups withtype I universal group C ∗ -algebras. Their study was initiated in the doctoral dissertationof Desmedt [5] where he has constructed a Plancherel measure on Irr( G ) and described its ∗ Email address: [email protected] D π ) π ∈ Irr( G ) and ( E π ) π ∈ Irr( G ) . They can be thought of as replacements ofthe operators ρ α from the compact theory; for example, the Haar integrals of b G are tracialif, and only if almost all operators D π , E π are multiples of the identity. One of the mainresults of our paper is a theorem which describes a relation between various properties of G and b G (unimodularity, traciality of the Haar integrals, trivial scaling group etc.) andproperties of operators D π , E π ( π ∈ Irr( G )) – this is accomplished in Section 6.In the next section we introduce objects used in the paper and set up the notation. Section3 is devoted to introducing a notion of matrix coefficients (in type I case) and recallingresults of Desmedt ([5]) and Caspers ([3]) which are used later in the paper. In Section4 we describe the polar decomposition of the map T ′ : Λ ψ ( x ) Λ ϕ ( x ∗ ) coming from theTomita-Takesaki theory and as a corollary we get an important relation between unitaryoperators Q L , Q R from the Desmedt’s theorem. In Section 5 we show how various opera-tors act on the level of direct integrals. We remark that a formula for ∇ it b ϕ from Theorem5.4 was recently used in [8] to deduce that the Toeplitz algebra is not an algebra of con-tinuous functions on a compact quantum group. Finally, in Section 7 we describe twointeresting examples of type I locally compact quantum groups: discrete group \ SU q (2) andthe quantum ” az + b ” group. Throughout the paper, G will be a locally compact quantum group in the senseof Kustermans and Vaes. W refer the reader to papers [11, 20] for an introduction tothe subject, here we will recall only necessary facts. Quantum group G comes togetherwith a number of objects: first of all we have a von Neumann algebra L ∞ ( G ), a normalunital ⋆ -homomorphism ∆ G : L ∞ ( G ) → L ∞ ( G ) ¯ ⊗ L ∞ ( G ) called comultiplication and twon.s.f. weights on L ∞ ( G ): ϕ and ψ . They are called respectively the left and the right Haarintegral as they satisfy certain invariance conditions. We will write Λ ϕ , Λ ψ for the GNSmaps. The GNS Hilbert spaces H ϕ , H ψ can be identified and will be denoted by L ( G ).We will write ( σ ϕt ) t ∈ R , ∇ ϕ , J ϕ for the group of modular automorphisms associated withthe weight ϕ , the modular operator and the modular conjugation – an analogousnotation will be used for other weights. The predual of L ∞ ( G ) will be denoted by L ( G ).With every locally compact quantum group G one can associate its dual b G . The objectsassociated with b G will be decorated with hats. The Hilbert spaces L ( G ), L ( b G ) can(and will) be identified. We will use a C ∗ -algebra C ( G ) ⊆ B(L ( G )). It is a σ -wot dense subalgebra of L ∞ ( G ). An important role in the theory plays the Kac-Takesakioperator W ∈ M(C ( G ) ⊗ C ( b G )). It is a unitary operator characterized by the property(( ω ⊗ id)W ∗ )Λ ϕ ( x ) = Λ ϕ (( ω ⊗ id)∆ G ( x )) ( ω ∈ L ( G ) , x ∈ N ϕ ). We note that the right legof W generates C ( b G ) – this means that the map λ : L ( G ) ∋ ω ( ω ⊗ id)W ∈ C ( b G )satisfies λ (L ( G )) = C ( b G ). There is also a unitary V ∈ L ∞ ( b G ) ′ ¯ ⊗ L ∞ ( G ) related to the Symbol ⊗ stands for the minimal tensor product of C ∗ -algebras or the tensor product of Hilbert spaces. ψ . With G one can associate yet another C ∗ -algebra, C u ( G ) called theuniversal version of C ( G ). It is related to C ( G ) via so called reducing morphism Λ G : C u ( G ) → C ( G ). We remark that C u ( b G ) plays a role of the full group C ∗ -algebraand its representations are in bijection with unitary representations of G (see [10, 16]).To be more precise, there exists a unitary operator W ∈ M(C ( G ) ⊗ C u ( b G )) such thatevery unitary representation of G on a Hilbert space H π is of the form U π = (id ⊗ π ) W for a nondegenerate representation π : C u ( b G ) → B( H π ). This correspondence preservesirreducibility – consequently the spectrum of C u ( b G ) will be denoted by Irr( G ).Besides the groups of modular automorphisms ( σ ϕt ) t ∈ R , ( σ ψt ) t ∈ R there is also a third groupof automorphisms of L ∞ ( G ) – ( τ t ) t ∈ R , called the scaling group . It is implemented by astrictly positive selfadjoint operator P : τ t ( x ) = P it xP − it ( x ∈ L ∞ ( G ) , t ∈ R ). We remarkthat P is selfdual: we have ˆ P = P . The Haar integrals are relatively invariant underthe scaling group: we have ϕ ◦ τ t = ν − t ϕ, ψ ◦ τ t = ν − t ψ ( t ∈ R ) for a number ν > scaling constant . The scaling constant relates the modular conjugations for ϕ and ψ : we have J ψ = ν i J ϕ . An important role in our paper is played by the so called modular element δ . It is a strictly positive selfadjoint operator affiliated with L ∞ ( G )which apppears as (a part of) the Radon–Nikodym derivative between ψ and ϕ . Thereis a plethora of formulas which relates the above objects. Let us end this part of theintroduction with a collection of them – we will use it a lot throught the paper: J b ϕ J ϕ = ν i J ϕ J b ϕ , ∇ itψ = J b ϕ ∇ − itϕ J b ϕ , J b ϕ δ it = δ it J b ϕ , ∇ itψ = ˆ δ − it P − it ∇ isϕ δ it = ν ist δ it ∇ isϕ , ∇ isψ δ it = ν ist δ it ∇ isψ , J ϕ P it = P it J ϕ , P it δ is = δ is P it P it ∇ isϕ = ∇ isϕ P it , P it ∇ isψ = ∇ isψ P it , (2.1)where t and s are arbitrary reals numbers. The above properties belong to the standardtheory of locally compact quantum groups, their proofs can be found in [10, 11, 20].Throught the paper, we will extensively use the theory of direct integrals – we refer thereader to [6, 12] for basic notions and properties. Let us mention here only that if ( H x ) x ∈ X is a measurable field o Hilbert spaces, then R ⊕ X H x d µ ( x ) is a Hilbert space which consists of(classes of) measurable vector fields ξ = ( ξ x ) x ∈ X satisfying R X k ξ x k d µ ( x ) < + ∞ . Usuallyone also writes ξ = R ⊕ X ξ x d µ ( x ). For two closed operators A and B , the symbol A ◦ B willstand for the operator given by A ◦ B ( ξ ) = A ( B ( ξ )) on the domain Dom( A ◦ B ) = { ξ ∈ Dom( B ) | Bξ ∈ Dom( A ) } . Whenever A ◦ B is closable, we will denote its closure by AB .The complex conjugate of a Hilbert space H will be denoted by H . For an operator A on H , A T will be an operator on H given by A T ξ = A ∗ ξ ( ξ ∈ H ). If π is a representation of G on H π then we associate with it a representation π c = · T ◦ π ◦ b R u on H π , where b R u is theunitary antipode on C u ( b G ) ([10, 16]). All scalar products are linear on the right. This means that ( H x ) x ∈ X comes together with a choice of a fundamental sequence { ( ξ nx ) x ∈ X | n ∈ N } and for each n ∈ N , the function X ∋ x
7→ h ξ nx | ξ x i ∈ C is measurable. We will neglect mentioning thefundamental sequence and simply say that ( H x ) x ∈ X is a measurable field of Hilbert spaces. Preliminaries
Let us introduce two notions: we say that G is second countable if C u ( b G ) separable and type I if C u ( b G ) is of type I. Our work is based on the work of Desmedt [5] and Caspersand Koelink [3, 4]. First of all, we will use the fundamental result of Desmedt which statesexistence of the Plancherel measure and its properties (see also [7] and discussion therein).We recall only parts that will be used in the paper. Theorem 3.1.
Let G be a second countable, type I locally compact quantum group. Thereexists a standard measure µ on Irr( G ) , a measurable field of Hilbert spaces ( H π ) π ∈ Irr( G ) ,measurable field of representations , measurable field of strictly positive self-adjoint opera-tors ( D π ) π ∈ Irr( G ) and a unitary operator Q L : L ( G ) → R ⊕ Irr( G ) HS( H π ) d µ ( π ) such that:1) For all α ∈ L ( G ) such that λ ( α ) ∈ N b ϕ and µ -almost every π ∈ Irr( G ) the operator ( α ⊗ id)( U π ) ◦ D − π is bounded and its closure ( α ⊗ id)( U π ) D − π is Hilbert-Schmidt.2) The operator Q L is the isometric extension of Λ b ϕ ( λ (L ( G )) ∩ N b ϕ ) ∋ Λ b ϕ ( λ ( α )) Z ⊕ Irr( G ) ( α ⊗ id)( U π ) D − π d µ ( π ) ∈ Z ⊕ Irr( G ) HS( H π ) d µ ( π ) ,
3) The operator Q L satisfies the following equations: Q L ( ω ⊗ id)W = (cid:0)Z ⊕ Irr( G ) ( ω ⊗ id) U π ⊗ H π d µ ( π ) (cid:1) Q L and Q L ( ω ⊗ id) χ (V) = (cid:0)Z ⊕ Irr( G ) H π ⊗ π c (( ω ⊗ id) W ) d µ ( π ) (cid:1) Q L for every ω ∈ L ( G ) .4) Haar integrals on b G are tracial if and only if almost all D π are multiples of theidentity.5) The operator Q L transforms L ∞ ( b G ) ∩ L ∞ ( b G ) ′ into diagonalisable operators.6) We can assume that ( H π ) π ∈ Irr( G ) is the canonical measurable field of Hilbert spaces. We have also the right version of the above theorem. This condition is equivalent to number of other separability assumptions, see [7, Lemma 14.6]. Wenote that these conditions are satisfied for G if and only they are satisfied for b G . We use the same symbol π for a class of representations and its representative chosen according to afixed measurable field of representations. heorem 3.2. Let G be a second countable, type I locally compact quantum group. Thereexists a standard measure µ R on Irr( G ) a measurable field of Hilbert space ( K π ) π ∈ Irr( G ) ,measurable field of representations, measurable field of strictly positive self-adjoint operators ( E π ) π ∈ Irr( G ) and a unitary operator Q R : L ( G ) → R ⊕ Irr( G ) HS( K π ) d µ R ( π ) such that:1) For all α ∈ L ( G ) such that λ ( α ) ∈ N b ψ and µ R -almost every π ∈ Irr( G ) the operator ( α ⊗ id)( U π ) ◦ E − π is bounded and its closure ( α ⊗ id)( U π ) E − π is Hilbert-Schmidt.2) The operator Q R is the isometric extension of J b ϕ J ϕ Λ b ψ ( λ (L ( G )) ∩ N b ψ ) ∋ J b ϕ J ϕ Λ b ψ ( λ ( α )) Z ⊕ Irr( G ) ( α ⊗ id)( U π ) E − π d µ R ( π ) ∈ Z ⊕ Irr( G ) HS( K π ) d µ R ( π ) ,
3) The operator Q R satisfies the following equations: Q R J b ϕ J ϕ ( ω ⊗ id)W = (cid:0)Z ⊕ Irr( G ) ( ω ⊗ id) U π ⊗ H π d µ R ( π ) (cid:1) Q R J b ϕ J ϕ and Q R J b ϕ J ϕ ( ω ⊗ id) χ (V) = (cid:0)Z ⊕ Irr( G ) H π ⊗ π c (( ω ⊗ id) W ) d µ R ( π ) (cid:1) Q R J b ϕ J ϕ for every ω ∈ L ( G ) .4) Haar inegrals on b G are tracial if and only if almost all E π are multiples of the identity.5) The operator Q R transforms L ∞ ( b G ) ∩ L ∞ ( b G ) ′ into diagonalisable operators.6) We can choose µ R = µ and K π = H π (and the same field of representations as inTheorem 3.1). From now on, let G be a second countable, type I locally compact quantum group andchoose all the objects provided by theorems 3.1, 3.2. The last point of the above theoremallows us to assume µ R = µ and K π = H π . Let us introduce two strictly positive, selfadjointoperators D = R ⊕ Irr( G ) D π d µ ( π ) and E = R ⊕ Irr( G ) E π d µ ( π ). We will use plenty of times thefollowing easily derived property: Proposition 3.3.
Define an antiunitary operator
Σ = R ⊕ Irr( G ) J H π d µ ( π ) , where J H π : H π ⊗ H π ∋ ξ ⊗ η η ⊗ ξ ∈ H π ⊗ H π ( π ∈ Irr( G )) . We have ν i J b ψ = J b ϕ = Q ∗ L Σ Q L = Q ∗ R Σ Q R . roof. Let b ϕ u be the left Haar weight on the universal C ∗ -algebra C u ( b G ). Its GNS con-struction is (L ( G ) , Λ b G , Λ b ϕ ◦ Λ b G ) (see [10]), hence J b ϕ is the modular conjugation for b ϕ u . Itis transformed to Σ by Q L – it is a part of the Desmedt’s result.Similarly, Q R, transforms J b ψ to Σ: J b ψ = Q ∗ R, Σ Q R, . Operator Q R is defined as Q R = Q R, J ϕ J b ϕ (see [7, Theorem 3.4]). Consequently, we get J b ψ = J ϕ J b ϕ Q ∗ R Σ Q R J b ϕ J ϕ . Using thecommutation relation J b ϕ J ϕ = ν i J ϕ J b ϕ (see equation (2.1)) and formula J b ψ = ν − i J b ϕ (thescaling constant of b G is ˆ ν = ν − ) we arrive at Q ∗ R Σ Q R = J b ϕ J ϕ ( ν − i J b ϕ ) J ϕ J b ϕ = ν − i ν i J ϕ J b ϕ J b ϕ J ϕ J b ϕ = J b ϕ . Let us note in the next proposition how Q L , Q R transform L ∞ ( b G ) and its comutant. Proposition 3.4.
We have the following equalities of von Neumann algebras: Q L L ∞ ( b G ) Q ∗ L = Z ⊕ Irr( G ) B( H π ) ⊗ H π d µ ( π ) , Q L L ∞ ( b G ) ′ Q ∗ L = Z ⊕ Irr( G ) H π ⊗ B( H π ) d µ ( π ) Q R L ∞ ( b G ) Q ∗ R = Z ⊕ Irr( G ) H π ⊗ B( H π ) d µ ( π ) , Q R L ∞ ( b G ) ′ Q ∗ R = Z ⊕ Irr( G ) B( H π ) ⊗ H π d µ ( π )The first part of the above result is a result of Desmedt. The second one can be derivedas in the proof of Proposition 3.3, using equation Q R = Q R, J ϕ J b ϕ . Let us now defineanalogs of the matrix coefficients U αi,j used in the theory of compact quantum groups.Elements of this form were already considered in [3]. Definition 3.5.
For ξ, η ∈ R ⊕ Irr( G ) H π d µ ( π ) we define elements of L ∞ ( G ): M Lξ,η = Z Irr( G ) (id ⊗ ω ξ π ,η π )( U π ∗ ) d µ ( π ) , M Rξ,η = Z Irr( G ) (id ⊗ ω ξ π ,η π )( U π ) d µ ( π ) . The above elements will be referred to as left (resp. right) matrix coefficients.Note that the above (weak) integrals converge in σ -wot and we have ( M Lξ,η ) ∗ = M Rη,ξ .Our further reasoning is based on results derived by Caspers and Koelink in [3, 4]. Weremark that one needs to be careful when taking equations from these papers as thereis a difference in convention: we prefer to use inner products linear on the right andfunctionals ω ξ,η defined accordingly. That is why we choose to state explicitly used resultswith necessary changes, which we do in this section.First, we can transport a left (resp. right) matrix coefficient via Q L (resp. Q R ). Thefollowing is a reformulation of [4, Lemma 3.7, Lemma 3.9].6 emma 3.6.
1) If ξ, η ∈ R ⊕ Irr( G ) H π d µ ( π ) , ξ ∈ Dom( D ) and the vector field ( η π ⊗ D π ξ π ) π ∈ Irr( G ) is squareintegrable, then M Lξ,η ∈ N ϕ and Q L Λ ϕ ( M Lξ,η ) = R ⊕ Irr( G ) η π ⊗ D π ξ π d µ ( π ) .2) If ξ, η ∈ R ⊕ Irr( G ) H π d µ ( π ) , ξ ∈ Dom( E ) and the vector field ( η π ⊗ E π ξ π ) π ∈ Irr( G ) is squareintegrable, then M Rξ,η ∈ N ψ and Q R Λ ψ ( M Rξ,η ) = R ⊕ Irr( G ) η π ⊗ E π ξ π d µ ( π ) . Using the above result and the fact that Q L , Q R are unitary, one can easily derive thefollowing density results: Lemma 3.7.
1) Set { Λ ϕ ( M Lξ,η ) } , where ξ, η run over vectors in R ⊕ Irr( G ) H π d µ ( π ) such that ξ ∈ Dom( D ) and ( η π ⊗ D π ξ π ) π ∈ Irr( G ) is square integrable, is lineary dense in L ( G ) .2) Set { Λ ψ ( M Rξ,η ) } , where ξ, η run over vectors in R ⊕ Irr( G ) H π d µ ( π ) such that ξ ∈ Dom( E ) and ( η π ⊗ E π ξ π ) π ∈ Irr( G ) is square integrable, is lineary dense in L ( G ) . Consider an antilinear map Λ ψ ( N ψ ∩ N ϕ ∗ ) ∋ Λ ψ ( x ) Λ ϕ ( x ∗ ) ∈ L ( G ) (3.1)and define T ′ to be its closure. Let T ′ = J ′ ∇ ′ be the polar decomposition of T ′ . It iswell known that J ′ is antiunitary and ∇ ′ is strictly positive and selfadjoint. In the nextsection we will describe these operators, for now let us recall how they look on the level ofdirect integrals. Proposition 3.8.
We have Q L J ′ Q ∗ R = Σ and Q R ∇ ′ Q ∗ R = R ⊕ Irr( G ) D π ⊗ ( E − π ) T d µ ( π ) . The above proposition is a combination of [4, Proposition 4.4, Proposition 4.5, Theorem4.6]. We finish this section with formulas expressing the action of modular automorphismgroups on the matrix coefficients.
Proposition 3.9.
For each ξ, η ∈ R ⊕ Irr( G ) H π d µ ( π ) , t ∈ R the following holds: σ ψt ( M Rξ,η ) = ν it δ it M RE it ξ,D it η , σ ϕt ( M Rξ,η ) = ν it M RE it ξ,D it η δ it ,σ ψt ( M Lξ,η ) = ν − it M LD it ξ,E it η δ − it , σ ϕt ( M Lξ,η ) = ν − it δ − it M LD it ξ,E it η . The formulas expressing the action of σ ϕ , σ ψ on M Rξ,η are stated in [3, Remark 2.2.11].The other two follow by taking the adjoint. We note that they can be derived using theformula for ∇ ′ (Proposition 3.8) and equation ν it δ it = ∇ itψ ∇ ′− it (see [17, Equations (29),(30), page 112] and the proof of [20, Theorem 3.11]). This map appears during a construction of the Radon-Nikodym derivative between ψ and ϕ , see [17]. Relation between Q L and Q R In this section we will describe the polar decompostion of the closed operator T ′ : Λ ψ ( x ) Λ ϕ ( x ∗ ) (see equation (3.1)), namely we will derive a equation T ′ = ( ν i J ϕ )( J ϕ ν i ∇ − ϕ δ − J ϕ ).As a corollary we get an important relation between Q L and Q R . Before we do that, letus justify through a formal calculation, why the above formula for T ′ should hold: T ′ Λ ψ ( x ) = Λ ϕ ( x ∗ ) = J ϕ ∇ ϕ Λ ϕ ( x ) = J ϕ ∇ ϕ J ϕ σ ϕi/ ( δ − ) ∗ J ϕ Λ ϕ ( xδ )= ∇ − ϕ ( ν − i δ − ) ∗ J ϕ Λ ψ ( x ) = ( ν i J ϕ )( J ϕ ν i ∇ − ϕ δ − J ϕ )Λ ψ ( x ) . (4.1)We need to include the factor ν i due to the following lemma: Lemma 4.1.
For all s, t ∈ R operators ∇ sϕ ◦ δ t , δ t ◦ ∇ sϕ are closable. We have equality ν ist ∇ sϕ δ t = ν − ist δ t ∇ sϕ of strictly positive, selfadjoint operators, moreover ( ν ist ∇ sϕ δ t ) ir = ν − ist r ∇ isrϕ δ itr = ν ist r δ itr ∇ isrϕ ( r ∈ R ) . The above result is a consequence of the commutation relation ∇ isϕ δ it = ν ist δ it ∇ isϕ ( s, t ∈ R ). Indeed, it follows that operators ∇ sϕ , δ t satisfy the Weyl relation. Then Lemma 4.1follows from [24, Example 3.1, Theorem 3.1]. The next lemma describes the action of theunbounded operator δ t . Lemma 4.2.
1) Let t ∈ R , x ∈ N ϕ be such that x ◦ δ t is closable and xδ t ∈ N ϕ . Then J ϕ Λ ϕ ( x ) ∈ Dom( δ t ) and ν it J ϕ δ t J ϕ Λ ϕ ( x ) = Λ ϕ ( xδ t ) .2) Let t ∈ R , x ∈ N ψ be such that x ◦ δ t is closable and xδ t ∈ N ψ . Then J ϕ Λ ψ ( x ) ∈ Dom( δ t ) and ν it J ϕ δ t J ϕ Λ ψ ( x ) = Λ ψ ( xδ t ) .Proof. We prove only the first assertion, the second one can be derived analogously. Take x ∈ N ϕ , t ∈ R which satisfy conditions of the lemma and define x n = q nπ Z R e − np xδ ip d p ∈ L ∞ ( G ) ( n ∈ N )(the above weak integral converges in σ - wot ). Operator x n ◦ δ t is closable and we have x n δ t = q nπ Z R e − np ( xδ t ) δ ip d p = q nπ Z R e − n ( p + it ) xδ ip d p. (4.2)Clearly x n , x n δ t ∈ N ϕ and due to the Hille’s theoremΛ ϕ ( x n ) = q nπ Z R e − np Λ ϕ ( xδ ip ) d p = q nπ J ϕ Z R e − np ν − p δ − ip J ϕ Λ ϕ ( x ) d p, ϕ ( x n δ t ) = q nπ J ϕ Z R e − np ν − p δ − ip J ϕ Λ ϕ ( xδ t ) d p = q nπ J ϕ Z R e − n ( p − it ) ν − p δ − ip J ϕ Λ ϕ ( x ) d p. Consequently, Λ ϕ ( x n ) −−−→ n →∞ Λ ϕ ( x ) and Λ ϕ ( x n δ t ) −−−→ n →∞ Λ ϕ ( xδ t ). For each r ∈ R we have δ ir J ϕ Λ ϕ ( x n ) = q nπ Z R e − np ν − p δ − i ( p − r ) J ϕ Λ ϕ ( x ) d p = f n ( r ) , where f n is an entire function f n : C ∋ z q nπ Z R e − n ( p + z ) ν − p + z δ − ip J ϕ Λ ϕ ( x ) d p ∈ L ( G ) . From the above follows that J ϕ Λ ϕ ( x n ) ∈ Dom( δ z ) for all z ∈ C and δ z J ϕ Λ ϕ ( x n ) = f n ( − iz ).Let us show that the sequence ( δ t J ϕ Λ ϕ ( x n )) n ∈ N converges to ν it J ϕ Λ ϕ ( xδ t ): δ t J ϕ Λ ϕ ( x n ) = f n ( − it ) = q nπ Z R e − n ( p − it ) ν − p − it δ − ip J ϕ Λ ϕ ( x ) d p = ν it q nπ Z R e − n ( p − it ) ν − p δ − ip J ϕ Λ ϕ ( x ) d p = ν it J ϕ Λ ϕ ( x n δ t ) −−−→ n →∞ ν it J ϕ Λ ϕ ( xδ t ) . Norm closedness of δ t implies J ϕ Λ ϕ ( x ) ∈ Dom( δ t ) and δ t J ϕ Λ ϕ ( x ) = ν it J ϕ Λ ϕ ( xδ t ).In what follows we introduce a space D of sufficiently nice vectors on which calculation(4.1) is justified and which forms a core for the operators involved. First, define δ n,z = q nπ Z R e − nt ν zt δ it d t ∈ L ∞ ( G ) ( n ∈ N , z ∈ C ) . Note that for each z ∈ C , the sequence ( δ n,z ) n ∈ N is bounded and converges to in sot .Next, for x ∈ N ϕ ∩ N ϕ ∗ ∩ N ψ ∩ N ψ ∗ , k ∈ N , A = ( A , A ) ∈ C define x k,A = kπ Z R Z R e − k ( t − A ) − k ( s − A ) σ ϕt ◦ σ ψs ( x ) d t d s ∈ L ∞ ( G ) . Finally, define a subspace D via D = span { Λ ψ ( δ n,z x k,A δ m,w ) | x, x ∗ ∈ N ϕ ∩ N ψ , n, m, k ∈ N , A ∈ C , z, w ∈ C } . Lemma 4.3. • The subspace D is a core for ∇ − ϕ . Moreover, for ξ ∈ Dom( ∇ − ϕ ) we can find asequence ( ξ p ) p ∈ N in { Λ ψ ( x k,A δ m,w ) | x, x ∗ ∈ N ϕ ∩ N ψ , m, k ∈ N , A ∈ C , w ∈ C } such that ξ p −−−→ p →∞ ξ and ∇ − ϕ ξ p −−−→ p →∞ ∇ − ϕ ξ Each element of D can be written as Λ ψ ( x ) for some x ∈ L ∞ ( G ) such that x, x ∗ ∈ N ϕ ∩ N ψ ∩ T z ∈ C Dom( σ ψz ) . Moreover, σ ψz ( x ) ∈ N ψ and Λ ψ ( x ∗ ) , Λ ψ ( σ ψz ( x )) ∈ D .Next, Λ ψ ( x ) ∈ T z ∈ C Dom( ∇ zψ ) and ∇ izψ Λ ψ ( x ) = Λ ψ ( σ ψz ( x )) . • For all z, w ∈ C , Λ ψ ( x ) ∈ D the operator δ z ◦ x ◦ δ w is closable and after closurebelongs to N ψ ∩ N ϕ . • We have J ϕ D = D . A proof of the above lemma requires only standard reasoning, hence will be skipped.In the next two lemmas we prove properties of D which allows us to derive the polardecomposition of T ′ . Lemma 4.4.
The subspace D is a core for ν − i J ϕ δ − J ϕ . We have ν − i J ϕ δ − J ϕ Λ ψ ( x ) = Λ ϕ ( x ) for all x ∈ N ϕ ∩ N ψ such that x ◦ δ − is closable and xδ − ∈ N ψ . Moreover, the operator ( J ϕ ∇ ϕ ) ◦ ( ν − i J ϕ δ − J ϕ ) = ν i ( ∇ − ϕ ◦ δ − ) J ϕ is closable and D is a core for its closure ν i ∇ − ϕ δ − J ϕ .Proof. It is clear that span S n ∈ N δ n, L ( G ) is a core for δ − . Take ξ = δ n, η ∈ Dom( δ − )for some n ∈ N and let ( η p ) p ∈ N be a sequence of vectors of the form Λ ψ ( x k,A,B δ m,w ) (seethe first point of the Lemma 4.3) converging to η . We have δ n, η p ∈ D , k ξ − δ n, η p k ≤ k η − η p k −−−→ p →∞ k δ − ξ − δ − δ n, η p k ≤ k δ − δ n, kk η − η p k −−−→ p →∞ , which shows that D is a core for δ − . Since D is invariant under J ϕ , it is also a core for ν − i J ϕ δ − J ϕ .Take x ∈ N ϕ ∩ N ψ such that x ◦ δ − is closable and xδ − ∈ N ψ . Lemma 4.2 gives us J ϕ Λ ψ ( x ) ∈ Dom( δ − ) and ν − i J ϕ δ − J ϕ Λ ψ ( x ) = Λ ψ ( xδ − ) = Λ ϕ ( x ) . Equality from the claim ( J ϕ ∇ ϕ ) ◦ ( ν − i J ϕ δ − J ϕ ) = ν i ( ∇ − ϕ ◦ δ − ) J ϕ is a straightforwardconsequence of the relation J ϕ ∇ ϕ = ∇ − ϕ J ϕ .To deduce the last assertion let us observe that Lemma 4.1 gives us an equality ν i/ ∇ − ϕ δ − = ν − i/ δ − ∇ − ϕ . It follows that the closure of ν − i/ δ − ◦ ∇ − ϕ is ∇ − ϕ δ − . Take ξ ∈ Dom( ν − i/ δ − ◦ ∇ − ϕ ). For each n ∈ N we have δ n, ξ ∈ Dom( ν − i/ δ − ◦ ∇ − ϕ ), δ n, ξ −−−→ n →∞ ξ (4.3)and ν − i/ δ − ◦ ∇ − ϕ ( δ n, ξ ) = ν − i/ σ ϕi/ ( δ n, ) δ − ◦ ∇ − ϕ ( ξ )= ν − i/ δ n, − / δ − ◦ ∇ − ϕ ( ξ ) −−−→ n →∞ ν − i/ δ − ◦ ∇ − ϕ ( ξ ) . (4.4)10s previously, since D is invariant for J ϕ , it is enough to check that D is a core for ∇ − ϕ δ − . Take ξ ∈ Dom( ∇ − ϕ δ − ). The above reasoning and equations (4.3), (4.4) showthat it is enough to take vector of the form ξ = δ n, η for η ∈ Dom( ν − i/ δ − ◦ ∇ − ϕ ) andsome n ∈ N . Let ( η p ) p ∈ N be a sequence of vectors of the form Λ ψ ( x k,A,B δ m,w ) such that η p −−−→ p →∞ η and ∇ − ϕ η p −−−→ p →∞ ∇ − ϕ η. We have δ n, η p ∈ D , δ n, η p −−−→ p →∞ δ n, η = ξ and k ν − i/ δ − ◦ ∇ − ϕ ( δ n, η − δ n, η p ) k = k δ n, − / δ − ◦ ∇ − ϕ ( η − η p ) k≤ k δ n, − / δ − kk∇ − ϕ ( η − η p ) k −−−→ p →∞ . Lemma 4.5.
The subspace D is a core for T ′ .Proof. Take x ∈ N ψ ∩ N ϕ ∗ and define x n as x n = nπ R R R R e − n ( r + p ) δ ip xδ ir d r d p ( n ∈ N ) . We have x n , x ∗ n ∈ N ϕ ∩ N ψ . Next, define x n,n = nπ R R R R e − n ( t + s ) σ ϕt ◦ σ ψs ( x n ) d s d t. We have δ n, x n,n δ n, ∈ N ψ ∩ N ϕ ∗ , Λ ψ ( δ n, x n,n δ n, ) ∈ D , Λ ψ ( δ n, x n,n δ n, ) −−−→ n →∞ Λ ψ ( x ) and T ′ Λ ψ ( δ n, x n,n δ n, ) = Λ ϕ ( δ n, x ∗ n,n δ n, ) −−−→ n →∞ Λ ϕ ( x ∗ ) = T ′ Λ ϕ ( x ) . Now we can derive the main results of this section.
Proposition 4.6.
We have ( J ϕ ∇ ϕ ) ◦ ( ν − i J ϕ δ − J ϕ ) = ν i ( ∇ − ϕ ◦ δ − ) J ϕ and after closure ν i ∇ − ϕ δ − J ϕ = T ′ . Proof.
The first equality was justified in Lemma 4.4. Take Λ ψ ( x ) ∈ D . Lemmas 4.3, 4.4justify the following calculation:( J ϕ ∇ ϕ ) ◦ ( ν − i J ϕ δ − J ϕ ) Λ ψ ( x ) = J ϕ ∇ ϕ Λ ϕ ( x ) = Λ ϕ ( x ∗ ) = T ′ Λ ψ ( x ) . In lemmas 4.4, 4.5 we have shown that D is a core for T ′ and ν i ∇ − ϕ δ − J ϕ , which shows T ′ = ν i ∇ − ϕ δ − J ϕ .The above result has a number of interesting corollaries. Corollary 4.7.
The polar decomposition of T ′ is T ′ = ( ν i J ϕ ) ( J ϕ ν i ∇ − ϕ δ − J ϕ ). Moreover,we have ( J ϕ ν i ∇ − ϕ δ − J ϕ ) it = ν i t J ϕ ∇ it/ ϕ δ it/ J ϕ ( t ∈ R ) . (4.5)11 roof. The first equality follows directly from Proposition 4.6. Let us justify that it isindeed the polar decomposition. First, it is clear that ν i J ϕ is antiunitary. Next, Lemma4.1 implies that ν i ∇ − ϕ δ − is selfadjoint and strictly positive. Consequently, the operator J ϕ ν i ∇ − ϕ δ − J ϕ has the same properties. Uniqueness of the polar decomposition gives usthe first claim. The second formula follows from Lemma 4.1:( J ϕ ν i ∇ − ϕ δ − J ϕ ) it = f ( ν i ∇ − ϕ δ − ) it = f (( ν i ∇ − ϕ δ − ) it )= J ϕ ( ν i ∇ − ϕ δ − ) − it J ϕ = J ϕ ν − i t ∇ it/ ϕ δ it/ J ϕ , where f : a J ϕ a ∗ J ϕ .Now we combine our polar decomposition of T ′ with the result of Caspers (Proposition3.8) and Proposition 3.3. Corollary 4.8.
We have Q L ν i J ϕ Q ∗ R = Σ and Q ∗ R Q L = Q ∗ L Q R = ν − i J b ϕ J ϕ .Formula Q ∗ R Q L = ν − i J b ϕ J ϕ is of great importance and will be used numerous timesthrought the paper. In this section we will derive several equations, which express important operators onL ( G ) via Q L , Q R as direct integrals. The first result of this type comes from the polardecomposition of T ′ . Proposition 5.1.
For all t ∈ R we have ∇ itψ δ − it = J ϕ ∇ itϕ δ it J ϕ = ν − i t Q ∗ R (cid:0)Z ⊕ Irr( G ) D itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q R ,J ϕ ∇ itψ δ − it J ϕ = ∇ itϕ δ it = ν i t Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ ( D − itπ ) T d µ ( π ) (cid:1) Q L , ∇ − itϕ δ − it = J ϕ ∇ − itψ δ it J ϕ = ν i t Q ∗ R (cid:0)Z ⊕ Irr( G ) E itπ ⊗ ( D − itπ ) T d µ ( π ) (cid:1) Q R ,J ϕ ∇ − itϕ δ − it J ϕ = ∇ − itψ δ it = ν − i t Q ∗ L (cid:0)Z ⊕ Irr( G ) D itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q L . Proof.
First, observe that we have ∇ itψ = J b ϕ ∇ − itϕ J b ϕ = δ it ( J ϕ δ it J ϕ ) ∇ itϕ (see [20, Theorem5.18] and equation (2.1)). It follows that ∇ itψ δ − it = ν − it δ − it ∇ itψ = ν − it J ϕ δ it ∇ itϕ J ϕ = J ϕ ∇ itϕ δ it J ϕ , J ϕ ∇ itϕ δ it J ϕ via directintegral of operators follows from equation (4.5) combined with Proposition 3.8. The secondequation can be found using already derived relation Q ∗ L Q R = ν − i J b ϕ J ϕ : J ϕ ∇ itϕ δ it J ϕ = ν − i t ν i J ϕ J b ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) D itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q L ν − i J b ϕ J ϕ = ν − i t J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ ( D − itπ ) T d µ ( π ) (cid:1) Q L J ϕ , which implies ∇ itϕ δ it = ν i t Q ∗ L ( R ⊕ Irr( G ) E itπ ⊗ ( D − itπ ) T d µ ( π )) Q L . The last two equationscomes from applying the operation J b ϕ · J b ϕ to both sides of already derived formulas.Let us now derive an interesting corollary of these results. Corollary 5.2.
There exists a unique measurable function f : Irr( G ) → R > such that J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) D itπ ⊗ H π d µ ( π ) (cid:1) ∗ Q R J ϕ = Q ∗ R (cid:0)Z ⊕ Irr( G ) f ( π ) it E itπ ⊗ H π d µ ( π ) (cid:1) Q R ,J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) H π ⊗ ( E itπ ) T d µ ( π ) (cid:1) ∗ Q R J ϕ = Q ∗ R (cid:0)Z ⊕ Irr( G ) f ( π ) it H π ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q R ,J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) H π ⊗ ( D itπ ) T d µ ( π ) (cid:1) ∗ Q L J ϕ = Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it H π ⊗ ( E itπ ) T d µ ( π ) (cid:1) Q L ,J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ H π d µ ( π ) (cid:1) ∗ Q L J ϕ = Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it D itπ ⊗ H π d µ ( π ) (cid:1) Q L for all t ∈ R .We note that the function f might depend on the choice of a measure µ . Proof.
Fix t ∈ R . The first and the third row in Proposition 5.1 implies J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) D itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q R J ϕ = Q ∗ R (cid:0)Z ⊕ Irr( G ) E − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q R . Since J ϕ L ∞ ( b G ) J ϕ = L ∞ ( b G ), J ϕ L ∞ ( b G ) ′ J ϕ = L ∞ ( b G ) ′ and the center of R ⊕ Irr( G ) H π ⊗ B( H π ) d µ ( π )is R ⊕ Irr( G ) C HS( H π ) d µ ( π ), Proposition 3.4 implies that there exists a measurable function f t : Irr( G ) → T such that J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) D itπ ⊗ H π d µ ( π ) (cid:1) Q R J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) E itπ ⊗ H π d µ ( π ) (cid:1) Q R = J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) H π ⊗ ( E itπ ) T d µ ( π ) (cid:1) Q R J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) H π ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q R = Z ⊕ Irr( G ) f t ( π ) HS( H π ) d µ ( π ) . J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) D itπ ⊗ H π d µ ( π ) (cid:1) ∗ Q R J ϕ = Q ∗ R (cid:0)Z ⊕ Irr( G ) f t ( π ) E itπ ⊗ H π d µ ( π ) (cid:1) Q R (5.1)and J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) H π ⊗ ( E itπ ) T d µ ( π ) (cid:1) ∗ Q R J ϕ = Q ∗ R (cid:0)Z ⊕ Irr( G ) f t ( π ) H π ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q R . (5.2)Equation (5.1) together with relation Q ∗ L Q R = ν − i J b ϕ J ϕ (Corollary 4.8) gives us J ϕ J ϕ J b ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) D itπ ⊗ H π d µ ( π ) (cid:1) ∗ Q L J b ϕ J ϕ J ϕ = J ϕ J b ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) f t ( π ) E itπ ⊗ H π d µ ( π ) (cid:1) Q L J b ϕ J ϕ , hence also (thanks to Q L J b ϕ Q ∗ L = Σ, see Proposition 3.3) Q ∗ L (cid:0)Z ⊕ Irr( G ) H π ⊗ ( D − itπ ) T d µ ( π ) (cid:1) ∗ Q L = J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) f t ( π ) H π ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q L J ϕ . The last equation can be derived from equation (5.2) in a similar manner. Clearly we have f t ( π ) = f ( π ) it for a measurable function f : Irr( G ) → R > .In the second part of this section, we will transport operators ∇ it b ϕ , ∇ it b ψ , ˆ δ it ( t ∈ R ) to R ⊕ Irr( G ) HS( H π ) d µ ( π ). We start with a formula expressing the action of ( τ t ) t ∈ R on matrixcoefficients. Lemma 5.3.
For ξ, η ∈ R ⊕ Irr( G ) H π d µ ( π ) and t ∈ R we have τ t ( M Lξ,η ) = ν − it δ − it Z Irr( G ) (id ⊗ ω D itπ π (ˆ δ − itu ) ξ π ,E itπ η π )( U π ∗ ) d µ ( π )= ν − it Z Irr( G ) (id ⊗ ω D − itπ ξ π ,E − itπ π (ˆ δ − itu ) η π )( U π ∗ ) d µ ( π ) δ it ,τ t ( M Rξ,η ) = ν it Z Irr( G ) (id ⊗ ω E itπ ξ π ,D itπ π (ˆ δ − itu ) η π )( U π ) d µ ( π ) δ it = ν it δ − it Z Irr( G ) (id ⊗ ω E − itπ π (ˆ δ − itu ) ξ π ,D − itπ η π )( U π ) d µ ( π ) . Later on in Proposition 5.5 we will get simpler expressions for this action (once we findout what π (ˆ δ itu ) is). 14 roof. The proof is based on several facts from the theory of locally compact quantumgroups. First of all, we know that ˆ δ it = P − it ∇ − itψ (equation (2.1)). Next, [20, Lemma 5.14]gives us( σ ϕt ⊗ id)W = ( ⊗ P − it )W( ⊗ ∇ − itψ ) , ( σ ψt ⊗ id)W = ( ⊗ ∇ − itψ )W( ⊗ P − it ) , and ( τ t ⊗ id)W = (id ⊗ ˆ τ − t )W. We note also that ˆ δ it ∈ M(C ( b G )), ˆ δ itu ∈ M(C u ( b G )) andΛ b G (ˆ δ itu ) = ˆ δ it ([10]). Fix t ∈ R , a representation π ∈ Irr( G ) which factorises through C ( G )(i.e. π = π ′ ◦ Λ b G for a representation π ′ : C ( b G ) → B( H π )) and arbitrary vectors ξ π , η π ∈ H π .We have τ t ((id ⊗ ω ξ π ,η π )( U π ∗ )) = (id ⊗ ω ξ π ,η π )( τ t ⊗ id)(id ⊗ π )( W ∗ )= (id ⊗ ω ξ π ,η π ◦ π ′ )( τ t ⊗ id)(W ∗ ) = (id ⊗ ω ξ π ,η π ◦ π ′ )(id ⊗ ˆ τ − t )(W ∗ )= (id ⊗ ω ξ π ,η π ◦ π ′ )(( ⊗ P − it )(W ∗ )( ⊗ P it )) . Now we write the above expression in two different ways: we have τ t ((id ⊗ ω ξ π ,η π )( U π ∗ )) = (id ⊗ ω ξ π ,η π ◦ π ′ )(( ⊗ P − it ∇ − itψ )( ⊗ ∇ itψ )(W ∗ )( ⊗ P it ))= (id ⊗ ω ξ π ,η π ◦ π ′ )(( ⊗ ˆ δ it ) ( σ ϕt ⊗ id)(W ∗ )) = σ ϕt ((id ⊗ ω ξ π ,η π ◦ π )(( ⊗ ˆ δ itu ) ( W ∗ )))= σ ϕt ((id ⊗ ω π (ˆ δ − itu ) ξ π ,η π )( U π ∗ )) (5.3)and τ t ((id ⊗ ω ξ π ,η π )( U π ∗ )) = (id ⊗ ω ξ π ,η π ◦ π ′ )(( ⊗ P − it )(W ∗ )( ⊗ ∇ − itψ )( ⊗ ∇ itψ P it ))= (id ⊗ ω ξ π ,η π ◦ π ′ )(( σ ψ − t ⊗ id)(W ∗ ) ( ⊗ ˆ δ − it )) = (id ⊗ ω ξ π ,η π ◦ π )(( σ ψ − t ⊗ id)( W ∗ ) ( ⊗ ˆ δ − itu ))= (id ⊗ ω ξ π ,π (ˆ δ − itu ) η π )(( σ ψ − t ⊗ id)( U π ∗ )) = σ ψ − t ((id ⊗ ω ξ π ,π (ˆ δ − itu ) η π )( U π ∗ )) . (5.4)Let now ξ, η be vectors in R ⊕ Irr( G ) H π d µ ( π ). Then fields ( π (ˆ δ − itu ) ξ π ) π ∈ Irr( G ) , ( π (ˆ δ − itu ) η π ) π ∈ Irr( G ) are also square integrable. Using equations (5.3), (5.4) and Proposition 3.9 we arrive at τ t ( M Lξ,η ) = τ t ( Z Irr( G ) (id ⊗ ω ξ π ,η π )( U π ∗ ) d µ ( π )) = Z Irr( G ) τ t ((id ⊗ ω ξ π ,η π )( U π ∗ )) d µ ( π )= Z Irr( G ) σ ϕt ((id ⊗ ω π (ˆ δ − itu ) ξ π ,η π )( U π ∗ )) d µ ( π ) = σ ϕt ( Z Irr( G ) (id ⊗ ω π (ˆ δ − itu ) ξ π ,η π )( U π ∗ ) d µ ( π ))= ν − it δ − it Z Irr( G ) (id ⊗ ω D itπ π (ˆ δ − itu ) ξ π ,E itπ η π )( U π ∗ ) d µ ( π )and τ t ( M Lξ,η ) = τ t ( Z Irr( G ) (id ⊗ ω ξ π ,η π )( U π ∗ ) d µ ( π )) = Z Irr( G ) τ t ((id ⊗ ω ξ π ,η π )( U π ∗ )) d µ ( π )= Z Irr( G ) σ ψ − t ((id ⊗ ω ξ π ,π (ˆ δ − itu ) η π )( U π ∗ )) d µ ( π ) = σ ψ − t ( Z Irr( G ) (id ⊗ ω ξ π ,π (ˆ δ − itu ) η π )( U π ∗ ) d µ ( π ))= ν − it Z Irr( G ) (id ⊗ ω D − itπ ξ π ,E − itπ π (ˆ δ − itu ) η π )( U π ∗ ) d µ ( π ) δ it . Theorem 5.4.
For every t ∈ R we have ∇ − it b ψ = δ it P it = Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q L = Q ∗ R (cid:0)Z ⊕ Irr( G ) D − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q R , ∇ it b ϕ = J ϕ δ it P it J ϕ = Q ∗ L (cid:0)Z ⊕ Irr( G ) D − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L = Q ∗ R (cid:0)Z ⊕ Irr( G ) E itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q R . Next, we show that the modular element for b G can be expressed using operators( D π ) π ∈ Irr( G ) , ( E π ) π ∈ Irr( G ) . Proposition 5.5.
For all t ∈ R we have ˆ δ it = ν − i t Q ∗ L (cid:0)Z ⊕ Irr( G ) D itπ E − itπ ⊗ H π d µ ( π ) (cid:1) Q L = ν − i t Q ∗ R (cid:0)Z ⊕ Irr( G ) H π ⊗ ( D − itπ E itπ ) T d µ ( π ) (cid:1) Q R . Moreover, π (ˆ δ itu ) = ν it E − itπ D itπ and ν ist D isπ E itπ = E itπ D isπ for all s, t ∈ R and almost all π ∈ Irr( G ) . We also get better expressions for the action of ( τ t ) t ∈ R : τ t ( M Lξ,η ) = δ − it M LE it ξ,E it η = M LD − it ξ,D − it η δ it ,τ t ( M Rξ,η ) = M RE it ξ,E it η δ it = δ − it M RD − it ξ,D − it η for all t ∈ R and ξ, η ∈ R ⊕ Irr( G ) H π d µ ( π ) .Proof. Let ξ, η be vector fields satisfying conditions from the first point of Lemma 3.6.Note that vector fields ( D − itπ ξ π ) π ∈ Irr( G ) , ( E − itπ π (ˆ δ − itu ) η π ) π ∈ Irr( G ) also satisfy conditions of16his lemma. Using the second equation from Lemma 5.3 we get: Q L P it Λ ϕ ( M Lξ,η ) = ν t Q L Λ ϕ ( τ t ( M Lξ,η ))= ν t − it Q L Λ ϕ ( Z Irr( G ) (id ⊗ ω D − itπ ξ π ,E − itπ π (ˆ δ − itu ) η π )( U π ∗ ) d µ ( π ) δ it )= ν t − it Q L J ϕ σ ϕi/ ( δ it ) ∗ J ϕ Λ ϕ ( Z Irr( G ) (id ⊗ ω D − itπ ξ π ,E − itπ π (ˆ δ − itu ) η π )( U π ∗ ) d µ ( π ))= ν − it Q L J ϕ δ − it J ϕ Q ∗ L Z ⊕ Irr( G ) E − itπ π (ˆ δ − itu ) η π ⊗ D π D − itπ ξ π d µ ( π )= ν − it Q L J ϕ δ − it J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) E − itπ π (ˆ δ − itu ) ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L Λ ϕ ( M Lξ,η ) . Since the set of Λ ϕ ( M Lξ,η ) with ξ, η as above form a lineary dense set (Lemma 3.7), we get J ϕ δ it J ϕ P it = Q ∗ L (cid:0)Z ⊕ Irr( G ) ( ν − it E − itπ π (ˆ δ − itu )) ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L . (5.5)Since ( J ϕ δ it J ϕ P it ) t ∈ R , (( D itπ ) T ) t ∈ R are strongly continuous groups (see equation (2.1)) thesame is true for ( ν − it E − itπ π (ˆ δ − itu )) t ∈ R (see point 2) of Lemma 8.1). Using relations gatheredin equation (2.1) one easily checks that J b ϕ commutes with J ϕ δ it J ϕ P it . Since J b ϕ = Q ∗ L Σ Q L ,point 3) of Lemma 8.1 implies ν − it E − itπ π (ˆ δ − itu ) = D − itπ ⇒ π (ˆ δ itu ) = ν it E − itπ D itπ ( π ∈ Irr( G ) , t ∈ R ) (5.6)Let us choose s, t ∈ R and use the fact that ( π (ˆ δ ipu )) p ∈ R is a group: we have ν i ( t + s )22 E − i ( t + s ) π D i ( t + s ) π = π (ˆ δ i ( t + s ) u ) = π (ˆ δ itu ) π (ˆ δ isu ) = ν it E − itπ D itπ ν is E − isπ D isπ , and formula ν ist E − isπ D itπ = D itπ E − isπ easily follows. Equations expressing the action of( τ t ) t ∈ R on matrix coefficients follows from the equation π (ˆ δ itu ) = ν i t E − itπ D itπ , commutationrelation between E itπ and D isπ and Lemma 5.3. Let us now plug in the above results toequation (5.5): J ϕ δ it J ϕ P it = ν − it Q ∗ L (cid:0)Z ⊕ Irr( G ) E − itπ π (ˆ δ − itu ) ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L = ν − it Q ∗ L (cid:0)Z ⊕ Irr( G ) E − itπ ν it E itπ D − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L = Q ∗ L (cid:0)Z ⊕ Irr( G ) D − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L , (5.7)which is the third equation of Theorem 5.4. If we use formula Q ∗ R Q L = ν − i J b ϕ J ϕ , we readilyget the second equation. Now we can derive the first pair of equations of Proposition 5.5.17ince for all t ∈ R we have ∇ itψ = ˆ δ − it P − it and J ϕ ˆ δ it = ˆ δ it J ϕ , it follows that ˆ δ it = J ϕ ˆ δ it J ϕ =( J ϕ P − it δ − it J ϕ )( J ϕ δ it ∇ − itψ J ϕ ) , which we can express using equation (5.7) and Proposition5.1: Q L ˆ δ it Q ∗ L = (cid:0)Z ⊕ Irr( G ) D itπ ⊗ ( D − itπ ) T d µ ( π ) (cid:1) ν − i t (cid:0)Z ⊕ Irr( G ) E − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) = ν − i t Z ⊕ Irr( G ) D itπ E − itπ ⊗ H π d µ ( π ) . On the other hand, we also have ˆ δ it = ( ∇ − itψ δ it )( δ − it P − it ), hence Q R ˆ δ it Q ∗ R = ν − i t (cid:0)Z ⊕ Irr( G ) D − itπ ⊗ ( E itπ ) T d µ ( π ) (cid:1)(cid:0)Z ⊕ Irr( G ) D itπ ⊗ ( D − itπ ) T d µ ( π ) (cid:1) , which implies the second equation for ˆ δ it and ends the proof of Proposition 5.5. In orderto finish the proof of Theorem 5.4 we have to derive a lemma concerning the function f introduced in Corollary 5.2. Lemma 5.6.
For all t ∈ R we have J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it HS( H π ) d µ ( π ) (cid:1) ∗ Q L J ϕ = Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) − it HS( H π ) d µ ( π ) (cid:1) Q L ,J ϕ Q ∗ R (cid:0)Z ⊕ Irr( G ) f ( π ) it HS( H π ) d µ ( π ) (cid:1) ∗ Q R J ϕ = Q ∗ R (cid:0)Z ⊕ Irr( G ) f ( π ) − it HS( H π ) d µ ( π ) (cid:1) Q R . Proof of Lemma 5.6.
Recall that J ϕ ˆ δ it J ϕ = ˆ δ it , hence ν i t J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) D itπ E − itπ ⊗ H π d µ ( π ) (cid:1) Q L J ϕ = ν − i t Q ∗ L (cid:0)Z ⊕ Irr( G ) D itπ E − itπ ⊗ H π d µ ( π ) (cid:1) Q L . Using the above relation and the fourth equation of Corollary 5.2 we get J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) D itπ ⊗ H π d µ ( π ) (cid:1) ∗ Q L J ϕ = J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) D − itπ E itπ ⊗ H π d µ ( π ) (cid:1) Q L J ϕ J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ H π d µ ( π ) (cid:1) ∗ Q L J ϕ = ν − it Q ∗ L (cid:0)Z ⊕ Irr( G ) D − itπ E itπ ⊗ H π d µ ( π ) (cid:1) Q L Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it D itπ ⊗ H π d µ ( π ) (cid:1) Q L = Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it E itπ ⊗ H π d µ ( π ) (cid:1) Q L , Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ H π d µ ( π ) (cid:1) Q L = J ϕ (cid:0) J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ H π d µ ( π ) (cid:1) ∗ Q L J ϕ (cid:1) ∗ J ϕ = J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it D itπ ⊗ H π d µ ( π ) (cid:1) ∗ Q L J ϕ = Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it E itπ ⊗ H π d µ ( π ) (cid:1) Q L J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it HS( H π ) d µ ( π ) (cid:1) ∗ Q L J ϕ and J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it HS( H π ) d µ ( π ) (cid:1) ∗ Q L J ϕ = Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) − it HS( H π ) d µ ( π ) (cid:1) Q L . The second equation can be proved analogously or using equation Q ∗ R Q L = ν − i J b ϕ J ϕ .Using the above lemma and Corollary 5.2 we can derive the first equation of Theorem5.4 out of the third one: δ it P it = J ϕ J ϕ δ it P it J ϕ J ϕ = J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) D − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L J ϕ = J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it HS( H π ) d µ ( π ) (cid:1) Q L J ϕ J ϕ Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) − it D − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L J ϕ = Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) it HS( H π ) d µ ( π ) (cid:1) Q L Q ∗ L (cid:0)Z ⊕ Irr( G ) f ( π ) − it E itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q L = Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q L . Now, the last equation of Theorem 5.4 follows as usual from the formula relating Q L and Q R . This concludes the proof of Theorem 5.4 and Proposition 5.5.The commutation relation ν ist D isπ E itπ = E itπ D isπ ( t, s ∈ R ) derived in the previousproposition has the following consequence. Corollary 5.7. If ν = 1 then for almost all π ∈ Irr( G ), operators D π , E π have emptypoint spectrum. In particular, if ν = 1 then the set of finite dimensional irreduciblerepresentations is of measure zero. In this section we show how the properties of operators ( E π ) π ∈ Irr( G ) , ( D π ) π ∈ Irr( G ) are re-lated to the modular theory of a type I, second countable locally compact quantum group(i.e. properties of the modular element, scaling group, modular automorphism groups etc.).First, let us mention three lemmas which are probably well known to experts and whichhold for a general locally compact group. 19 emma 6.1. The following conditions are equivalent:1) P it ∈ L ∞ ( G ) ′ for all t ∈ R ,2) the scaling group of G is trivial,3) P it = for all t ∈ R .Proof. Implications 1) ⇔ ⇐
3) follow from the equation τ t ( x ) = P it xP − it ( x ∈ L ∞ ( G )).For all x ∈ N ϕ and t ∈ R we have P it Λ ϕ ( x ) = ν t Λ ϕ ( τ t ( x )), hence 2) implies P it = ν t .Taking the norm of both sides gives us 1 = ν t hence ν = 1. Lemma 6.2.
1) The Haar integrals on G are tracial if, and only if P = ˆ δ = .2) b G is unimodular if, and only if ∇ itϕ = ∇ − itψ ( t ∈ R ) .Proof. We will use formulas gathered in equation (2.1). Equality ∇ itψ = ˆ δ − it P − it ( t ∈ R )shows that P = ˆ δ = implies ∇ itψ = and the traciality of ψ . Let us prove the converseimpliation. If ∇ itψ = then P it = ˆ δ − it ∈ L ∞ ( b G ) for all t ∈ R . Since P it commutes with J b ϕ ,we have P it = J b ϕ P it J b ϕ ∈ L ∞ ( b G ) ′ and by the previous lemma P it = = ˆ δ − it .If b G is unimodular, then we have J b ϕ ∇ − itϕ J b ϕ = ∇ itψ = P − it for all t ∈ R . Since P − it commutes with J b ϕ , it follows that ∇ itψ = ∇ − itϕ . On the other hand, if ∇ itψ = ∇ − itϕ for all t ∈ R , then ˆ δ − it P − it = ∇ itψ = ∇ − itϕ = J b ϕ ∇ itψ J b ϕ = J b ϕ ˆ δ − it P − it J b ϕ = J b ϕ ˆ δ − it J b ϕ P − it and we get ˆ δ it = J b ϕ ˆ δ it J b ϕ . This in particular means that ˆ δ it ∈ Z (L ∞ ( b G )) and [17, Proposi-tion 1.23] implies ˆ δ it = J b ϕ ˆ δ − it J b ϕ , unimodularity of b G follows.Although we will not use this result, let us mention here that if G is unimodular then σ b ϕt ( x ) = ˆ τ t ( x ) = ˆ δ − it x ˆ δ it and ∆ b G ( σ b ϕt ( x )) = ( σ b ϕt ⊗ σ b ϕt )∆ b G ( x ) for all t ∈ R , x ∈ L ∞ ( b G ). It isa consequence of the formula P − it = δ it ( J ϕ δ it J ϕ )ˆ δ it ( J b ϕ ˆ δ it J b ϕ ) and ∆ b G (ˆ δ it ) = ˆ δ it ⊗ ˆ δ it (see[20, Theorem 5.20, Proposition 5.15]. Lemma 6.3.
For all t, s ∈ R , if σ ϕt = σ ψs then ∇ itϕ = ∇ isψ . If ( s, t ) = (0 , then also ν = 1 .Proof. For all x ∈ N ϕ we have ∇ − isψ ∇ itϕ Λ ϕ ( x ) = ν s Λ ϕ ( σ ψ − s ( σ ϕt ( x ))) = ν s Λ ϕ ( x ) (see [20,Remark 5.2 ii)]), hence ∇ − isψ ∇ itϕ = ν s . Taking the norm of both sides implies ν s = 1and proves the first claim. If s = 0 then we get ν = 1, if s = 0 and ( s, t ) = (0 ,
0) then t = 0 and we get ∇ itϕ = . Formula ∇ itϕ Λ ψ ( y ) = ν t Λ ψ ( σ ϕt ( y )) = ν t Λ ψ ( y ) ( y ∈ N ψ ) implies ν = 1.The next theorem is the main result of this section. It presents a web of connectionsbetween various properties of a type I, second countable locally compact quantum group(and its dual). 20 heorem 6.4. Let G be a second countable, type I locally compact quantum group. Con-sider the following conditions:1) D itπ ∈ C H π for all t ∈ R and almost all π ∈ Irr( G ) ,2) E itπ ∈ C H π for all t ∈ R and almost all π ∈ Irr( G ) ,3) the Haar integrals on b G are tracial ( left ⇔ right ⇔ both),4) the Haar integrals on G are tracial ( left ⇔ right ⇔ both),5) ˆ δ it ∈ Z (L ∞ ( b G )) for all t ∈ R ,6) G is unimodular,7) E itπ D − itπ ∈ C H π for all t ∈ R and almost all π ,8) E itπ = D itπ for all t ∈ R and almost all π ∈ Irr( G ) ,9) b G is unimodular,10) E itπ D itπ ∈ C H π for all t ∈ R and almost all π ∈ Irr( G ) ,11) δ it ∈ Z (L ∞ ( G )) for all t ∈ R ,12) σ ϕt = σ ψt for all t ∈ R .The following implications hold:
1) 2) 3) 4)6) 10) 11) 12) 8) 9)7) 5)
Moreover, each of the above conditions implies ν = 1 .Proof. First, let us note that ϕ is tracial if and only ψ is tracial: it is a consequenceof the equation ∇ itψ = J b ϕ ∇ − itϕ J b ϕ ( t ∈ R ). Equivalence 1) ⇔ ⇔
3) is a part of theDesmedt’s theorem, one can also deduce this from formulas for ∇ b ϕ , ∇ b ψ – see Theorem 5.4.Equivalence 7) ⇔
5) follows from the formula for ˆ δ it in Proposition 5.5 and Q L L ∞ ( b G ) Q ∗ L = R ⊕ Irr( G ) B( H π ) ⊗ H π d µ ( π ) (see Proposition 3.4). Equivalence 8) ⇔
9) is a straightforwardconsequence of Proposition 5.5. 21ssume 6), i.e. that G is unimodular and let us derive 10). Fix t ∈ R . Theorem 5.4 givesus P it = Q ∗ L (cid:0)Z ⊕ Irr( G ) E itπ ⊗ ( E − itπ ) T d µ ( π ) (cid:1) Q L = Q ∗ L (cid:0)Z ⊕ Irr( G ) D − itπ ⊗ ( D itπ ) T d µ ( π ) (cid:1) Q L , which implies E itπ ⊗ ( E − itπ ) T = D − itπ ⊗ ( D itπ ) T ( π ∈ Irr( G )). Consequently, D itπ E itπ S = SD itπ E itπ for all S ∈ HS( H π ). This means that D itπ E itπ = λ t H π for some λ t ∈ C and wearrive at the point 10). On the other hand, point 10) implies that there exists λ t,π ∈ T such that E itπ = λ t,π D − itπ . It follows that ν = 1, moreover the first and the third row ofTheorem 5.4 implies δ it = J ϕ δ it J ϕ . This in particular means that δ it belongs to the centerof L ∞ ( G ) – we have δ it = J ϕ ( δ it ) ∗ J ϕ [17, Proposition 1.23]. These two equations togetherimply δ = .The last equivalence, 11) ⇔ σ ψt ( x ) = δ it σ ϕt ( x ) δ − it ( x ∈ L ∞ ( G ) , t ∈ R , see [20, Theorem 3.11]).The remaining implications are trivial. Let us now argue why all of the above conditionsimply ν = 1. Clearly we only need to justify this for 7) and 11). If E itπ D − itπ ∈ C H π then ν ist D isπ E itπ = E itπ D isπ forces ν = 1. If δ it ∈ Z (L ∞ ( G )) then ν it δ it = σ ϕt ( δ it ) = δ it for all t ∈ R ([17, Proposition 1.23]), hence also in this case ν = 1.Let us now show how certain classes of quantum groups fit into the above diagram. Inparticular, these examples show that one-sided implications in the above theorem cannotbe reversed. Proposition 6.5.
Let G be a type I, second countable locally compact quantum group. • If G is classical and non-unimodular, then it satisfies and does not satisfy . • If b G is classical and non-unimodular, then G satisfies and does not satisfy . • If G is compact and not of Kac type, then it satisfies and does not satisfy . • If G is discrete and non-unimodular, then it satisfies and does not satisfy . The numbering in the above proposition corresponds to the numbering introduced inTheorem 6.4. Clearly each of the above classes is non-empty: examples are given by theclassical ax + b group, its dual, the SU q (2) group and its dual (see Example 7.1). At theend of this section let us derive a corollary of Theorem 6.4. Corollary 6.6.
Let G be a type I, second countable locally compact quantum group. TheHaar integrals on G and b G are tracial if, and only if G and b G are unimodular. Proof.
The right implication is an easy corollary of Lemma 6.2. Assume that G and b G are unimodular. Equivalences 8) ⇔
9) and 6) ⇔
10) of Theorem 6.4 imply that E π = D π ∈ C H π for almost all π ∈ Irr( G ). Then 2) ⇔
3) of the same theorem impliesthat the Haar integrals on b G are tracial. Equalities ∇ it b ψ = δ − it P − it , ∇ itψ = ˆ δ − it P − it ( t ∈ R )end the proof. 22 Examples \ SU q (2) Fix a real number q ∈ ] − , \ { } . Let G = SU q (2) be the compact quantum groupintroduced by Woronowicz in [22] and let L be the dual discrete quantum group L = \ SU q (2).The C ∗ -algebra of continuous functions on the quantum space SU q (2), C(SU q (2)) is theuniversal unital C ∗ -algebra generated by elements α, γ satisfying the following relations: α ∗ α + γ ∗ γ = , αγ = qγα, αγ ∗ = qγ ∗ α,αα ∗ + q γγ ∗ = , γγ ∗ = γ ∗ γ. The Haar integral of SU q (2) is faithful on C(SU q (2)) and we have C u (SU q (2)) = C(SU q (2))(SU q (2) is coamenable, see [1, Theorem 2.12]). Furthermore, the C ∗ -algebra C(SU q (2)) isseparable and type I (see [22, Theorem A2.3]) hence L is an interesting example of a secondcountable, type I discrete quantum group . We will describe the Plancherel measure forthis group and show how various operators related to L act on the level of direct integrals.Let us start with describing the measurable space Irr( L ) (i.e. the spectrum of C(SU q (2))).The following result is a reformulation of [18, Theorem 3.2]: Proposition 7.1.
Measurable space
Irr( L ) can be identified with the disjoint union of twocircles T ⊔ T = { ψ ,ρ | ρ ∈ T } ∪ { ψ ,λ | λ ∈ T } . Representations ψ ,ρ are one dimensionaland given by ψ ,ρ ( α ) = ρ, ψ ,ρ ( α ∗ ) = ρ, ψ ,ρ ( γ ) = 0 , ψ ,ρ ( γ ∗ ) = 0 ( ρ ∈ T ) . Representations ψ ,λ act on a separable Hilbert space H λ = ℓ ( Z + ) with an orthonormalbasis { φ k | k ∈ Z + } via ψ ,λ ( α ) φ k = p − q k φ k − , ψ ,λ ( α ∗ ) φ k = p − q k +1) φ k +1 ,ψ ,λ ( γ ) φ k = λq k φ k , ψ ,λ ( γ ∗ ) φ k = λq k φ k , ( ρ ∈ T , k ∈ Z + ) , with the convention φ − n = 0 ( n ∈ N ) . In the next proposition we calculate the Plancherel measure of L , the unitary operator Q L and operators ( D π ) π ∈ Irr( L ) . In what follows, ϕ, ψ are the Haar integrals on L and h isthe Haar integral on G = SU q (2). Proposition 7.2.
The Plancherel measure of L equals on { ψ ,ρ | ρ ∈ T } and the normal-ized Lebesgue measure on the second circle { ψ ,λ | λ ∈ T } . Consequently, we will identify Irr( L ) with T . Operators { D λ | λ ∈ T } are given by D λ = (1 − q ) − diag(1 , | q | − , | q | − , . . . ) ( λ ∈ T ) In this section L is the ”main” group and G is the ”dual” one. ith respect to the basis { φ k | k ∈ Z + } . Operator Q L is given by Q L : L ( G ) ∋ Λ h ( a ) Z ⊕ Irr( L ) ψ ,λ ( a ) D − λ d µ ( λ ) ∈ Z ⊕ Irr( L ) HS( H λ ) d µ ( λ ) ( a ∈ C(SU q (2))) . Proof.
Define µ to be the normalized Lebesgue measure on the second circle of Irr( L ) = T ⊔ T and let Q L be the operator given by the above formula. In order to show thatthese objects are the one given by Desmedt’s theorem, we will use point 7) of [7, Theorem3.3]. Let us start with showing that Q L is well defined and unitary. First, it is clearthat for a ∈ C(SU q (2)) the field of operators ( ψ ,λ ( a ) D − λ ) λ ∈ T is measurable and squareintegrable. Consequently, we can introduce a densely defined linear map Q L : Λ h ( a ) R ⊕ Irr( L ) ψ ,λ ( a ) D − λ d µ ( λ ). Since kQ L Λ h ( a ) k ≤ k a k ( a ∈ C(SU q (2))), the linear map Q L ◦ Λ h isbounded. Let us now show that Q L is isometry, i.e. hQ L Λ h ( a ′ ) | Q L Λ h ( a ) i = h Λ h ( a ′ ) | Λ h ( a ) i for all a, a ′ ∈ C(SU q (2)). Since hQ L Λ h ( a ′ ) | Q L Λ h ( a ) i = (cid:10)Z ⊕ Irr( L ) ψ ,λ ( a ′ ) D − λ d µ ( λ ) (cid:12)(cid:12) Z ⊕ Irr( L ) ψ ,λ ( a ) D − λ d µ ( λ ) (cid:11) = (cid:10)Z ⊕ Irr( L ) ψ ,λ ( ) D − λ d µ ( λ ) (cid:12)(cid:12) Z ⊕ Irr( L ) ψ ,λ ( a ′∗ a ) D − λ d µ ( λ ) (cid:11) = hQ L Λ h ( ) | Q L Λ h ( a ′∗ a ) i and h Λ h ( a ′ ) | Λ h ( a ) i = h Λ h ( ) | Λ h ( a ′∗ a ) i , it is enough to consider the case a ′ = . Next,as maps Q L ◦ Λ h , Λ h are bounded and linear, it is enough to consider a in a basis ofPol(SU q (2)), { α l γ n γ ∗ m , α ∗ l ′ γ n γ ∗ m | l, n, m ∈ Z + , l ′ ∈ N } (see [22, Theorem 1.2]).In order to calculate h Λ h ( ) | Λ h ( a ) i we need to introduce a faithful representation π : C(SU q (2)) → B( ℓ ( Z + × Z )) defined in [22]. One can express the Haar integral h as h ( a ) = (1 − q ) ∞ X k =0 q k h φ k, | π ( a ) φ k, i ( a ∈ C(SU q (2))) , where { φ k,p | ( k, p ) ∈ Z + × Z } is the standard basis of ℓ ( Z + × Z ). Now, for l, n, m ∈ Z + we have h Λ h ( ) | Λ h ( α l γ n γ ∗ m ) i = h ( α l γ n γ ∗ m ) = δ l, (1 − q ) ∞ X k =0 q k δ n,m q ( n + m ) k = δ l, δ n,m − q − q n ) This result is formulated only for type I quantum groups with finite dimensional irreducible represen-tations. However, its proof is based on [7, Lemma 3.2] and proof of this lemma works just as well for moregeneral groups with bounded operators D − π , such as second countable, type I discrete quantum groups. h Λ h ( ) | Λ h ( α ∗ l γ n γ ∗ m ) i = δ l, δ n,m − q − q n ) . On the other hand hQ L Λ h ( ) | Q L Λ h ( α l γ n γ ∗ m ) i = (cid:10)Z ⊕ Irr( L ) D − λ d µ ( λ ) (cid:12)(cid:12) Z ⊕ Irr( L ) ψ ,λ ( α l γ n γ ∗ m ) D − λ d µ ( λ ) (cid:11) = δ l, (1 − q ) Z Irr( L ) ∞ X k =0 h φ k | λ n − m q ( n + m ) k q k φ k i d µ ( λ )= δ l, δ n,m (1 − q ) ∞ X k =0 q ( n + m ) k q k = δ l, δ n,m − q − q n ) . In an analogous manner we check hQ L Λ h ( ) | Q L Λ h ( α ∗ l γ n γ ∗ m ) i = δ l, δ n,m − q − q n ) . Thisshows that Q L is isometry and consequently extends to the whole of L ( G ). Let us nowargue that Q L is surjective. Fix λ ∈ T , k, l ∈ Z + . We have ψ ,λ ( γγ ∗ ) φ k = q k φ k , hence ψ ,λ ( χ { q l } ( γγ ∗ )) φ k = δ k,l φ k (note that operator χ { q l } ( γγ ∗ ) belongs to C(SU q (2)) because q l is an isolated point in the spectrum of γγ ∗ ). Next, for n ∈ Z + the following holds ψ ,λ ( α n χ q l ( γγ ∗ )) φ k = δ k,l ( n − Y a =0 (1 − q k − a ) ) ) φ k − n = δ k,l ( n − Y a =0 (1 − q k − a ) ) ) φ l − n which (together with a similar reasoning for α ∗ ) implies that for all l, n ∈ Z + there existsan operator E n,l ∈ C(SU q (2)) such that ψ ,λ ( E n,l ) φ k = δ l,k φ n ( k ∈ Z + , λ ∈ T ). Next, for m ∈ Z + we have ψ ,λ ( q − lm E n,l γ m ) φ k = δ l,k λ m φ n , ψ ,λ ( q − lm E n,l γ ∗ m ) φ k = δ l,k λ − m φ n ( k ∈ Z + , λ ∈ T )and consequently for any polynomial function P in λ, λ and n, l ∈ Z + an operator R ⊕ Irr( L ) P ( λ ) ψ ,λ ( E n,l ) d µ ( λ ) belongs to the image of Q L . By density of such polynomials inL ( T ) it follows that for all f ∈ L ( T ) Z ⊕ Irr( L ) f ( λ ) ψ ,λ ( E n,l ) d µ ( λ ) ∈ Q L (L ( G )) . (7.1)We have an isomorphism (given by choice of bases) R ⊕ Irr( L ) HS( H λ ) d µ ( λ ) ≃ L ( T ) ⊗ HS( ℓ ( Z + )) , hence it is clear that operators as in (7.1) span a dense subspace in R ⊕ Irr( L ) HS( H λ ) d µ ( λ ),and consequently Q L is unitary. Let us now check the first commutation relation. We have Q L λ L ( ω ) Q ∗ L ( Q L Λ h ( a )) = Q L Λ h ( λ L ( ω ) a ) = Z ⊕ Irr( L ) ψ ,λ ( λ L ( ω ) a ) D − λ d µ ( λ )= Z ⊕ Irr( L ) ψ ,λ ( λ L ( ω )) ψ ,λ ( a ) D − λ d µ ( λ ) = (cid:0)Z ⊕ Irr( L ) ψ ,λ ( λ L ( ω )) ⊗ H λ d µ ( λ ) (cid:1) Q L Λ h ( a ) , for all ω ∈ ℓ ( L ) , a ∈ C(SU q (2)) where λ L ( ω ) = ( ω ⊗ id)W L , hence Q L λ L ( ω ) Q ∗ L = Z ⊕ Irr( L ) ψ ,λ ( λ L ( ω )) ⊗ H λ d µ ( λ ) ( ω ∈ ℓ ( L )) . (7.2)25n order to show the second commutation relation, let us show that Q L transports J h tothe direct integral of adjoints. For a ∈ Pol(SU q (2)) we have Q L J h Λ h ( a ) = Q L Λ h ( σ h − i/ ( a ∗ )) = Z ⊕ Irr( L ) ψ ,λ ( σ h − i/ ( a ∗ )) D − λ d µ ( λ ) . Next, observe that ψ ,λ ( σ ht ( a )) = D − itλ ψ ,λ ( a ) D itλ for all λ ∈ T , t ∈ R , a ∈ Pol(SU q (2)).Indeed, we have σ ht ( α ) = | q | − it α, σ ht ( γ ) = γ ( t ∈ R ) ([13, Example 1.7.4]) and consequently ψ ,λ ( σ ht ( γ )) = ψ ,λ ( γ ) = D − itλ ψ ,λ ( γ ) D itλ ( t ∈ R )and similarly for all k ∈ Z + , t ∈ R D − itλ ψ ,λ ( α ) D itλ φ k = (1 − q k ) | q | − ikt | q | i ( k − t φ k − = | q | − it ψ ,λ ( α ) φ k = ψ ,λ ( σ ht ( α )) φ k . It follows that for all a ∈ Pol(SU q (2)) Q L J h Λ h ( a ) = Z ⊕ Irr( L ) D − λ ψ ,λ ( a ∗ ) D λ D − λ d µ ( λ ) = Z ⊕ Irr( L ) ( ψ ,λ ( a ) D − λ ) ∗ d µ ( λ ) , hence Q L J h Q ∗ L equals Σ = R ⊕ Irr( L ) J H λ d µ ( λ ). Now we can show the second commutationrelation. Formula χ (V L ) = ( J h ⊗ J h )(W L ) ∗ ( J h ⊗ J h ) ([20, Proposition 5.9]) implies thatfor all ω ∈ ℓ ( L ) we have ( ω ⊗ id) χ (V L ) = J h (( ω ◦ R L ⊗ id)W L ) ∗ J h and consequently Q L ( ω ⊗ id) χ (V L ) Q ∗ L = Q L J h Q ∗ L (cid:0)Z ⊕ Irr( L ) ψ ,λ ( λ L ( ω ◦ R L )) ⊗ H λ d µ ( λ ) (cid:1) ∗ Q L J h Q ∗ L = Z ⊕ Irr( L ) H λ ⊗ ψ ,λ ( λ L ( ω ◦ R L )) T d µ ( λ ) = Z ⊕ Irr( L ) H λ ⊗ ( ψ ,λ ) c ( λ L ( ω )) d µ ( λ ) , which is the second commutation relation. We are left to show Q L (L ∞ ( G ) ∩ L ∞ ( G ) ′ ) Q ∗ L = Diag( Z ⊕ Irr( L ) HS( H λ ) d µ ( λ )) , let us first argue that Q L L ∞ ( G ) Q ∗ L = Z ⊕ Irr( L ) B( H λ ) ⊗ H λ d µ ( λ )) . (7.3)Inclusion ⊆ follows from the commutation relation (7.2). On the other hand, equation(7.2) and reasoning similar to the one showing that Q L is unitary, implies that for anypolynomial P in λ, λ and n, l ∈ Z + we have Z ⊕ Irr( L ) P ( λ ) ψ ,λ ( E n,l ) ⊗ H λ d µ ( λ ) ∈ Q L L ∞ ( G ) Q ∗ L . -wot density of polynomials in L ∞ ( T ) and isomorphism R ⊕ Irr( L ) B( H λ ) ⊗ H λ d µ ( λ ) ≃ L ∞ ( T ) ¯ ⊗ B( ℓ ( Z + )) gives us (7.3). Consequently Q L (L ∞ ( G ) ∩ L ∞ ( G ) ′ ) Q ∗ L = (cid:0)Z ⊕ Irr( L ) B( H λ ) ⊗ H λ d µ ( λ ) (cid:1) ∩ (cid:0)Z ⊕ Irr( L ) H λ ⊗ B( H λ ) d µ ( λ ) (cid:1) = Diag( Z ⊕ Irr( L ) HS( H λ ) d µ ( λ )) . In the next proposition we find an action of the operator P it on the level of directintegrals. Proposition 7.3.
For each t ∈ R , operator Q L P it Q ∗ L acts on R ⊕ Irr( L ) HS( H λ ) d µ ( λ ) asfollows: Q L P it Q ∗ L : Z ⊕ Irr( L ) T λ d µ ( λ ) Z ⊕ Irr( L ) T λ | q | it d µ ( λ ) . Note that the above result implies that Q L P it Q ∗ L is not decomposable. Proof.
Let ˜ P it be the operator in the claim, i.e. ˜ P it : R ⊕ Irr( L ) T λ d µ ( λ ) R ⊕ Irr( L ) T λ | q | it d µ ( λ ).Clearly it is well defined and bounded. The scaling group of G = SU q (2) acts as follows([13, Example 1.7.8]) τ G t ( α ) = α, τ G t ( α ∗ ) = α ∗ , τ G t ( γ ) = | q | it γ, τ G t ( γ ∗ ) = | q | − it γ ∗ ( t ∈ R ) . Recall that P it satisfies P it Λ h ( a ) = Λ h ( τ G t ( a )) for all t ∈ R , a ∈ C( G ). Fix l, k, n, m ∈ Z + , λ ∈ T and corresponding operator α l γ n γ ∗ m in the basis of Pol( G ). We have ψ ,λ ( α l γ n γ ∗ m ) φ k = ( l − Y a =0 (1 − q k − a ) ) ) λ n − m q k ( n + m ) φ k − l = | q | − it ( n − m ) ( l − Y a =0 (1 − q k − a ) ) )( λ | q | it ) n − m q k ( n + m ) φ k − l = | q | − it ( n − m ) ψ ,λ | q | it ( α l γ n γ ∗ m ) φ k , (recall that we use convention φ − p = 0 for p ∈ N ) and consequently Q L P it Λ h ( α l γ n γ ∗ m ) = | q | it ( n − m ) Q L Λ h ( α l γ n γ ∗ m )= Z ⊕ Irr( L ) ψ ,λ | q | it ( α l γ n γ ∗ m ) D − λ d µ ( λ ) = ˜ P it Q L Λ h ( α l γ n γ ∗ m ) . In a similar manner we check Q L P it Λ h ( α ∗ l γ n γ ∗ m ) = ˜ P it Q L Λ h ( α ∗ l γ n γ ∗ m ). The claim followsbecause Λ h (Pol( G )) is dense in L ( G ). 27he last result of this section describes the action of an operator Q L J ϕ Q ∗ L . Proposition 7.4.
Operator Q L J ϕ Q ∗ L acts on R ⊕ Irr( L ) HS( H λ ) d µ ( λ ) as follows: Q L J ϕ Q ∗ L : Z ⊕ Irr( L ) T λ d µ ( λ ) Z ⊕ Irr( L ) j λ T − sgn( q ) λ j λ d µ ( λ ) , where j λ is the antilinear operator on H λ = ℓ ( Z + ) given by j λ φ k = φ k ( λ ∈ T , k ∈ Z + ) . Note that this result implies that operator Q L J ϕ Q ∗ L is not decomposable if q > Proof.
Using formula R G = S G τ G i/ and [22, Equation 1.14] one easily checks that R G ( α ) = α ∗ , R G ( α ∗ ) = α, R G ( γ ) = − sgn( q ) γ, R G ( γ ∗ ) = − sgn( q ) γ ∗ . On the other hand we have R G ( a ) = J ϕ a ∗ J ϕ for all a ∈ C(SU q (2)), hence J ϕ α = αJ ϕ , J ϕ α ∗ = α ∗ J ϕ , J ϕ γ = − sgn( q ) γ ∗ J ϕ , J ϕ γ ∗ = − sgn( q ) γJ ϕ . Denote by ˜ J ϕ the operator from the claim and fix λ ∈ T , k, n, m, l ∈ Z + . We have ψ ,λ ( α l γ m γ ∗ n ) φ k = λ m − n q ( m + n ) k ( l − Y a =0 (1 − q k − a ) ) ) φ k − l = ( − sgn( q )) m + n ( − sgn( q ) λ ) m − n q ( m + n ) k ( l − Y a =0 (1 − q k − a ) ) ) φ k − l = ( − sgn( q )) m + n j λ ψ , − sgn( q ) λ ( α l γ n γ ∗ m ) j λ φ k , consequently Q L J ϕ Λ h ( α l γ n γ ∗ m ) = Q L α l ( − sgn( q )) n γ ∗ n ( − sgn( q )) m γ m J ϕ Λ h ( )= ( − sgn( q )) n + m Z ⊕ Irr( L ) ψ ,λ ( α l γ m γ ∗ n ) D − λ d µ ( λ )= Z ⊕ Irr( L ) j λ ψ , − sgn( q ) λ ( α l γ n γ ∗ m ) j λ D − λ d µ ( λ )= ˜ J ϕ Z ⊕ Irr( L ) ψ ,λ ( α l γ n γ ∗ m ) D − λ d µ ( λ ) = ˜ J ϕ Q L Λ h ( α l γ n γ ∗ m ) . Equation Q L J ϕ Λ h ( α ∗ l γ n γ ∗ m ) = ˜ J ϕ Q L Λ h ( α ∗ l γ n γ ∗ m ) can be checked similarly. Remark 7.5.
In propositions 7.3, 7.4 we have expressed operators P it ( t ∈ R ) and J ϕ on R ⊕ Irr( L ) HS( H λ ) d µ ( λ ). Theorem 5.4 and Proposition 5.1 allow us to do the same for δ it , ∇ itϕ , ∇ itψ ( t ∈ R ) – operators obtained in this way are not decomposable.28 .2 Quantum group az + b In this section we will describe some aspects of the theory of the quantum az + b group.We begin by introducing a complex number q and an abelian group Γ q ⊆ C × . We willconsider three cases:1) q = e πiN for a natural number N ∈ N \ { } and Γ q = { q k r | k ∈ Z , r ∈ R > } ,2) q is a real number in ]0 ,
1[ and Γ q = { q iθ + k | θ ∈ R , k ∈ Z } ,3) q = e ρ , where Re( ρ ) < , Im( ρ ) = N π and N ∈ Z \ { } . In this caseΓ q = { e k + itρ | k ∈ Z , t ∈ R } .It will be more convenient for us to work in the dual picure : let b G be the quantum az + b group associated with the parameter q . We refer the reader to papers [25, 14, 26]for construction of these groups, here we will recall only necessary properties.We treat all three cases simultaneously. The group Γ q has closure given by Γ q = Γ q ∪ { } and is selfdual. This duality is implemented by a certain bicharacter χ : Γ q × Γ q → T .We choose a Haar measure on Γ q in such a way that the Fourier transform F ( f )( γ ) = R Γ q χ ( γ, γ ′ ) f ( γ ′ ) d µ ( γ ′ ) is a unitary operator on L (Γ q ). Next, the group Γ q acts on C (Γ q )by translations: σ γ ( f )( γ ′ ) = f ( γγ ′ ) ( f ∈ C (Γ q ) , γ ∈ Γ q , γ ′ ∈ Γ q ). Let C (Γ q ) ⋊ σ Γ q ⊆ B(L (Γ q )) be the associated crossed product C ∗ -algebra (note that since Γ q is abelian, thereduced crossed product is universal). It turns out that the C ∗ -algebra C ( b G ) is isomor-phic to the crossed product C (Γ q ) ⋊ σ Γ q . Furthermore, it is known that b G is coamenable.Indeed, it was pointed in [14, 15]. It follows from an easy observation that the universalproperty of C (Γ q ) ⋊ σ Γ q together with the trivial representation of Γ q and the characterC (Γ q ) ∋ f f (0) ∈ C give rise to a character of C (Γ q ) ⋊ σ Γ q ≃ C ( b G ). Then [2,Theorem 3.1] implies that b G is coamenable.One easily checks that the quotient space Γ q / Γ q consists of two points and is not an-tidiscrete. Consequently, [21, Proposition 7.30] implies that G is second countable andtype I. Using [21, Theorem 8.39] one can describe the spectrum of C ( b G ) ≃ C (Γ q ) ⋊ σ Γ q :there is a family of one dimensional representations indexed by b Γ q and one faithful irre-ducible infinite dimensional representation given by the inclusion into B(L (Γ q )). Denotethis representation by π . Proposition 7.6.
The Plancherel measure of G equals the Dirac measure at π , a represen-tation corresponding to the inclusion π : C ( b G ) ≃ −→ C (Γ q ) ⋊ σ Γ q ֒ → B(L (Γ q )) . Consequentlywe have Q L , Q R : L ( G ) → HS(L (Γ q )) .Proof. It is observed in [26] that we have b ψ ◦ τ b G t = | q − it | b ψ for all t ∈ R , hence thescaling constant of G equals ν = ˆ ν − = | q − i | . In the first and the third case q is not real In fact, G is isomorphic to the quantum group opposite to quantum az + b . ν is nontrivial. Corollary 5.7 implies that the set of one dimensionalrepresentations is of measure zero, and the claim follows . Let us now consider the secondcase, i.e. q ∈ ]0 , ∞ ( b G ) is isomorphic to the von Neumann algebra M associated with a pair ( a, b ) ofadmissible normal operators (see [19, Definition 5.1]). Moreover, up to an isomorphism Mdoes not depend on the choice of ( a, b ), in particular we can take a pair ( a, b ) introduced in[19, Proposition 5.2]. In this case one easily sees that the resulting von Neumann algebraequals the whole B( ℓ ( Z )). In particular it is a factor, hence Proposition 3.4 implies thatthe Plancherel measure of G must be the Dirac measure at π .Now we turn to the problem of identifying operators D π , E π . To simplify the nota-tion, we will call these operators respectively D and E . Let us start with introducingtwo normal (unbounded) operators on L (Γ q ): a and b . Operator b acts by multiplication:( bf )( γ ) = γf ( γ ) ( f ∈ Dom( b ) , γ ∈ Γ q ) and has the obvious domain. The second operator a can be defined as a = F b F ∗ .Note that there exists an isomorphism of von Neumann algebras Φ R : L ∞ ( b G ) → B(L (Γ q ))induced by Q R J b ϕ J ϕ , such that Φ R ( x ) = π ( x ) for x ∈ C ( b G ) (see Theorem 3.2 and Proposi-tion 3.4). Under this isomorphism, the right Haar integral b ψ is transformed to Tr( E − · E − )– it follows from the construction of the Plancherel measure in [5]. On the other hand,we have b ψ ( x ) = Tr( | b | π ( x ) | b | ) for all x ∈ C ( b G ) + ([26, Theorem 3.1]). This means thatthe weights Tr( E − · E − ) , Tr( | b | · | b | ) are equal on Φ R (C ( b G )). Let θ be the restriction ofthese weights to Φ R (C ( b G )). The modular automorphism group of Tr( E − · E − ) is givenby σ Tr E − t ( A ) = E − it AE it , similarly σ Tr | b | t ( A ) = | b | it A | b | − it ( A ∈ B(L (Γ q )) , t ∈ R ).Next, the weight θ satisfies the KMS condition for both groups ( σ Tr E − t | Φ R (C ( b G )) ) t ∈ R and( σ Tr | b | t | Φ R (C ( b G )) ) t ∈ R and as this weight is faithful, [9, Corollary 6.35] implies E − it AE it = | b | it A | b | − it for all A ∈ Φ R (C ( b G )) , t ∈ R . By the σ -wot density of Φ R (C ( b G )) inB(L (Γ q )) we get E = c | b | − for some c >
0. Equality Tr( E − · E − ) = Tr( | b | · | b | ) onΦ R (C ( b G )) forces c = 1 and consequently E = | b | − .The next step is to identify the operator D . Observe that Lemma 5.6 implies f ( π ) = 1.Recall ([14, Section 6.2], [25, Equation 3.18]) that operator a − ◦ b is closable and its clo-sure a − b is normal. Moreover, we have R b G ( π − ( b )) = π − ( − qa − b ). If we combine thisinformation together with Corollary 5.2 and the equality E = | b | − we arrive at Q ∗ L ( D it ⊗ L (Γ q ) ) Q L = R b G ( Q ∗ L ( E it ⊗ L (Γ q ) ) Q L ) = R b G ( Q ∗ L ( | b | − it ⊗ L (Γ q ) ) Q L )= Q ∗ L ( | − qa − b | − it ⊗ L (Γ q ) ) Q L = Q ∗ L ( | qa − b | − it ⊗ L (Γ q ) ) Q L , which implies D = | qa − b | − . Proposition 7.7.
We have D = | qa − b | − and E = | b | − . We remark that it was already observed in [19] that in the first case, L ∞ ( b G ) is isomorphic to thealgebra of bounded operators on a separable Hilbert space. Appendix
Lemma 8.1.
Let H be a Hilbert space and J : H ⊗ H → H ⊗ H an antilinear map given by J : ξ ⊗ η η ⊗ ξ ( ξ, η ∈ H ) .1) If x, y ∈ B( H ) are unitaries such that x ⊗ ( y ∗ ) T = y ⊗ ( x ∗ ) T ∈ B( H ⊗ H ) , then operators xy ∗ , y ∗ x are selfadjoint.2) Let ( a t ) t ∈ R , ( b t ) t ∈ R be families of unitary operators on H . Define c t = a t ⊗ b T t ( t ∈ R ) .If ( b t ) t ∈ R and ( c t ) t ∈ R are strongly continuous groups, then ( a t ) t ∈ R is also a stronglycontinuous group.3) Let ( a t ) t ∈ R , ( b t ) t ∈ R be strongly continuous groups of unitary operators on H . Define c t = a t ⊗ b T t ( t ∈ R ) . If J c t = c t J for all t ∈ R then a t = b − t ( t ∈ R ) .Proof.
1) Equality from the assumption gives us xSy ∗ = ySx ∗ for all S ∈ HS( H ). We canapproximate the unit by Hilbert-Schmidt operators hence xy ∗ = yx ∗ , i.e. xy ∗ is selfadjoint.Multiplying this equation from the left by x ∗ and from the right by x gives us y ∗ x = x ∗ y ,i.e. y ∗ x is selfadjoint.2) For all t, s ∈ R we have a t + s ⊗ b T t + s = c t + s = c t c s = a t a s ⊗ b T t b T s hence a t + s = a t a s ,i.e. ( a t ) t ∈ R is a group. Equation a t ⊗ H = c t ( H ⊗ b T − t ) implies that R ∋ t a t ∈ B( H ) isstrongly continuous.3) We have J c t J = b − t ⊗ a T − t for all t ∈ R . Consequently, for s, t ∈ R , x ∈ B( H ) we have c s J c t J ( x ⊗ H ) J c − t J c − s = ( a s b − t ⊗ b T s a T − t )( x ⊗ H )( b t a − s ⊗ a T t b T − s ) = a s b − t xb t a − s ⊗ H , and on the other hand J c t J c s ( x ⊗ H ) c − s J c − t J = ( b − t a s ⊗ a T − t b T s )( x ⊗ H )( a − s b t ⊗ b T − s a T t ) = b − t a s xa − s b t ⊗ H , hence a s b − t xb t a − s = b − t a s xa − s b t ⇒ a − s b t a s b − t x = xa − s b t a s b − t . The above equation holds for all x ∈ B( H ), hence there exists λ t,s ∈ C such that a − s b t a s b − t = λ t,s H and consequently a − s b t = λ t,s b t a − s ( t, s ∈ R ). Clearly we have | λ t,s | = 1. Since a t ⊗ b T t = c t = J J c t = J c t J = b − t ⊗ ( a − t ) T , for all t ∈ R , the first point implies that a t b t , b t a t are selfadjoint. For s = − t we have a t b t = λ t, − t b t a t , and since a t b t , b t a t are selfad-joint we have λ t, − t ∈ R ∩ T = {− , } . As the function t λ t, − t ∈ R is continuous and λ , = 1, we have λ t, − t = 1 for all t ∈ R . Consequently b t a t = a t b t = ( a t b t ) ∗ = b − t a − t and b t = a − t ( t ∈ R ) . Acknowledgements
The author would like to express his gratitude towards Piotr M. So ltan for many helpfuldiscussions and suggestions. 31he author was partially supported by the Polish National Agency for the Academic Ex-change, Polonium grant PPN/BIL/2018/1/00197, FWO–PAS project VS02619N: von Neu-mann algebras arising from quantum symmetries and NCN (National Centre of Science)grant 2014/14/E/ST1/00525.
References [1] E. B´edos, G. J. Murphy, and L. Tuset. Co-amenability of compact quantum groups.
J. Geom. Phys. , 40(2):130–153, 2001.[2] E. B´edos and L. Tuset. Amenability and co-amenability for locally compact quantumgroups.
Internat. J. Math. , 14(8):865–884, 2003.[3] M. Caspers.
Non-commutative integration on locally compact quantum groups: Fouriertheory, Gelfand pairs, non-commutative L p -spaces . PhD thesis, Radboud UniversiteitNijmegen, 2012.[4] M. Caspers and E. Koelink. Modular properties of matrix coefficients of corepresen-tations of a locally compact quantum group. J. Lie Theory , 21(4):905–928, 2011.[5] P. Desmedt.
Aspects of the theory of locally compact quantum groups: Amenability -Plancherel measure . PhD thesis, Katholieke Universiteit Leuven, 2003.[6] J. Dixmier.
Von Neumann algebras , volume 27 of
North-Holland Mathematical Li-brary . North-Holland Publishing Co., Amsterdam-New York, 1981.[7] J. Krajczok. Coamenability of type I locally compact quantum groups. arXiv e-prints ,page arXiv:2001.06740, 2020.[8] J. Krajczok and P. M. So ltan. The quantum disk is not a quantum group. arXive-prints , page arXiv:2005.02967, 2020.[9] J. Kustermans. KMS-weights on C ∗ -algebras, 1997.[10] J. Kustermans. Locally compact quantum groups in the universal setting. Internat.J. Math. , 12(3):289–338, 2001.[11] J. Kustermans and S. Vaes. Locally compact quantum groups in the von Neumannalgebraic setting.
Math. Scand. , 92(1):68–92, 2003.[12] C. Lance. Direct integrals of left Hilbert algebras.
Math. Ann. , 216:11–28, 1975.[13] S. Neshveyev and L. Tuset.
Compact quantum groups and their representation cate-gories , volume 20 of
Cours Sp´ecialis´es [Specialized Courses] . Soci´et´e Math´ematiquede France, Paris, 2013. 3214] P. M. So ltan. New quantum “ az + b ” groups. Rev. Math. Phys. , 17(3):313–364, 2005.[15] P. M. So ltan. Quantum Bohr compactification.
Illinois J. Math. , 49(4):1245–1270,2005.[16] P. M. So ltan and S. L. Woronowicz. From multiplicative unitaries to quantum groups.II.
J. Funct. Anal. , 252(1):42–67, 2007.[17] M. Takesaki.
Theory of operator algebras. II , volume 125 of
Encyclopaedia of Math-ematical Sciences . Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6.[18] L. L. Vaksman and Ya. S. Soibelman. An algebra of functions on the quantum groupSU(2).
Funktsional. Anal. i Prilozhen. , 22(3):1–14, 96, 1988.[19] A. van Daele. The Haar measure on some locally compact quantum groups, 2001.[20] A. van Daele. Locally compact quantum groups. A von Neumann algebra approach.
SIGMA Symmetry Integrability Geom. Methods Appl. , 10:Paper 082, 41, 2014.[21] D. P. Williams.
Crossed products of C ∗ -algebras , volume 134 of Mathematical Surveysand Monographs . American Mathematical Society, Providence, RI, 2007.[22] S. L. Woronowicz. Twisted SU(2) group. An example of a noncommutative differentialcalculus.
Publ. Res. Inst. Math. Sci. , 23(1):117–181, 1987.[23] S. L. Woronowicz. Compact quantum groups. In
Sym´etries quantiques (Les Houches,1995) , pages 845–884. North-Holland, Amsterdam, 1998.[24] S. L. Woronowicz. Quantum exponential function.
Rev. Math. Phys. , 12(6):873–920,2000.[25] S. L. Woronowicz. Quantum “ az + b ” group on complex plane. Internat. J. Math. ,12(4):461–503, 2001.[26] SL Woronowicz. Haar weight on some quantum groups. In JP Gazeau, R Kerner,JP Antoine, S Metens, and JY Thibon, editors,
Group 24 : Physical And MathematicalAspects Of Symmetries , volume 173 of