Center of the algebra of functions on the quantum group S U q (2) and related topics
aa r X i v : . [ m a t h . OA ] F e b CENTER OF THE ALGEBRA OF FUNCTIONS ON THE QUANTUMGROUP SU q (2) AND RELATED TOPICS
JACEK KRAJCZOK AND PIOTR M. SO LTAN
Abstract.
The center of the algebra of continuous functions on the quantum group SU q (2) isdetermined as well as centers of other related algebras. Several other results concerning thisquantum group are given with direct proofs based on concrete realization of these algebras asalgebras of operators on a Hilbert space. Dedicated to Marek Bo˙zejko on the occasion of his 70th birthday. Introduction
The aim of this paper is to provide very direct and relatively elementary proofs of certain factsconcerning the quantum SU(2) group introduced by S.L. Woronowicz in the seminal paper [10].The issues addressed in this paper are the following ◮ faithfulness of the representation π introduced in [10, Proof of Theorem 1.2], ◮ determining the center of the algebras Pol(SU q (2)), C(SU q (2)) and L ∞ (SU q (2)) as well asthe commutant of π (C(SU q (2))), ◮ giving direct proofs of faithfulness of Haar measure and continuity of the counit.The above tasks are interrelated and the relations between them will be explained in detail.Most results of this work are taken from the first author’s BSc thesis submitted at the Facultyof Physics, University of Warsaw. These results are known and in most cases proofs are published,but our approach is rather elementary and direct.The quantum groups SU q (2) were introduced in [10] and later studied in numerous papers inmathematics and theoretical physics. Apart from [10] our approach will be based on fundamentaltexts [9, 12] and more specialized [1, 4, 5, 11]. Methods of functional analysis and operator algebrasare covered in textbooks such as [2, 6, 8].The paper is organized as follows: in the next subsections we briefly introduce terminologyand notation needed in the remainder of the paper. Section 2 is devoted to a detailed proof offaithfulness of a particular representation of the algebra of functions on the quantum group SU q (2)defined in [10]. In Section 3 we introduce the additional structure on the C ∗ -algebra studied inSection 2 which defines the quantum group SU q (2). We also list some objects needed for latersections and recall the formula for the Haar measure. Section 4 provides the proof of the mainresult of the paper, namely that the center of the algebra of continuous functions on SU q (2) istrivial. This is achieved by examining the commutant of this algebra in the faithful representationstudied earlier. These results are used in Section 5 to prove that the Haar measure of SU q (2) isfaithful and its co-unit is continuous (the latter fact is justified in two different ways). Section6 is devoted to determining the center of the von Neumann algebra generated by the image ofthe algebra of continuous functions on SU q (2) in the GNS representation for the Haar measure,i.e. the center of the algebra L ∞ (SU q (2)). Finally in Section 7 we sketch a way to use some of ourresults and some major results from the literature to obtain an alternative proof of faithfulness of π . Mathematics Subject Classification.
Primary: 16T20, Secondary: 46L89, 20G42.
Key words and phrases.
Compact quantum group, center, Haar measure, counit.Partially supported by the National Science Center (NCN) grant no. 2015/17/B/ST1/00085.
Terminology and notation of compact quantum group theory.
We will follow someof the modern texts on quantum groups in declaring a compact quantum group to be an abstractobject of the category dual to the category of C ∗ -algebras related to a unital C ∗ -algebra withadditional structure (see e.g. [5] and Section 3). In particular for a compact quantum group G we write C( G ) for the corresponding C ∗ -algebra which we refer to as the algebra of continuousfunctions on G . It is important to note that the actual set G does not exist.1.2. Notation for spectral subspaces and polar decompositions.
In the proofs of some ofthe results we will employ very useful, yet rather non-standard notation for spectral subspaces ofoperators introduced in [13, Section 0] and put to use e.g. in [7, Section 3.1]. Let H be a Hilbertspace and T be a normal operator on H . Let f be a function on the spectrum Sp T of T withvalues in { true , false } . Our notation will be to write H ( f ( T )) for the spectral subspace for T corresponding to the subset (cid:8) λ ∈ Sp T f ( λ ) = true (cid:9) . (1.1)(we assume that the function f is such that (1.1) is measurable). This notation allows us to writee.g. ◮ H ( T = λ ) for the spectral subspace for T corresponding to { λ } , ◮ H ( | T | > ε ) for the spectral subspace for | T | corresponding to ] ε, + ∞ [, ◮ H ( T = λ ) for the spectral subspace for T corresponding to Sp T \{ λ } , i.e. the orthogonalcomplement of H ( T = λ )and many other similar expressions. Note that the Hilbert space on which the operator acts isexplicitly included in the notation. For example, if an operator S acts on a Hilbert space K then we accordingly use notation of the form K ( f ( S )), where f is again a { true , false } -valuedfunction. Apart from H ( · · · ) we will also use the symbol χ ( · · · ) to denote the projection ontothe corresponding spectral subspace.We will also use the following convention for polar decompositions: if, as above, T is a boundedoperator on H , we write | T | for the operator √ T ∗ T and the partial isometry entering the polardecomposition of T will be denoted by Phase T . Thus T = (Phase T ) | T | will always denote the polar decomposition of T .2. The algebra of functions on the quantum group SU q (2)Let A be the universal C ∗ -algebra generated by two elements α and γ subject to relations α ∗ α + γ ∗ γ = , αγ = qγα,αα ∗ + q γ ∗ γ = , γ ∗ γ = γγ ∗ . (2.1)where q is a parameter in ] − , \{ } . We remark that due to the Fuglede-Putnam theorem([6, Section 12.16]) the relation αγ ∗ = qγ ∗ α follows from (2.1) (cf. [7, Sections 1.3 & 3.1]). It isa matter of simple computation to see that the relations (2.1) are equivalent to unitarity of thematrix (cid:20) α − qγ ∗ γ α ∗ (cid:21) . This fact immediately shows that the universal C ∗ -algebra generated by α and γ with relations(2.1) exists. Indeed any C ∗ -seminorm on the ∗ -algebra generated by symbols α and γ subjectto (2.1) must be less or equal to 1 on entries of a unitary matrix. This implies that for anynon-commutative polynomial a in α, α ∗ , γ, γ ∗ and the quantity k a k = sup ̺ (cid:13)(cid:13) ̺ ( a ) (cid:13)(cid:13) (where the supremum is taken over all ∗ -representations of the ∗ -algebra generated by α and γ onHilbert spaces) is finite. This is clearly a C ∗ -seminorm, but due to [10, Theorem 1.2] it is in facta norm. In the proof of this result S.L. Woronowicz introduced a special representation π of the ∗ -algebra generated by α and γ which was shown to be injective. Just before statement of [10, ENTER OF C(SU q (2)) 3 Theorem A2.3] it is mentioned that the representation π is faithful on A . We will now give a proofof this result. Before proceeding let us mention that instead of giving a direct proof of faithfulnessof π one can use a combination of results of [11] or [4] and [1] together with our results to arriveat the same conclusion (cf. Section 7).2.1. Faithfulness of π . The universal property of A is the following: for any unital C ∗ -algebra B containing two elements α and γ such that α ∗ α + γ ∗ γ = , α γ = qγ α ,α α ∗ + q γ ∗ γ = , γ ∗ γ = γ γ ∗ . (2.2)there is a unique unital ∗ -homomorphism ̺ : A → B such that ̺ ( α ) = α and ̺ ( γ ) = γ . (2.3)By the Gelfand-Naimark theorem this property is equivalent to a simpler one: for any Hilbert space H and any pair ( α , γ ) of operators on H satisfying (2.2) there exists a unique representation ̺ of A on H such that (2.3) holds.Following [10, Proof of Theorem 1.2] will now introduce the representation π mentioned aboveusing this universal property: let H = ℓ ( Z + × Z ) and let { e n,k } n ∈ Z + k ∈ Z be the standard orthonormalbasis of H . Define operators α and γ on H by α e n,k = p − q n e n − ,k , γ e n,k = q n e n,k +1 . (2.4)The operators α and γ satisfy the defining relations of A , so there exists a representation π of A on H such that π ( α ) = α and π ( γ ) = γ . For future reference let us note the action of α ∗ and γ ∗ : α ∗ e n,k = p − q n +1) e n +1 ,k , γ ∗ e n,k = q n e n,k − . (2.5)Our aim is to show that π is faithful. Let C ∗ ( α , γ ) be the smallest C ∗ -algebra of operatorson H containing α and γ . It is easy to see that C ∗ ( α , γ ) is the image of the representation π .Presently let us note that H is isomorphic to ℓ ( Z + ) ⊗ ℓ ( Z ) with the isomorphism mapping e n,k to e n ⊗ e k and under this isomorphism the operators α and γ are transformed to α = s p − q N ⊗ , γ = q N ⊗ u , (2.6)where ◮ s is the unilateral shift s : e n ( e n − n > , n = 0 , (2.7) ◮ N is the unbounded self-adjoint operator of multiplication by the sequence (1 , , , . . . ): N : e n ne n , n ∈ Z + , (2.8) ◮ u is the bilateral shift u : e k e k +1 , k ∈ Z . In the proof of Theorem 2.1 below we will establish a similar decomposition for an arbitrary pairof operators satisfying relations (2.1). Let T be the algebra of operators on ℓ ( Z + ) generated by s (the Toeplitz algebra, cf. [2, Section V.1]) and similarly let U be the algebra of operators on ℓ ( Z )generated by u . Then clearly α and γ belong to T ⊗ U ⊂ B( ℓ ( Z + ) ⊗ ℓ ( Z )) and consequently π ( A ) ⊂ T ⊗ U . JACEK KRAJCZOK AND PIOTR M. SO LTAN
Theorem 2.1.
Let H be a Hilbert space and ( α , γ ) a pair of bounded operators on H satisfyingrelations (2.2) . Then there exists a unique unital ∗ -homomorphism ̺ : C ∗ ( α , γ ) → B( H ) suchthat ̺ ( α ) = α and ̺ ( γ ) = γ . Proof.
We have γ ∗ γ = − α ∗ α , (2.9a) q γ ∗ γ = − α α ∗ . (2.9b)The spectrum of (2.9a) is 1 − Sp( α ∗ α ) = (cid:8) − λ λ ∈ Sp( α ∗ α ) (cid:9) while the spectrum of (2.9b) is (cid:8) − λ λ ∈ Sp( α α ∗ ) (cid:9) . Now recall that Sp( α ∗ α ) \ { } = Sp( α α ∗ ) \ { } , soSp( γ ∗ γ ) \ { } = (cid:8) − λ λ ∈ Sp( α ∗ α ) \ { } (cid:9) = (cid:8) − λ λ ∈ Sp( α α ∗ ) \ { } (cid:9) = (cid:8) − λ λ ∈ Sp( α α ∗ ) (cid:9) \ { } = q Sp( γ ∗ γ ) \ { } . Finally note that since k γ k ≤ α ∗ α + γ ∗ γ = ) and | q | < q Sp( γ ∗ γ ) is contained in [0 , q ] and so it does not contain 1. It follows thatSp( γ ∗ γ ) \ { } = q Sp( γ ∗ γ ) . It is now easy to see that either Sp( γ ∗ γ ) = { } (in which case γ = 0) orSp( γ ∗ γ ) = { } ∪ (cid:8) q n n ∈ Z + (cid:9) . In the former case the relations on α and γ mean simply that α is unitary. In the latterwe find that the operator | γ | has discrete spectrum and the space H can be decomposed intoeigenspaces of | γ | : H = (ker γ ) ⊕ (cid:18) ∞ M n =0 H ( | γ | = | q | n ) (cid:19) . The operator Phase γ is zero on ker γ and is unitary on ∞ L n =0 H ( | γ | = | q | n ). Moreover, sincePhase γ commutes with | γ | we see that Phase γ must map each subspace H ( | γ | = | q | n ) intoitself: for ψ ∈ H ( | γ | = | q | n ) we have | γ | (Phase γ ) ψ = (Phase γ ) | γ | ψ = | q | n (Phase γ ) ψ. Next let us analyze the partial isometry Phase α . Its initial projection (Phase α ) ∗ (Phase α ) isthe projection onto H ( | α | 6 = 0) = H ( | γ | 6 = 1), while its final projection (Phase α )(Phase α ) ∗ is the projection onto the closure of the range of α . Note that it follows from α α ∗ = − q | γ | that the range of α is all of H , so(Phase α )(Phase α ) ∗ = . (2.10)Since | α | = − | γ | , the relation α α ∗ = − q | γ | can be rewritten as(Phase α ) (cid:0) − | γ | (cid:1) (Phase α ) ∗ = − q | γ | which in view of (2.10) is (Phase α ) | γ | (Phase α ) ∗ = | q || γ | . (2.11)Now, multiplying (2.11) from the right by Phase α yields(Phase α ) | γ | χ ( | γ | 6 = 1) = | q || γ | (Phase α ) , so on H ( | γ | 6 = 1) we have (Phase α ) | γ | = | q || γ | (Phase α ) . (2.12)It follows that Phase α maps H ( | γ | = 1) to zero and H ( | γ | = | q | n ) into H ( | γ | = | q | n − ) for n >
0. Moreover the mapPhase α (cid:12)(cid:12) H ( | γ | = | q | n ) : H ( | γ | = | q | n ) −→ H ( | γ | = | q | n − ) ENTER OF C(SU q (2)) 5 is onto because the range of Phase α is all of H .Now let us rewrite α γ = qγ α in terms of the respective polar decompositions:(Phase α ) p − | γ | (Phase γ ) | γ | = q (Phase γ ) | γ | (Phase α ) p − | γ | or (Phase α )(Phase γ ) | γ | p − | γ | = q (Phase γ ) | γ | (Phase α ) p − | γ | . On H ( | γ | 6 = 1) the operator p − | γ | is invertible, so(Phase α )(Phase γ ) | γ | = q (Phase γ ) | γ | (Phase α )and using (2.12) we obtain(Phase α )(Phase γ ) | γ | = sgn( q )(Phase γ )(Phase α ) | γ | on H ( | γ | 6 = 1), where sgn( q ) = +1 if q > q ) = − α )(Phase γ ) = sgn( q )(Phase γ )(Phase α ) on H ( | γ | 6 = 1), while on H ( | γ | = 1) wehave Phase α = 0, so (Phase α )(Phase γ ) = sgn( q )(Phase γ )(Phase α ) (2.13)on all of H .We now see that the pair ( α , γ ) is specified uniquely by the normal operator γ and a partialisometry Phase α which is zero on H ( | γ | = 1), maps H ( | γ | = | q | n ) onto H ( | γ | = | q | n − ) for n > γ restricts to a unitary map H ( | γ | = | q | n ) −→ H ( | γ | = | q | n )for each n , we see that all these spaces are isomorphic (the isomorphism H ( | γ | = | q | n ) → H ( | γ | = | q | n +1 ) being provided by (Phase α ) ∗ ) and the action of Phase γ on each of thesespaces is unitarily equivalent to e.g. the one on H ( | γ | = 1). In particular writing K for H ( | γ | = 1) and u for Phase γ (cid:12)(cid:12) K we have a unitary operator U : H ( | γ | 6 = 0) −→ ℓ ( Z + ) ⊗ K with U γ U ∗ = q N ⊗ u ,U α U ∗ = s p − q N ⊗ where s and N are the operators described by (2.7) and (2.8) respectively.Now by universal properties of the C ∗ -algebras T and U ([2, Theorem V.2.2]) there exist uniquerepresentations ̺ : T −→
B(ker γ ) ,̺ : U −→ B( K ) ,̺ : U −→ C such that ̺ ( s ) = (Phase α ) (cid:12)(cid:12) ker γ ,̺ ( u ) = u and ̺ ( u ) = 1. Now let ̺ be the restriction of the mapping T ⊗ U ∋ x (cid:0) ( ̺ ⊗ ̺ )( x ) , U ∗ (id ⊗ ̺ )( x ) U (cid:1) ∈ B(ker γ ) ⊕ B( H ( γ = 0)) ⊂ B( H ) . to π ( A ) ⊂ T ⊗ U . It satisfies ̺ ( α ) = α and ̺ ( γ ) = γ . Uniqueness of ̺ is clear, as C ∗ ( α , γ ) is the smallest C ∗ -algebra of operators on H containing α and γ and the value of ̺ on these operators is specified. (cid:3) Theorem 2.1 immediately implies the following corollary:
Corollary 2.2.
The representation π is faithful. In particular A is isomorphic to the C ∗ -algebragenerated by the operators α and γ . JACEK KRAJCZOK AND PIOTR M. SO LTAN The quantum group SU q (2)In [10] S.L. Woronowicz found that the algebra A described in Section 2 possesses very richstructure. In particular there is a unique ∗ -homomorphism ∆ : A → A ⊗ A called a comultiplication such that ∆( α ) = α ⊗ α − qγ ∗ ⊗ γ, ∆( γ ) = γ ⊗ α + α ∗ ⊗ γ (the tensor product A ⊗ A is unambiguous because A can be shown to be nuclear, cf. [10, AppendixA2]). Moreover one easily sees that ∆ satisfies(∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆i.e. it is coassociative . The pair ( A , ∆) satisfies the conditions of [12, Definition 2.1], so that A is an algebra of functions on a compact quantum group. This quantum group is denoted by thesymbol SU q (2). Hence, the algebra A is also denoted by the symbol C(SU q (2)).The isomorphism π of C(SU q (2)) onto C ∗ ( α , γ ) provides a comultiplication on the latter C ∗ -algebra. However its existence may be proved directly without knowing that π is faithful. In factthe existence of ∆ on C ∗ ( α , γ ) together with several other results can be used to prove that π isa faithful representation (see Section 7).The ∗ -algebra generated by α and γ with appropriate restriction of ∆ is a Hopf ∗ -algebraand we denote it by Pol(SU q (2)). A convenient basis of Pol(SU q (2)) is given by the set { a k,m,n } k ∈ Z , m,n ∈ Z + , where a k,m,n = ( α k γ m γ ∗ n k ≥ ,α ∗− k γ m γ ∗ n k < Z × Z + × Z + labeling the basisby Γ.As any other compact quantum group, the quantum group SU q (2) possesses the Haar measure which is the unique state h on C(SU q (2)) which satisfies( h ⊗ id)∆( a ) = h ( a ) = (id ⊗ h )∆( a ) , a ∈ C(SU q (2)) . (see [10, 9, 12]). The state h was found by S.L. Woronowicz who produced an explicit formula h ( a ) = (1 − q ) ∞ X n =0 q n h e n, a e n, i , a ∈ C(SU q (2)) , where C(SU q (2)) is identified with the C ∗ -algebra of operators on ℓ ( Z + × Z ) generated by α and γ . As noted already in [9] the Haar measure of a compact quantum group need not be faithful.However it is faithful for the quantum SU(2) groups, a fact whose proof we will give in Section 5.4. Center of
C(SU q (2))4.1. Commutant of
C(SU q (2)) . By results of Section 2.1 we can identify C(SU q (2)) with analgebra of operators on the Hilbert space ℓ ( Z + × Z ) which we will continue to denote by H . Tokeep the notation lighter we will denote the set Z + × Z by Λ.We need to introduce a measurable and bounded function f : f : Sp( γ ∗ γ ) = { } ∪ (cid:8) q n (cid:12)(cid:12) n ∈ Z + (cid:9) → C : ( ,q n (sgn( q )) n , n ∈ Z + , and an operator Phase γ = f ( γ ∗ γ ) Phase γ . It’s easy to see that Phase γ acts on H as verticalbilateral shift: (Phase γ ) e n,k = e n,k +1 , ( n, k ) ∈ Λ . We begin by describing the commutant C(SU q (2)) ′ of this algebra of operators. Theorem 4.1.
The commutant of
C(SU q (2)) in B( H ) coincides with the von Neumann algebragenerated by Phase γ . ENTER OF C(SU q (2)) 7 Proof.
Let T ∈ C(SU q (2)) ′ . Define matrix elements { T n,kn ′ ,k ′ } ( n,k ) , ( n ′ ,k ′ ) ∈ Λ of T by T e n,k = X ( n ′ ,k ′ ) ∈ Λ T n,kn ′ ,k ′ e n ′ ,k ′ , ( n, k ) ∈ Λ . (4.1)Fixing ( n, k ) ∈ Λ we compute T γ e n,k = T q n e n,k +1 = X ( n ′ ,k ′ ) ∈ Λ T n,k +1 n ′ ,k ′ q n e n ′ ,k ′ = X ( n ′ ,k ′ ) ∈ Λ T n,k +1 n ′ ,k ′ +1 q n e n ′ ,k ′ +1 , γ T e n,k = γ X ( n ′ ,k ′ ) ∈ Λ T n,kn ′ ,k ′ e n ′ ,k ′ = X ( n ′ ,k ′ ) ∈ Λ T n,kn ′ ,k ′ q n ′ e n ′ ,k ′ +1 . As T commutes with γ , we obtain T n,k +1 n ′ ,k ′ +1 q n = T n,kn ′ ,k ′ q n ′ , ( n, k ) , ( n ′ , k ′ ) ∈ Λ . (4.2)Similarly the computation T γ ∗ e n,k = T q n e n,k − = X ( n ′ ,k ′ ) ∈ Λ T n,k − n ′ ,k ′ q n e n ′ ,k ′ = X ( n ′ ,k ′ ) ∈ Λ T n,k − n ′ ,k ′ − q n e n ′ ,k ′ − , γ ∗ T e n,k = γ ∗ X ( n ′ ,k ′ ) ∈ Λ T n,kn ′ ,k ′ e n ′ ,k ′ = X ( n ′ ,k ′ ) ∈ Λ T n,kn ′ ,k ′ q n ′ e n ′ ,k ′ − . and the fact that T commutes with γ ∗ give T n,k − n ′ ,k ′ − q n = T n,kn ′ ,k ′ q n ′ , ( n, k ) , ( n ′ , k ′ ) ∈ Λ . (4.3)Now take ( n, k ) , ( n ′ , k ′ ) ∈ Λ. Combining (4.2) and (4.3) we obtain T n,kn ′ ,k ′ (4.2) = q n − n ′ T n,k +1 n ′ ,k ′ +1 (4.3) = q n − n ′ ) T n,kn ′ ,k ′ . which implies that T n,kn ′ ,k ′ = 0 , ( n, k ) , ( n ′ , k ′ ) ∈ Λ , n = n ′ . (4.4)Using this we compute for ( n, k ) ∈ Λ T α ∗ e n,k = T p − q n +1) e n +1 ,k = X k ′ ∈ Z T n +1 ,kn +1 ,k ′ p − q n +1) e n +1 ,k ′ , α ∗ T e n,k = α ∗ X k ′ ∈ Z T n,kn,k ′ e n,k ′ = X k ′ ∈ Z T n,kn,k ′ p − q n +1) e n +1 ,k ′ , and using this together with (4.4) and (4.3) we get T n,kn ′ ,k ′ = δ n,n ′ T ,k ,k ′ = δ n,n ′ T ,k − k ′ , , ( n, k ) , ( n ′ , k ′ ) ∈ Λ . (4.5)We are now ready to finish the proof. Denote by vN(Phase γ ) the von Neumann algebragenerated by Phase γ . The inclusionvN(Phase γ ) ⊂ C(SU q (2)) ′ is clear, as Phase γ commutes with α and γ (cf. (2.6)).Now take T ∈ C(SU q (2)) ′ and S ∈ vN(Phase γ ) ′ with matrix elements { S n,kn ′ ,k ′ } ( n,k ) , ( n ′ ,k ′ ) ∈ Λ .For k ∈ N and ( n, k ) ∈ Λ we have S (Phase γ ) k e n,k = X ( n ′ ,k ′ ) ∈ Λ S n,k + k n ′ ,k ′ e n ′ ,k ′ = X ( n ′ ,k ′ ) ∈ Λ S n,k + k n ′ ,k ′ + k e n ′ ,k ′ + k , (Phase γ ) k Se n,k = X ( n ′ ,k ′ ) ∈ Λ S n,kn ′ ,k ′ e n ′ ,k ′ + k , so as S commutes with Phase γ , we obtain S n,kn ′ ,k ′ = S n,k + k n ′ ,k ′ + k ( n, k ) , ( n ′ , k ′ ) ∈ Λ , k ∈ Z . (4.6) JACEK KRAJCZOK AND PIOTR M. SO LTAN
Now (4.5) and (4.6) show that T commutes with S : h e l,p T Se n,k i = * e l,p T X ( n ′ ,k ′ ) ∈ Λ S n,kn ′ ,k ′ e n ′ ,k ′ + = * e l,p X ( n ′ ,k ′ ) ∈ Λ S n,kn ′ ,k ′ X k ′′ ∈ Z T ,k ′ − k ′′ , e n ′ ,k ′′ + = X k ′ ∈ Z S n,kl,k ′ T ,k ′ − p , = X k ′ ∈ Z S n,kl,k + p − k ′ T ,k − k ′ , = X k ′ ∈ Z S n,k ′ l,p T ,k − k ′ , and h e l,p ST e n,k i = * e l,p S X k ′′ ∈ Z T ,k − k ′′ , e n,k ′′ + = * e l,p X k ′′ ∈ Z T ,k − k ′′ , X ( n ′ ,k ′ ) ∈ Λ S n,k ′′ n ′ ,k ′ e n ′ ,k ′ + = X k ′′ ∈ Z T ,k − k ′′ , S n,k ′′ l,p . Since S is arbitrary in vN(Phase γ ) ′ , this means that T ∈ vN(Phase γ ) ′′ = vN(Phase γ ), sovN(Phase γ ) ⊃ C(SU q (2)) ′ . (cid:3) Remark 4.2.
Let us point out that the proof of Theorem 4.1 does not depend on the analysisof the commutation relations (2.1) presented in the proof of Theorem 2.1. Using this analysis onecan obtain information on the center of the von Neumann algebra generated by the image of arepresentation of C(SU q (2)) (cf. Section 6 and Theorem 6.2).4.2. Center of
C(SU q (2)) . As C(SU q (2)) is a unital algebra, its center is non-zero because itmust contain all scalar multiples of the unit . We will show that there are no other centralelements in C(SU q (2)) or, in other words, that the center of C(SU q (2)) is trivial .In the proof of the next theorem we will identify C(SU q (2)) with the C ∗ -algebra of operatorson ℓ ( Z + × Z ) generated by α and γ . Theorem 4.3.
The center of
C(SU q (2)) is trivial.Proof. Let T be a central element of C(SU q (2)). As Pol(SU q (2)) is dense in C(SU q (2)). Theelement T can be approximated in norm by finite linear combinations of elements of the basis(3.1): T = lim λ →∞ X ( k,m,n ) ∈ Γ C λk,m,n a k,m,n . Given ε > λ ∈ N such that for any λ ≥ λ we have (cid:13)(cid:13)(cid:13)(cid:13) X ( k,m,n ) ∈ Γ C λk,m,n a k,m,n − T (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε . As in the proof of Theorem 4.1 we will use matrix elements { T n,kn ′ ,k ′ } ( n,k ) , ( n ′ ,k ′ ) ∈ Λ of T as definedby (4.1).Take λ ≥ λ and l ∈ Z + . Using (4.5), (2.4) and (2.5) we obtain (cid:0) ε (cid:1) ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) X ( k,m,n ) ∈ Γ C λk,m,n a k,m,n − T (cid:19) e l, (cid:13)(cid:13)(cid:13)(cid:13) q (2)) 9 = (cid:13)(cid:13)(cid:13)(cid:13) X k ∈{ ,...,l } m,n ∈ Z + C λk,m,n q l ( m + n ) k − Y i =0 p − q l − i ) e l − k,m − n ++ X k ∈− N m,n ∈ Z + C λk,m,n q l ( m + n ) | k |− Y i =0 p − q l +1+ i ) e l + | k | ,m − n − X k ∈ Z T , − k , e l,k (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) X k ∈{ ,...,l } s ∈ Z (cid:18) X m,n ∈ Z + m − n = s C λk,m,n q l ( m + n ) k − Y i =0 p − q l − i ) (cid:19) e l − k,s + X k ∈− N s ∈ Z (cid:18) X m,n ∈ Z + m − n = s C λk,m,n q l ( m + n ) | k |− Y i =0 p − q l +1+ i ) (cid:19) e l + | k | ,s + X s ∈ Z (cid:18) X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) − T , − s , (cid:19) e l,s (cid:13)(cid:13)(cid:13)(cid:13) = X k ∈{ ,...,l } s ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) X m,n ∈ Z + m − n = s C λk,m,n q l ( m + n ) k − Y i =0 p − q l − i ) (cid:12)(cid:12)(cid:12)(cid:12) + X k ∈− N s ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) X m,n ∈ Z + m − n = s C λk,m,n q l ( m + n ) | k |− Y i =0 p − q l +1+ i ) (cid:12)(cid:12)(cid:12)(cid:12) + X s ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) − T , − s , (cid:12)(cid:12)(cid:12)(cid:12) ≥ X s ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) − T , − s , (cid:12)(cid:12)(cid:12)(cid:12) . Thus ∀ ε> ∃ λ ∈ N ∀ l ∈ Z + ∀ s ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) − T , − s , (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε , so in particular ∀ ε> ∀ s ∈ Z ∃ λ ∈ N ∀ l ∈ Z + (cid:12)(cid:12)(cid:12)(cid:12) X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) − T , − s , (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . (4.7)Let us take s ∈ Z \ { } , ε > λ ∈ N be determined by (4.7). Sincelim l →∞ X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) = 0 , there exists l ∈ N such that (cid:12)(cid:12)(cid:12)(cid:12) X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε
20 JACEK KRAJCZOK AND PIOTR M. SO LTAN (we can pass to the limit under the sum because only a finite number of its terms are non-zero).Combining this with (4.7) we obtain | T , − s , | ≤ (cid:12)(cid:12)(cid:12)(cid:12) T , − s , − X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) X m,n ∈ Z + m − n = s C λ ,m,n q l ( m + n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε + ε = ε, so for s ∈ Z \ { } we have | T , − s , | = 0. Consequently T = T , , which ends the proof. (cid:3) An immediate consequence of Theorem 4.3 is the following:
Corollary 4.4.
The center of
Pol(SU q (2)) is trivial. Let us note that Corollary 4.4 can also be easily proved directly by writing a central elementof Pol(SU q (2)) as a linear combination of the basis (3.1) and checking the conditions implied bycommutation with the generators α and γ .5. Faithfulness of Haar measure and continuity of counit
Faithfulness of the Haar measure.Proposition 5.1.
The Haar measure of the quantum group SU q (2) is faithful.Proof. Recall that for a ∈ C(SU q (2)) ⊂ B( ℓ ( Z + × Z )) we have h ( a ) = (1 − q ) ∞ X n =0 q n h e n, a e n, i . Assume now that h ( b ∗ b ) = 0 for some b ∈ C(SU q (2)). Then for all n we have k b e n, k = 0.Moreover for any k ∈ Z we have k b e n,k k = h b e n,k b e n,k i = (cid:10) b (Phase γ ) k e n, b (Phase γ ) k e n, (cid:11) = (cid:10) (Phase γ ) k b e n, (Phase γ ) k b e n, (cid:11) = h b e n, b e n, i = k b e n, k = 0because b commutes with Phase γ (cf. Theorem 4.1). It follows that b is zero on all elements ofthe standard basis of ℓ ( Z + × Z ), so b = 0. (cid:3) The counit.
The counit of SU q (2) is the ∗ -character ε of Pol(SU q (2)) determined uniquelyby ε ( α ) = 1 and ε ( γ ) = 0. By universal property of the C ∗ -algebra A the counit extends to acharacter A → C . By Theorem 2.1 ε can be extended from the ∗ -algebra generated by α and γ to C ∗ ( α , γ ). In this section we will provide two different proofs of this fact independent ofTheorem 2.1 (faithfulness of π ). This may be used to give an alternative proof of faithfulness ofthe representation π (cf. Section 7). Theorem 5.2.
There exists a character e ε : C ∗ ( α , γ ) → C such that e ε ( α ) = 1 and e ε ( γ ) = 0 . (5.1)5.2.1. Direct method.
First note that we can easily determine ε on elements of the basis (3.1) ofPol(SU q (2)): ε ( a k,m,n ) = ( m = n = 0 , . Thus in order to show that the counit extends from Pol(SU q (2)) to C ∗ ( α , γ ) it will be enoughto exhibit a continuous linear functional ω on B( H ) such that ω ( α k γ m γ ∗ n ) = δ m, δ n, = ω ( α ∗ k γ m γ ∗ n ) for all k, m, n ∈ Z + . Indeed, as the counit is multiplicative on Pol(SU q (2)) itis easy to see that the restriction e ε of ω to C ∗ ( α , γ ) is a character which maps α to 1 and γ to 0. ENTER OF C(SU q (2)) 11 Define a sequence ( ω L ) L ∈ N of functionals on B( H ): ω L ( T ) = L +1 * L X l =0 e l, T L X l =0 e l, + , T ∈ B( H ) . Clearly each ω L is positive, and so k ω L k = ω L ( ) = 1 for all L . Standard facts about weaktopologies ([8, Chapter 1]) show that there exists a subsequence ( ω L p ) p ∈ N of ( ω L ) L ∈ N weak ∗ convergent to ω ∈ B( H ) ∗ : ω = w ∗ - lim p →∞ ω L p (moreover k ω k = 1). We will show that e ε = ω (cid:12)(cid:12) C ∗ ( α , γ ) satisfies (5.1). Lemma 5.3.
For k ∈ Z \ { } and L ∈ N define Ξ k,L = L − k P l =0 k − Q i =0 p − q l + k − i ) k > , L −| k | P l =0 | k |− Q i =0 p − q l +1+ i ) k < . Then for any k ∈ Z \ { } we have lim L →∞ Ξ k,L L +1 = 1 .Proof. Fix k ∈ Z \ { } and ( θ, ± ) = ( ( k, − ) k > , (1 , +) k < . The series ∞ X l =0 (cid:18) − | k |− Y i =0 p − q l + θ ± i ) (cid:19) is convergent to some G k ∈ R . Indeed, using in the first step L’Hˆopital’s rule we obtainlim l →∞ − | k |− Q i =0 p − q l +1+ θ ± i ) − | k |− Q i =0 p − q l + θ ± i ) H = lim l →∞ − | k |− P j =0 (cid:18) Q i ∈{ ,..., | k |− }\{ j } p − q l +1+ θ ± i ) (cid:19) − q l +1+ θ ± j ) log( q )2 √ − q l +1+ θ ± j ) − | k |− P j =0 (cid:18) Q i ∈{ ,..., | k |− }\{ j } p − q l + θ ± i ) (cid:19) − q l + θ ± j ) log( q )2 √ − q l + θ ± j ) = lim l →∞ | k |− P j =0 (cid:18) Q i ∈{ ,..., | k |− }\{ j } p − q l +1+ θ ± i ) (cid:19) q θ ± j ) √ − q l +1+ θ ± j ) | k |− P j =0 (cid:18) Q i ∈{ ,..., | k |− }\{ j } p − q l + θ ± i ) (cid:19) q θ ± j ) √ − q l + θ ± j ) = | k |− P j =0 q θ ± j ) | k |− P j =0 q θ ± j ) = q < , so the series is convergent by d’Alembert’s ratio test.Then for any L ≥ | k | the numbers R L,k = G k − L −| k | X l =0 (cid:18) − | k |− Y i =0 p − q l + θ ± i ) (cid:19) satisfy lim L →∞ R L,k = 0. After simple manipulation we arrive atΞ k,L = L −| k | X l =0 | k |− Y i =0 p − q l + θ ± i ) = L + 1 − | k | − G k + R kL , which proves the lemma. (cid:3) We can now check values of ω on basic elements: choose k, m, n ∈ Z + and L ∈ N . We have ω L ( α k γ m γ ∗ n ) = L +1 * L X l =0 e l, α k γ m γ ∗ n L X l ′ =0 e l ′ , + = L +1 L X l =0 L X l ′ =0 [ l ′ − k ≥ * e l, q l ′ ( m + n ) k − Y i =0 p − q l ′ − i ) e l ′ − k,m − n + = [ m = n ] L +1 L X l =0 (cid:2) k + l ∈ { , . . . , L } (cid:3) q k + l ) n k − Y i =0 p − q k + l − i ) = (cid:2) ( m = n ) ∧ ( L − k ≥ (cid:3) L +1 L − k X l =0 q k + l ) n k − Y i =0 p − q k + l − i ) , where [ · · · ] denotes 1 if the logical expression in brackets is true and 0 otherwise. It follows thatfor m = n ω ( α k γ m γ ∗ n ) = lim p →∞ ω L p ( α k γ m γ ∗ n ) = 0 . When m = n > p →∞ L p − k X l =0 q k + l ) n k − Y i =0 p − q k + l − i ) ≤ q kn ∞ X l =0 q ln = q kn − q n < ∞ for all p , so ω ( α k γ m γ ∗ n ) = lim p →∞ ω L p ( α k γ m γ ∗ n )= lim p →∞ L p +1 L p − k X l =0 q k + l ) n k − Y i =0 p − q k + l − i ) = 0 . Similarly ω L ( α ∗ k γ m γ ∗ n ) = L +1 * L X l =0 e l, α ∗ k γ m γ ∗ n L X l ′ =0 e l ′ , + = L +1 L X l =0 L X l ′ =0 * e l, q l ′ ( m + n ) k − Y i =0 p − q l ′ +1+ i ) e l ′ + k,m − n + = [ m = n ] 1 L + 1 L X l =0 (cid:2) − k + l ∈ { , . . . , L } (cid:3) q − k + l ) n k − Y i =0 p − q − k + l +1+ i ) = (cid:2) ( m = n ) ∧ ( L ≥ k ) (cid:3) L +1 L X l = k q − k + l ) n k − Y i =0 p − q − k + l +1+ i ) = (cid:2) ( m = n ) ∧ ( L ≥ k ) (cid:3) L +1 L − k X l =0 q ln k − Y i =0 p − q l +1+ i ) . and arguing as before we obtain ω ( α ∗ k γ m γ ∗ n ) for ( m, n ) = (0 , ENTER OF C(SU q (2)) 13 Finally for m = n = 0 and any k ∈ N we have by Lemma 5.3 ω ( α k ) = lim p →∞ L p +1 L p − k X l =0 k − Y i =0 p − q l + k − i ) = lim p →∞ Ξ k,Lp L p +1 = 1 ,ω ( α ∗ k ) = lim p →∞ L p +1 L p − k X l =0 k − Y i =0 p − q l +1+ i ) = lim p →∞ Ξ − k,L p L p + 1 = 1 . As ω ( ) = k ω k = 1 we see that ω has the same values on basic elements of Pol(SU q (2)) (embeddedin C ∗ ( α , γ ) ⊂ B( H )) as the counit. Remark 5.4.
Consider now functionals { ω L } L ∈ N restricted to C ∗ ( α , γ ). Then the above argu-ments show in fact that the sequence ( ω L ) L ∈ N converges in the weak ∗ topology on C ∗ ( α , γ ) ∗ to e ε .Indeed, assuming that ( ω L ) L ∈ N is not convergent we would choose a subsequence ( ω L ′ p ) p ∈ N all ofwhose elements lie outside a fixed weak ∗ neighborhood O of e ε . But this subsequence would stillhave a convergent subsequence ( ω L ′ ps ) s ∈ N which by the same arguments as above can be shown toconverge on elements { α k γ m γ ∗ n } and { α ∗ k γ m γ ∗ n } to δ m, δ n, which by density of the span ofthese elements in C ∗ ( α , γ ) contradicts the fact that elements of ( ω L ′ p ) p ∈ N lie outside of O .5.2.2. Another method.
Instead of exhibiting a concrete sequence of vector functionals convergingto an extension of ε to C ∗ ( α , γ ) we can use the structure of the pair ( α , γ ) described in Section2.1 and the universal property of the Toeplitz algebra to show existence of e ε . Indeed, as we have α = s p − q N ⊗ , γ = q N ⊗ u , (when H is naturally identified with ℓ ( Z + ) ⊗ ℓ ( Z )) the C ∗ -algebra C ∗ ( α , γ ) is contained in T ⊗ U (cf. Section 2.1). By universal properties of T and U there are characters ϕ and ψ of thesealgebras such that ϕ ( s ) = 1 = ψ ( u ). Moreover, noting that q N = ∞ X n =0 q n s ∗ n ( − s ∗ s ) s n we find that( ϕ ⊗ ψ )( γ ) = ( ϕ ⊗ ψ )( q N ⊗ u (cid:1) = ϕ ( q N ) = ϕ (cid:18) ∞ X n =0 q n s ∗ n ( − s ∗ s ) s n (cid:19) = 0and ( ϕ ⊗ ψ )( α ) = ϕ (cid:0) s p − q N (cid:1) = ϕ ( s ) ϕ (cid:0)p − q N (cid:1) = ϕ ( s ) ϕ (cid:0) ) = 1 . Clearly restriction of ϕ ⊗ ψ to C ∗ ( α , γ ) is the extension of the counit of SU q (2).6. The GNS representation for h and center of L ∞ (SU q (2))We begin this section with a concrete realization of the GNS representation of C(SU q (2)) forthe Haar measure. Recall that h ( a ) = (1 − q ) ∞ X n =0 q n h e n, a e n, i , a ∈ C(SU q (2)) . Let ( H h , Ω h , π h ) be the GNS triple for h . We already know that h is faithful on C(SU q (2)), so π h is faithful.As before we write H for the carrier Hilbert space of the distinguished representation π con-sidered in Section 2.1. Consider c H = ∞ L n =0 H and a vector b Ω = p − q q e , q e , q e , ... ∈ c H . We have a representation b π of C(SU q (2)) on c H : b π ( a ) = a a a . . . . The subspace c K = (cid:8)b π ( a ) b Ω a ∈ C(SU q (2)) (cid:9) is invariant for the action of C(SU q (2)) and theresulting representation of C(SU q (2)) is equivalent to π h . The appropriate unitary operator H h → c K is given by π h ( a )Ω h b π ( a ) b Ω , a ∈ C(SU q (2)) . As an immediate corollary we thus get the following:
Proposition 6.1.
The operator π h ( γ ) has zero kernel.Proof. Since ker γ = { } , we have ker b π ( γ ) = { } and π h ( γ ) is unitarily equivalent to a restrictionof b π ( γ ) to the subspace c K . (cid:3) The algebra L ∞ (SU q (2)) is by definition the strong closure of π h (cid:0) C(SU q (2)) (cid:1) in B( H h ). It is avon Neumann algebra and it is referred to as the algebra of essentially bounded functions on thequantum group SU q (2).Recall that in Section 4 we’ve introduced a function f : f : Sp( π h ( γ ∗ γ )) = { } ∪ (cid:8) q n (cid:12)(cid:12) n ∈ Z + (cid:9) → C : ( ,q n (sgn( q )) n , n ∈ Z + . Let’s define an operator Phase π h ( γ ) = f ( π h ( γ ∗ γ )) Phase π h ( γ ) ∈ L ∞ (SU q (2)). Theorem 6.2.
The center of L ∞ (SU q (2)) is the von Neumann subalgebra of L ∞ (SU q (2)) gener-ated by Phase π h ( γ ) .Proof. Let α and γ be images of α and γ in the representation π h . We know already thatker γ = { } , so by the reasoning presented in the proof of Theorem 2.1 we have U γ U ∗ = q N ⊗ u ,U α U ∗ = s p − q N ⊗ , where U is a unitary H h → ℓ ( Z + ) ⊗ K for some Hilbert space K and a unitary u on K .Consequently the algebra generated by Phase γ = U ∗ ( ⊗ u ) U is contained in the center ofthe von Neumann algebra generated by α and γ .On the other hand, if T ∈ B( ℓ ( Z + ) ⊗ K ) = B( ℓ ( Z + )) ¯ ⊗ B( K ) commutes with U α U ∗ and U γ U ∗ then it must commute with q N ⊗ and s p − q N ⊗ . However, the C ∗ -algebragenerated by these two operators is K ⊗ where K is the algebra of compact operators on ℓ ( Z + ).It follows that T ∈ ⊗ B( H ).It follows that the center of vN( U α U ∗ , U γ U ∗ ) is the von Neumann algebra generated by ⊗ u which is obviously contained in vN( U α U ∗ , U γ U ∗ ). Consequently the center of vN( α , γ )is generated by U ∗ ( ⊗ u ) U = Phase γ . (cid:3) Another proof of faithfulness of π In this section we will outline an alternative way to prove faithfulness of the representation π considered in Section 2.1.First let us note the following facts ◮ π is an epimorphism from the universal C ∗ -algebra A generated by α and γ satisfying(2.1) onto C ∗ ( α , γ ) acting on H = ℓ ( Z + × Z ); ENTER OF C(SU q (2)) 15 ◮ the Haar functional on A factorizes through π and a faithful functional h : a (1 − q ) ∞ X n =0 q n h e n, a e n, i on C ∗ ( α , γ ) (by Proposition 5.1); ◮ there exists a comultiplication on C ∗ ( α , γ ) which agrees with the comultiplication on A (this follows from results of either [11] or [4] and is rather non-trivial), in particularC ∗ ( α , γ ) with this comultiplication carries the structure of an algebra of functions on acompact quantum group G ; ◮ the functional h is the Haar measure of this quantum group; ◮ the map π is injective on the ∗ -algebra Pol(SU q (2)) generated inside A by α and γ ([10,Theorem 1.2]), in particular we may view C ∗ ( α , γ ) as a completion of Pol(SU q (2)) for aquantum group norm ([3, Definition 7.1]); ◮ the algebra C ∗ ( α , γ ) admits a character which is a counit for G .By results of [1] a Hopf ∗ -algebra which admits a completion with continuous counit and faithfulHaar measure admits a unique compact quantum group completion. In particular A and C ∗ ( α , γ )must be the same and consequently π must be faithful. References [1] E. B´edos, G. J. Murphy and L. Tuset,
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