The Podles spheres converge to the sphere
aa r X i v : . [ m a t h . OA ] F e b THE PODLEŚ SPHERES CONVERGE TO THE SPHERE
KONRAD AGUILAR, JENS KAAD, AND DAVID KYED
Abstract.
We prove that the Podleś spheres S q converge in quantum Gromov-Hausdorffdistance to the classical 2-sphere as the deformation parameter q tends to . Moreover,we construct a q -deformed analogue of the fuzzy spheres, and prove that they converge to S q as their linear dimension tends to infinity, thus providing a quantum counterpart to aclassical result of Rieffel. Contents
1. Introduction 11.1. Acknowledgements 32. Preliminaries 32.1. Quantum SU (2) Introduction
The theory of C ∗ -algebras provides a vast noncommutative generalisation of the theoryof Hausdorff topological spaces, and by imposing suitable additional structure, one obtainsnoncommutative (or quantum) analogues of more sophisticated topological spaces. Two Mathematics Subject Classification.
Key words and phrases.
Quantum metric spaces, fuzzy spheres, Podleś sphere, spectral triples, quantumGromov-Hausdorff distance. very successful examples of this phenomenon are the theory of quantum groups [29, 49]which are noncommutative analogues of topological groups, and Connes’ notion of spectraltriples [10], which are the noncommutative counterparts to (spin) Riemannian manifolds.In the same vein, it is also very natural to ask for a noncommutative generalisation ofordinary metric spaces, and Rieffel’s seminal work [43, 44] provides a very satisfactoryanswer to this question. Rieffel’s fundamental insight is that the right noncommutativecounterpart to a metric on a compact topological space is a certain densely defined semi-norm on a unital C ∗ -algebra, and he dubbed these structures compact quantum metricspaces . Over the past 20 years, ample examples of compact quantum metric spaces haveemerged, and the theory has been developed in several different directions through theworks of many hands; see [3, 6, 21, 26, 33, 36, 42] and references therin. One of the mostimportant features of the theory is that it admits a generalisation of the classical Gromov-Hausdorff distance [15, 18], known as the quantum Gromov-Hausdorff distance [44], whichallows one to study the theory of quantum metric spaces from an analytic point of view,and thus ask questions pertaining to continuity and convergence. As an example of this,Rieffel showed in [44] that the noncommutative tori A θ admit a natural quantum metricstructure and that they vary continuously in the deformation parameter θ with respect tothe quantum Gromov-Hausdorff distance, and in [45] that the so-called fuzzy-spheres alsoadmit a compact quantum metric structure with respect to which they converge to theclassical 2-sphere S as their linear dimension tends to infinity; for many more examplesin this direction see for instance [1, 22, 30, 34].The definition of compact quantum metric spaces is inspired by Connes’ theory of non-commutative geometry, and the latter is therefore a natural source of many interestingexamples, which may be viewed as the noncommutative counterparts of Riemannian man-ifolds when these are considered merely as metric spaces with their Riemannian metric.Despite a continuous effort over at least 30 years, it has proven quite difficult to recon-cile the theory of quantum groups with Connes’ noncommutative geometry, [11, 38]. Infact, even for the most fundamental example, Woronowicz’ quantum SU (2) , there arestill several competing candidates for good spectral triples, and it is not known which ofthese provide quantum SU (2) with a quantum metric space structure, [7, 13, 23, 28, 40].However, just as SU (2) has the classical -sphere as a homogeneous space, its quantizedcounterpart SU q (2) also has a quantised -sphere S q , known as the standard Podleś sphere,as a “homogenous space” [41], and the work of Dąbrowski and Sitarz [14] provides S q witha spectral triple, which was shown in [2] to turn S q into a compact quantum metric space.This result provides the first genuine quantum analogue of a Riemannian geometry on S q ,and the most pertinent question to investigate at this point is therefore if the quantised -spheres S q converge to the classical round -sphere as the deformation parameter q tendsto 1. The present paper answers this questions in the affirmative: Theorem A (see Theorem 4.18) . As q tends to 1, the Podleś spheres S q converge to theclassical -sphere S in the quantum Gromov-Hausdorff distance. One of the main ingredients in the proof of Theorem A is a quantum analogue of Rieffel’sconvergence result for fuzzy spheres mentioned above. More precisely, we introduce a
HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 3 sequence of finite dimensional compact quantum metric spaces ( F Nq ) N ∈ N , which play therole of quantised counterparts to the classical fuzzy spheres, and prove the following result: Theorem B (See Theorem 4.17) . For each q ∈ (0 , the sequence of quantised fuzzyspheres (cid:0) F Nq (cid:1) N ∈ N converges to S q with respect to the quantum Gromov-Hausdorff distance. For q = 1 , Theorem B provides a variation of Rieffel’s result [45, Theorem 3.2], in thatit shows that the finite dimensional quantum metric spaces F N converge to the classicalround 2-sphere as N tends to infinity. Along the way, we also prove (see Proposition4.15) that for fixed N ∈ N , the 1-parameter family ( F Nq ) q ∈ (0 , of compact quantum metricspaces vary continuously in the quantum Gromov-Hausdorff distance.The rest of the paper is structured as follows: in Section 2 we give a detailed introductionto quantum SU (2) , the noncommutative geometry of the standard Podleś sphere andcompact quantum metric spaces, while Section 3 is devoted to introducing the quantisedfuzzy spheres mentioned above. In Section 4 we carry out the main analysis and prove ourconvergence results.1.1. Acknowledgements.
The authors gratefully acknowledge the financial support fromthe Independent Research Fund Denmark through grant no. 9040-00107B and 7014-00145B.2.
Preliminaries
Quantum SU (2) . Let us fix a q ∈ (0 , . We consider the universal unital C ∗ -algebra C ( SU q (2)) with two generators a and b subject to the relations ba = qab b ∗ a = qab ∗ bb ∗ = b ∗ ba ∗ a + q bb ∗ = 1 = aa ∗ + bb ∗ . This unital C ∗ -algebra is referred to as quantum SU (2) and was introduced by Woronowiczin [48]. Notice here that we are conforming with the notation from [2, 14] which is alsoknown as Majid’s lexicographic convention, see [37]. With these conventions the funda-mental corepresentation unitary takes the form u = (cid:18) a ∗ − qbb ∗ a (cid:19) . We let O ( SU q (2)) ⊆ C ( SU q (2)) denote the unital ∗ -subalgebra generated by a and b and refer to this unital ∗ -subalgebra as the coordinate algebra . The coordinate algebracan be given the structure of a Hopf ∗ -algebra where the coproduct ∆ : O ( SU q (2)) →O ( SU q (2)) ⊗O ( SU q (2)) , antipode S : O ( SU q (2)) → O ( SU q (2)) and counit ǫ : O ( SU q (2)) → C are defined on the fundamental unitary by ∆( u ) = u ⊗ u, S ( u ) = u ∗ and ǫ ( u ) = (cid:18) (cid:19) . KONRAD AGUILAR, JENS KAAD, AND DAVID KYED
For q = 1 , we also consider the universal unital ∗ -algebra U q ( su (2)) with generators e, f, k satisfying the relations kk − = 1 = k − k , ek = qke , kf = qf k and k − k − q − q − = f e − ef and with involution defined by e ∗ = f , f ∗ = e and k ∗ = k . We refer to this unital ∗ -algebraas the quantum enveloping algebra . The quantum enveloping algebra also becomes a Hopf ∗ -algebra with comultiplication, antipode and counit determined by ∆( e ) = e ⊗ k + k − ⊗ e S ( e ) = − q − e ǫ ( e ) = 0∆( f ) = f ⊗ k + k − ⊗ f S ( f ) = − qf ǫ ( f ) = 0 (2.1) ∆( k ) = k ⊗ k S ( k ) = k − ǫ ( k ) = 1 We are here again conforming with the notations from [14]. The quantum envelopingalgebra U q ( su (2)) is seen to be isomorphic, as a Hopf algebra, ˘U q ( sl ) with generators E, F, K from Klimyk and Schmüdgen [27, Chapter 3], by using the dictionary e F , f E , k K . For q = 1 we furthermore consider the universal enveloping Lie algebra U ( su (2)) with generators e, f, h satisfying the relations [ h, e ] = − e [ h, f ] = 2 f [ f, e ] = h and with involution defined by e ∗ = f , f ∗ = e and h ∗ = h . It too becomes a Hopf ∗ -algebrawith comultiplication, antipode and counit given by ∆( e ) = e ⊗ ⊗ e S ( e ) = − e ǫ ( e ) = 0∆( f ) = f ⊗ ⊗ f S ( f ) = − f ǫ ( f ) = 0 (2.2) ∆( h ) = h ⊗ ⊗ h S ( h ) = − h ǫ ( h ) = 0 Notice that O ( SU q (2)) agrees with classical coordinate algebra O ( SU (2)) for q = 1 . How-ever, for the quantum enveloping algebra the relationship is slightly more subtle: we obtainthe classical universal enveloping Lie algebra by formally putting h := 2 log( k ) / log( q ) sothat k = e log( q ) h/ and then letting log( q ) tend to zero. For more information on thesematters, we refer to [27, Section 3.1.3]. In order to unify our notation in the rest of thepaper we apply the convention that k = 1 ∈ U ( su (2)) =: U ( su (2)) .The algebras O ( SU q (2)) and U q ( su (2)) are linked by a non-degenerate dual pairing ofHopf ∗ -algebras h· , ·i : U q ( su (2)) × O ( SU q (2)) → C , which for q = 1 is given by h k, a i = q / h e, a i = 0 h f, a i = 0 h k, a ∗ i = q − / h e, a ∗ i = 0 h f, a ∗ i = 0 h k, b i = 0 h e, b i = − q − h f, b i = 0 h k, b ∗ i = 0 h e, b ∗ i = 0 h f, b ∗ i = 1 HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 5
In the case where q = 1 , this pairing is determined by h h, a i = 1 h e, a i = 0 h f, a i = 0 h h, a ∗ i = − h e, a ∗ i = 0 h f, a ∗ i = 0 h h, b i = 0 h e, b i = − h f, b i = 0 h h, b ∗ i = 0 h e, b ∗ i = 0 h f, b ∗ i = 1 See [27, Chapter 4, Theorem 21] for more details.The above dual pairing of Hopf ∗ -algebras yields a left and a right action of U q ( su (2)) on O ( SU q (2)) . For each η ∈ U q ( su (2)) the corresponding (linear) endomorphisms of O ( SU q (2)) are defined by δ η ( x ) := ( h η, ·i ⊗ x ) and ∂ η ( x ) := (1 ⊗ h η, ·i )∆( x ) , respectively.As it turns out, we shall need a rather detailed description of the irreducible representa-tions of U q ( su (2)) , and to this end the following notation is convenient: for q ∈ (0 , and n ∈ N , we define h n i := n − X m =0 q m . We also put h i := 0 . For q = 1 we of course have h n i = n , and for q = 1 , the relationshipwith the usual q -integers [ n ] := q n − q − n q − q − is given by h n i = q n − [ n ] .For each n ∈ N we have an irreducible ∗ -representation of U q ( su (2)) on the Hilbertspace C n +1 with standard orthonormal basis { e j } nj =0 . This irreducible ∗ -representation isgiven on generators by σ n ( k )( e j ) = q j − n/ · e j σ n ( e )( e j ) = q − n p h n − j + 1 ih j i · e j − σ n ( f )( e j ) = q − n p h n − j ih j + 1 i · e j +1 in the case where q = 1 and by σ n ( h )( e j ) = (2 j − n ) · e j σ n ( e )( e j ) = p ( n − j + 1) j · e j − σ n ( f )( e j ) = p ( n − j )( j + 1) · e j +1 in the case where q = 1 , see [27, Chapter 3, Theorem 13]. The above sequence of ir-reducible ∗ -representations together with the non-degenerate pairing h· , ·i : U q ( su (2)) ×O ( SU q (2)) → C gives rise to a complete set of irreducible corepresentation unitaries KONRAD AGUILAR, JENS KAAD, AND DAVID KYED u n ∈ M n +1 ( O ( SU q (2))) , n ∈ N . Indeed, the entries in u n are characterised by the identity σ n ( η )( e j ) = n X i =0 h η, u nij i · e i , which holds for all η ∈ U q ( su (2)) and all j ∈ { , , . . . , n } , see [27, Chapter 4, Proposition16 & 19] for this. We record that u = 1 and u = u . Notice here that we are applyinga different convention than Klimyk and Schmüdgen [27] who denote the unitary corep-resentations by { t l } l ∈ N . The relationship with our notation can be summarised by theidentities u nij = t n/ i − n/ ,j − n/ for n ∈ N and i, j ∈ { , , . . . , n } .When q = 1 , we obtain from the definition of the irreducible corepresentation unitariesthat we have the following formulae h k, u nij i = δ ij · q j − n/ h e, u nij i = δ i,j − · q − n p h n − j + 1 ih j ih f, u nij i = δ i,j +1 · q − n p h n − j ih j + 1 i (2.3)describing the pairing between the entries of the unitaries and the generators for U q ( su (2)) .For q = 1 , we have the same formulae for h e, u nij i and h f, u nij i but we moreover have that h h, u nij i = δ ij · (2 j − n ) .The left multiplication of the generators of O ( SU q (2)) on the entries of the irreducibleunitary corepresentations u n are computed explicitly here below, using the convention that u nij := 0 if n < or ( i, j ) / ∈ { , , . . . , n } : a ∗ · u nij = q i + j √ h n − i +1 ih n − j +1 ih n +1 i · u n +1 ij + √ h i ih j ih n +1 i · u n − i − ,j − b ∗ · u nij = q j √ h i +1 ih n − j +1 ih n +1 i · u n +1 i +1 ,j − q i +1 √ h n − i ih j ih n +1 i · u n − i,j − a · u nij = √ h i +1 ih j +1 ih n +1 i · u n +1 i +1 ,j +1 + q i + j +2 √ h n − i ih n − j ih n +1 i · u n − ij b · u nij = − q i − √ h j +1 ih n − i +1 ih n +1 i · u n +1 i,j +1 + q j √ h n − j ih i ih n +1 i · u n − i − ,j . (2.4)These formulae can be derived from the q -Clebsch-Gordan coefficients and we refer thereader to [13, Section 3] and [27, Chapter 3.4] for more information on these matters.The Haar state on the C ∗ -completion h : C ( SU q (2)) → C is determined by the identities h (1) = 1 and h ( u nij ) = 0 , for all n ∈ N and all i, j ∈ { , , . . . , n } , see [27, Chapter 4, Equation (50)]. The Haar stateis a twisted trace on O ( SU q (2)) with respect to the algebra automorphism ν : O ( SU q (2)) → HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 7 O ( SU q (2)) , which on the matrix units is given by ν ( u nij ) = q n − i − j ) · u nij . (2.5)That is, we have that h ( xy ) = h ( ν ( y ) x ) (2.6)for all x, y ∈ O ( SU q (2)) , see [27, Chapter 4, Proposition 15]. The Haar state is faithfuland we let L ( SU q (2)) denote the Hilbert space completion of the C ∗ -algebra C ( SU q (2)) with respect to the induced inner product h x, y i := h ( x ∗ y ) , x, y ∈ C ( SU q (2)) . The corresponding GNS-representation is denoted by ρ : C ( SU q (2)) → B ( L ( SU q (2))) . The entries from the irreducible unitary corepresentations u nij yield an orthogonal basis for L ( SU q (2)) . Their norms are determined by h u nij , u nij i = h (( u nij ) ∗ u nij ) = q n − i ) h n + 1 i , (2.7)see [27, Chapter 4, Theorem 17]. It is also convenient to record that h ( u nij ) ∗ , ( u nij ) ∗ i = h ( u nij ( u nij ) ∗ ) = q j h n + 1 i . (2.8)Finally, we remark that for each η ∈ U q ( su (2)) one has h ◦ δ η = h ◦ ∂ η = η (1) · h whichfollows directly from the bi-invariance of the Haar state.2.2. The Dąbrowski-Sitarz spectral triple.
In the previous section we saw that thedual pairing gives rise to an action of U q ( su (2)) on O ( SU q (2)) by linear endomorphisms,so in particular we obtain three linear maps ∂ e , ∂ f , ∂ k : O ( SU q (2)) → O ( SU q (2)) from the generators e, f, k ∈ U q ( su (2)) of the quantum enveloping algebra. Using thepairing between U q ( su (2)) and O ( SU q (2)) together with the formulas (2.1) one sees that ∂ k is an algebra automorphism and that ∂ e and ∂ f are twisted derivations, in the sense that ∂ e ( x · y ) = ∂ e ( x ) ∂ k ( y ) + ∂ − k ( x ) ∂ e ( y ) ,∂ f ( x · y ) = ∂ f ( x ) ∂ k ( y ) + ∂ − k ( x ) ∂ f ( y ) (2.9)for all x, y ∈ O ( SU q (2)) ; i.e. the twist is determined by the algebra automorphism ∂ k .The behaviour of our three operations with respect to the involution on O ( SU q (2)) is alsodetermined by (2.1) and the fact that we have a dual pairing of Hopf ∗ -algebras: ∂ e ( x ∗ ) = − q − · ∂ f ( x ) ∗ , ∂ f ( x ∗ ) = − q∂ e ( x ) ∗ and ∂ k ( x ∗ ) = ∂ − k ( x ) ∗ (2.10)for all x ∈ O ( SU q (2)) . For q = 1 we emphasise that our conventions imply that ∂ k = ∂ = id : O ( SU (2)) → O ( SU (2)) . In this case, both ∂ e and ∂ f are simply derivationson O ( SU (2)) , but we also have a third interesting derivation namely ∂ h : O ( SU (2)) → KONRAD AGUILAR, JENS KAAD, AND DAVID KYED O ( SU (2)) coming from the third generator h ∈ U ( su (2)) . The interaction between ∂ h andthe involution is encoded by the formula ∂ h ( x ∗ ) = − ∂ h ( x ) ∗ .It is convenient to specify the explicit formulae ∂ k ( a ) = q / a ∂ e ( a ) = b ∗ ∂ f ( a ) = 0 ∂ k ( a ∗ ) = q − / a ∗ ∂ e ( a ∗ ) = 0 ∂ f ( a ∗ ) = − qb (2.11) ∂ k ( b ) = q / b ∂ e ( b ) = − q − a ∗ ∂ f ( b ) = 0 ∂ k ( b ∗ ) = q − / b ∗ ∂ e ( b ∗ ) = 0 ∂ f ( b ∗ ) = a explaining the behaviour of the algebra automorphism ∂ k and the two twisted derivations ∂ e and ∂ f on the generators for the coordinate algebra O ( SU q (2)) . For q = 1 , our extraderivation ∂ h : O ( SU (2)) → O ( SU (2)) is given explicitly on the generators by ∂ h ( a ) = a , ∂ h ( b ) = b , ∂ h ( a ∗ ) = − a ∗ and ∂ h ( b ∗ ) = − b ∗ . For each n ∈ Z , we let A n ⊆ O ( SU q (2)) denote the n ’th spectral subspace coming fromthe strongly continuous circle action σ L : S × O ( SU q (2)) → O ( SU q (2)) determined on thegenerators by ( z, a ) z · a and ( z, b ) z · b . Thus, we let A n := (cid:8) x ∈ O ( SU q (2)) | σ L ( z, x ) = z n · x , ∀ z ∈ S (cid:9) . In particular, we have the fixed point algebra A ⊆ O ( SU q (2)) , which is referred to as the coordinate algebra for the standard Podleś sphere and we apply the notation O ( S q ) := A = (cid:8) x ∈ O ( SU q (2)) | σ L ( z, x ) = x , ∀ z ∈ S (cid:9) . The standard Podleś sphere is defined as the C ∗ -completion of O ( S q ) using the C ∗ -norminherited from C ( SU q (2)) and we apply the notation C ( S q ) ⊆ C ( SU q (2)) for this unital C ∗ -algebra. As the name suggest, the standard Podleś sphere was introduced by Podleśin [41] together with a whole range of “non-standard” Podleś spheres which we are notconsidering here. We shall also refer to the (standard) Podleś sphere as the quantised 2-sphere or the q -deformed 2-sphere , whenever linguistically convenient.The coordinate algebra O ( S q ) is generated by the elements A := bb ∗ B = ab ∗ B ∗ = ba ∗ and the following set (cid:8) A i B j , A i ( B ∗ ) k | i, j ∈ N , k ∈ N (cid:9) constitutes a vector space basis for this coordinate algebra, see [48, Theorem 1.2]. For q = 1 , it therefore follows that ∂ k fixes O ( S q ) , and another application of [48, Theorem1.2] shows that this is actually an alternative description of the coordinate algebra: O ( S q ) = (cid:8) x ∈ O ( SU q (2)) | ∂ k ( x ) = x (cid:9) . In terms of the irreducible unitary corepresentations { u n } ∞ n =0 , the coordinate algebra O ( S q ) can be described as span (cid:8) u mim | m ∈ N , i ∈ { , , . . . , m } (cid:9) . (2.12) HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 9
Indeed, one may use the description of the pairing with k ∈ U q ( su (2)) from (2.3) to obtainthe formula ∂ k ( u nij ) = q j − n/ u nij for all n ∈ N and all i, j ∈ { , , . . . , n } .We let H n denote the Hilbert space completion of A n with respect to the inner productinherited from L ( SU q (2)) and we put H + := H and H − := H − . We consider the directsum H + ⊕ H − as a Z / Z -graded Hilbert space with grading operator γ = (cid:18) − (cid:19) . Therestriction of the GNS-representation ρ : C ( SU q (2)) → B (cid:0) L ( SU q (2)) (cid:1) to the standardPodleś sphere then provides us with an injective, even ∗ -homomorphism π : C ( S q ) → B ( H + ⊕ H − ) given by π ( x ) := (cid:18) ρ ( x ) | H + ρ ( x ) | H − (cid:19) . The definition of the circle action together with the identities in (2.9) and (2.11) entailthat σ L ( z, ∂ e ( x )) = z − ∂ e (cid:0) σ L ( z, x ) (cid:1) and σ L ( z, ∂ f ( x )) = z ∂ f (cid:0) σ L ( z, x ) (cid:1) (2.13)for all z ∈ S and x ∈ O ( SU q (2)) . In particular, we obtain two unbounded operators E : A → H − and F : A − → H + agreeing with the restrictions ∂ e : A → A − and ∂ f : A − → A followed by the relevantinclusions. An application of the identities h ( ∂ e ( x )) = 0 = h ( ∂ f ( x )) x ∈ O ( SU q (2)) together with the identities in (2.9) and (2.10) shows that F ⊆ E ∗ and E ⊆ F ∗ . We let E : Dom ( E ) → L ( SU q (2)) and F : Dom ( F ) → L ( SU q (2)) denote the closures of E and F , respectively. The q -deformed Dirac operator D q : Dom ( D q ) → H + ⊕ H − is the odd unbounded operator given by D q := (cid:18) FE (cid:19) with Dom ( D q ) := Dom ( E ) ⊕ Dom ( F ) , and the main result from [14] is the following: Theorem 2.1. [14, Theorem 8]
The triple ( O ( S q ) , H + ⊕ H − , D q ) is an even spectral triple. The commutator with the Dirac operator D q : Dom ( D q ) → H + ⊕ H − induces a ∗ -derivation ∂ : O ( S q ) → B ( H + ⊕ H − ) , and since D q is odd the the × -matrix representing ∂ ( x ) is off-diagonal, and we denote it as follows: ∂ ( x ) = (cid:18) ∂ ( x ) ∂ ( x ) 0 (cid:19) . The ∗ -derivation ∂ : O ( S q ) → B ( H + ⊕ H − ) is closable since the Dirac operator D q isselfadjoint and therefore in particular closed. For x ∈ O ( S q ) , an application of the twisted Leibniz rule (2.9) yields that ∂ ( x ) = q / ρ ( ∂ e ( x )) | H + and ∂ ( x ) = q − / ρ ( ∂ f ( x )) | H − , andin the sequel we will therefore often think of ∂ and ∂ as the derivations q / ∂ e , q − / ∂ f : O ( S q ) → O ( SU q (2)) rather than their representations as bounded operators. Remark . In [14] the even spectral triple on O ( S q ) is also equipped with an extra anti-linear operator J : H + ⊕ H − → H + ⊕ H − (the reality operator). It is then verified that theeven spectral triple is in fact both real and U q ( su (2)) -equivariant and that these propertiesdetermine the spectral triple up to a non-trivial scalar z ∈ C \ { } . The description of theDirac operator in terms of the dual pairing of Hopf ∗ -algebras, which we are using here,can be found in [39, Section 3]. Remark . In the case where q = 1 , it can be proved that the direct sum of Hilbert spaces H + ⊕ H − agrees with the L -sections of the spinor bundle S + ⊕ S − → S on the -sphere.Letting Γ ∞ ( S + ⊕ S − ) denote the smooth sections of the spinor bundle it can moreover beverified that the unbounded selfadjoint operator D : Dom ( D ) → H + ⊕ H − agrees withthe closure of the Dirac operator D : Γ ∞ ( S + ⊕ S − ) → Γ ∞ ( S + ⊕ S − ) upon considering D asan unbounded operator on H + ⊕ H − . For more information on the spin geometry of the -sphere, we refer the reader to [17, Chapter 9A].2.3. Compact quantum metric spaces.
In this section we gather the necessary back-ground material concerning compact quantum metric spaces. These are the naturalnoncommutative analogues of classical compact metric spaces, and were introduced byRieffel [43, 44] around year 2000. The basic idea is that the noncommutative counterpartto a classical metric is captured by a certain seminorm, the domain of which can bechosen in several ways, leading to slight variations of the same theory. Rieffel’s originaltheory [44] is formulated in the language of order unit spaces, but one may equally welltake a C ∗ -algebraic setting as the point of departure [35, 36, 43]. We will here present ageneralisation of the C ∗ -algebraic setting and take concrete operator systems as our pointof departure.Let X be a concrete operator system; thus, for our purposes, X is a closed subspace of aspecified unital C ∗ -algebra such that X is stable under the adjoint operation and containsthe unit. Our operator system X gives rise to an order unit space X sa := (cid:8) x ∈ X | x = x ∗ (cid:9) where both the order and the unit are inherited from the ambient unital C ∗ -algebra. Theoperator system X has a state space S ( X ) consisting of all the positive linear functionalspreserving the units. This state space can be identified (via restriction) with the statespace S ( X sa ) of the associated order unit space. Definition 2.4. A compact quantum metric space is a concrete operator system X equipped with a seminorm L : X → [0 , ∞ ] satisfying the following:(i) One has L ( x ) = 0 if and only if x ∈ C · .(ii) The set Dom ( L ) := { x ∈ X | L ( x ) < ∞} is dense in X and L satisfies that L ( x ∗ ) = L ( x ) for all x ∈ X . HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 11 (iii) The function ρ L ( µ, ν ) := sup {| µ ( x ) − ν ( x ) | | L ( x ) } defines a metric on thestate space S ( X ) which metrises the weak ∗ -topology.In this case, the seminorm L is referred to as a Lip-norm and the corresponding metric isreferred to as the
Monge-Kantorovič metric .Remark that the requirement of having a Lip-norm which is invariant under the involu-tion implies that the identification of state spaces S ( X ) ∼ = S ( X sa ) becomes an isometry forthe Monge-Kantorovič metric. In Rieffel’s original work, the theory of compact quantummetric spaces is based on that of order unit spaces, and we remark that if ( X, L ) is a com-pact quantum metric space as defined above, then A := { x ∈ X sa | L ( x ) < ∞} becomesan order unit compact quantum metric space in Rieffel’s sense.The canonical commutative example upon which the above definition is modelled, arisesby considering a compact metric space ( M, d ) and its associated C ∗ -algebra C ( M ) , whichcan be endowed with a seminorm by setting L d ( f ) := sup (cid:26) | f ( p ) − f ( q ) | d ( p, q ) | p, q ∈ M, p = q (cid:27) . Then L d ( f ) is finite exactly when f is Lipschitz continuous, in which case L d ( f ) is theLipschitz constant. By results of Kantorovič and Rubinšte˘ın [24, 25], one has that ρ L d metrises the weak ∗ -topology on S ( C ( M )) , so that ( C ( M ) , L d ) is indeed a compact quan-tum metric space, and that the restriction of ρ L d to M ⊆ S ( C ( M )) agrees with the metric d .At first glance, it might seem like a difficult task to verify that the function ρ L metrisesthe weak ∗ -topology, but actually this can be reduced to a compactness question as thefollowing theorem shows. Theorem 2.5 ([42, Theorem 1.8]) . Let X be a concrete operator system and L : X → [0 , ∞ ] a seminorm satisfying (i) and (ii) from Definition 2.4. Then ( X, L ) is a compact quantummetric space if and only if the image of the Lip-unit ball { x ∈ X | L ( x ) } under thequotient map X → X/ C · is totally bounded for the quotient norm. Note, in particular, that this implies that if X is finite dimensional, then any seminormsatisfying (i) and (ii) from Definition 2.4 automatically provides X with a quantum metricstructure; we will use this fact repeatedly without further reference in the sequel.One of the many pleasant features of the theory of compact quantum metric spaces, isthat it allows for a noncommutative analogue of the classical Gromov-Hausdorff distancebetween compact metric spaces [18, 15], which we now recall. If ( X, L X ) and ( Y, L Y ) are twocompact quantum metric spaces, then a Lip-norm L : X ⊕ Y → [0 , ∞ ] is called admissible if the two quotient seminorms it defines on X and Y via the natural projections agree with L X and L Y , respectively. For any such L , one therefore obtains isometric embeddings ( S ( X ) , ρ L X ) ֒ → ( S ( X ⊕ Y ) , ρ L ) and ( S ( Y ) , ρ L Y ) ֒ → ( S ( X ⊕ Y ) , ρ L ) and hence one can consider their Hausdorff distance [20] dist ρ L H ( S ( X ) , S ( Y )) , and the quantum Gromov-Hausdorff distance between ( X, L X ) and ( Y, L Y ) is then defined as dist Q (( X, L X ); ( Y, L Y )) := inf { dist ρ L H ( S ( X ) , S ( Y )) | L : X ⊕ Y → [0 , ∞ ] admissible } . We remark that this is simply a rephrasing of Rieffels original definition from [44],where everything is formulated in terms of order unit spaces. More precisely, letting A = { x ∈ X sa | L X ( x ) < ∞} and B := { y ∈ Y | L Y ( y ) < ∞} we obtain order unit com-pact quantum metric spaces and dist Q (( X, L X ); ( Y, L Y )) = dist Q (( A, L X | A ); ( B, L Y | B )) .In [44], Rieffel showed that dist Q is symmetric and satisfies the triangle inequality, andthat distance zero is equivalent to the existence of a Lip-norm preserving isomorphism oforder unit spaces between the (completions of the) quantum metric spaces in question,see [44] for details on this. Over the past 20 years, several refinements of the quan-tum Gromov-Hausdorff distance have been proposed [26, 33, 35] for which distance zeroactually implies Lip-isometric isomorphism at the C ∗ -level, but the price for this is anincrease in the level of complexity in the definition. A very successful such refinement isLatrémolière’s notion of quantum propinquity , which has been developed in a series ofinfluential papers [30, 31, 32, 33]. In the present text, however, we will only consider Ri-effel’s original notion, and shall therefore not elaborate further on the quantum propinquity.Rieffel’s definition of compact quantum metric spaces is drawing inspiration from Connes’noncommutative geometry, and the latter is therefore, not surprisingly, a source of inter-esting examples of compact quantum metric spaces, [10, Chapter 6] and [9]. Concretely,if ( A , H, D ) is a unital spectral triple with A sitting as a dense unital ∗ -subalgebra of theunital C ∗ -algebra A ⊆ B ( H ) , then one obtains a seminorm L D : A → [0 , ∞ ] by setting L D ( a ) := (cid:26) k [ D, a ] k for a ∈ A∞ for a / ∈ A . (2.14)The main result of [2] is the following: Theorem 2.6.
For each q ∈ (0 , , the seminorm L D q arising from the Dąbrowski-Sitarzspectral triple ( O ( S q ) , H + ⊕ H − , D q ) turns C ( S q ) into a compact quantum metric space.Remark . For a unital spectral triple ( A , H, D ) there is also a maximal domain foran associated seminorm called the Lip-algebra and denoted by Lip D ( A ) ⊆ A . This denseunital ∗ -subalgebra consists of all elements in a ∈ A satisfying that a ( Dom ( D )) ⊆ Dom ( D ) and that [ D, a ] :
Dom ( D ) → H extends to a bounded operator on H . The correspondingseminorm is denoted by L max D . In general, the Lip-algebra does not agree with the domainof the closure of the derivation ∂ : A → B ( H ) , ∂ ( a ) := [ D, a ] . As a consequence, we do notknow whether it holds that, if ( A, L D ) a compact quantum metric space, then ( A, L max D ) isa compact quantum metric space. The converse can however be verified immediately byan application of Theorem 2.5. It is an interesting problem to investigate the relationshipbetween ( A, L D ) and ( A, L max D ) – in particular whether the quantum Gromov-Hausdorffdistance between the two is equal to zero. HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 13
In [2, Theorem 8.3] it is proved (for q ∈ (0 , ) that ( C ( S q ) , L max D q ) is a compact quantummetric space. Remark . In the case where q = 1 we have discussed earlier in Remark 2.3 that theunbounded selfadjoint operator D : Dom ( D ) → H + ⊕ H − agrees with the closure of theDirac operator D : Γ ∞ ( S + ⊕ S − ) → H + ⊕ H − coming from the spin geometry of the -sphere.We therefore know from [9, Proposition 1] that L D ( f ) agrees with the Lipschitz constantof f associated with the round metric on the -sphere (when f ∈ O ( S ) otherwise we getthe value ∞ ). In particular, we obtain that the Monge-Kantorovič metric coming from theseminorm L D : C ( S ) → [0 , ∞ ] agrees with the Monge-Kantorovič metric coming fromthe round metric on the -sphere. We may thus conclude that ( C ( S ) , L D ) is a compactquantum metric space as well.In the recent papers [12, 47] the notion of a unital spectral triple was extended byreplacing the C ∗ -algebra by an operator system. For such an operator system spectraltriple ( X , H, D ) one may again form a seminorm using the formula (2.14) and it thenmakes sense to ask whether the data ( X, L D ) is a compact quantum metric space. Weshall see examples of this phenomenon in our analysis of quantum fuzzy spheres herebelow, see Section 3.2. 3. Quantum fuzzy spheres
In this section we introduce the key ingredients needed to prove the convergence of thePodleś spheres towards the classical round sphere. Our proof of convergence proceeds viaa finite dimensional approximation procedure involving a quantum version of the fuzzyspheres. We are going to consider these quantum fuzzy spheres as finite dimensionaloperator system spectral triples sitting inside the Dąbrowski-Sitarz spectral triple for thecorresponding Podleś sphere. The operation which links the quantum fuzzy spheres to thePodleś sphere is then provided by a quantum analogue of the classical Berezin transform.We are now going to describe all these ingredients and once this is carried out the presentsection culminates with a proof of the Lip-norm contractibility of the quantum Berezintransform.Let us once and for all fix an N ∈ N and a deformation parameter q ∈ (0 , .3.1. The quantum Berezin transform.
Our first ingredient is a quantum analogue ofthe Berezin transform. This is going to be a positive unital map β N : C ( S q ) → C ( S q ) which has a finite dimensional image. The aim of this section is to introduce the Berezintransform and compute its image. We define the state h N : C ( S q ) → C by the formula h N ( x ) := h N + 1 i · h (cid:0) ( a ∗ ) N xa N (cid:1) . (3.1)Remark that h N is indeed unital since ( a ∗ ) N = u N and therefore h N (1) = h N + 1 i · h (cid:0) u N · ( u N ) ∗ (cid:1) = 1 by the identity in (2.8). In the definition here below, we let ∆ : C ( S q ) → C ( SU q (2)) ⊗ min C ( S q ) denote the left coaction of quantum SU (2) on the Podleś sphere. This coaction comes from the restriction of the coproduct ∆ : C ( SU q (2)) → C ( SU q (2)) ⊗ min C ( SU q (2)) to the Podleś sphere C ( S q ) ⊆ C ( SU q (2)) . Definition 3.1.
The quantum Berezin transform in degree N ∈ N is the positive unitalmap β N : C ( S q ) → C ( S q ) given by β N ( x ) := (1 ⊗ h N )∆( x ) .We notice that β N would a priori take values in C ( SU q (2)) , but the following lemmashows that β N : C ( S q ) → C ( SU q (2)) does indeed factorise through C ( S q ) . Recall to thisend that the elements u mim for m ∈ N and i ∈ { , , . . . , m } form a vector space basis forthe coordinate algebra O ( S q ) . Lemma 3.2.
For every m ∈ N and i ∈ { , , , . . . , m } we have the formula β N ( u mim ) = u mim · h N ( u mmm ) . Proof.
Let m ∈ N and i ∈ { , , , . . . , m } be given. We compute that β N ( u mim ) = (1 ⊗ h N )∆( u mim ) = m X l =0 u mil · h N ( u mlm )= m X l =0 u mil · h (cid:0) ( a ∗ ) N u mlm a N (cid:1) h N + 1 i = u mim · h N ( u mmm ) , where the last identity follows since the Haar state h : O ( SU q (2)) → C is invariant underthe algebra automorphism ν : O ( SU q (2)) → O ( SU q (2)) . (cid:3) For classical spaces, the Berezin transform is a well studied object (cf. [46] and referencestherein), and as we shall now see, our quantum Berezin transform exactly recovers theclassical ditto when q = 1 . So let us for a little while assume that q = 1 . Recall firstthat the irreducible corepresentation u N ∈ M N +1 ( C ( SU (2))) is actually the same as anirreducible representation u N : SU (2) → U ( C N +1 ) . Let Tr denote the trace on M N +1 ( C ) without normalisation so that Tr(1) = N + 1 . We choose P ∈ M N +1 ( C ) to be the rankone projection with P = 1 and all other entries equal to zero. Viewing C ( S ) as thefixpoint algebra C ( SU (2)) S , the classical Berezin transform b N : C ( S ) → C ( S ) is givenby (cf. [47, Section 3.3.1] or [45, Section 2]) b N ( f )( g ) := ( N + 1) Z SU (2) f ( g · x − ) H N ( x ) dλ ( x ) , g ∈ SU (2) , where H N denotes the density H N ( x ) := Tr (cid:0) P u N ( x ) P u N ( x ) ∗ (cid:1) and λ is the Haar probabil-ity measure on SU (2) . We remark that the trace property implies that H N ( x ) = H N ( x − ) which together with the unimodularity of SU (2) gives b N ( f )( g ) = ( N + 1) Z SU (2) f ( g · x ) H N ( x ) dλ ( x ) , g ∈ SU (2) . An element x in SU (2) is (in the first irreducible representation) given by a complex matrix (cid:18) ¯ z − z ¯ z z (cid:19) and the functions a and b then correspond to mapping the matrix to z and HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 15 z , respectively. Recall also that we have chosen the irreducible corepresentations u N sothat u N = ( a ∗ ) N . A direct computation now shows that Tr (cid:0) P u N ( x ) P u N ( x ) ∗ (cid:1) = ¯ z N z N = ( a ∗ ) N ( x ) a N ( x ) . Since the Haar state h at q = 1 is given by integration against λ , we obtain from this that ( N + 1) Z SU (2) f ( g · x ) H N ( x ) dλ ( x ) = ( N + 1) Z SU (2) ∆( f )( g, x )( a ∗ ) N ( x ) a N ( x ) dλ ( x )= h N + 1 i · h (cid:0) ∆( f )( g, − )( a ∗ ) N a N (cid:1) = (1 ⊗ h N )(∆( f ))( g ) , whenever g belongs to SU (2) . This shows that our quantum Berezin transform β N agreeswith the classical Berezin transform b N when q = 1 .We now return to the more general setting where the deformation parameter q belongsto (0 , . We apply the convention that u nij = 0 whenever n < or n ∈ N and ( i, j ) / ∈{ , , . . . , n } . Lemma 3.3.
The image of β N : C ( S q ) → C ( S q ) agrees with the linear span: span C (cid:8) u mim | m ∈ { , , . . . , N } , i ∈ { , , . . . , m } (cid:9) . In particular, we have that Im ( β N ) ⊆ O ( S q ) and that Dim ( Im ( β N )) = ( N + 1) .Proof. Let first n ∈ N and i, j ∈ { , , . . . , n } be given. Applying the formulae from (2.4)we obtain that a N · u nij = N X k =0 λ n,i,j ( k ) · u n +2 k − Ni + k,j + k and ( a ∗ ) N · u nij = N X k =0 µ n,i,j ( k ) · u n − k + Ni − k,j − k , where all the coefficients appearing are strictly positive. In particular, we may find strictlypositive coefficients such that a N ( a ∗ ) N · u nij = N X k = − N α n,i,j ( k ) · u n +2 ki + k,j + k . Let now m ∈ N be given. Since h ( u nij ) = 0 for all n > and h ( u ) = 1 we obtain that h N ( u mm,m ) = h (cid:0) ( a ∗ ) N u mm,m a N (cid:1) · h N + 1 i = h (cid:0) a N ( a ∗ ) N · u mm,m (cid:1) q − N · h N + 1 i = N X k = − N α m,m,m ( k ) · h ( u m +2 km + k,m + k ) q − N · h N + 1 i = (cid:26) for m > Nα m,m,m ( − m ) q − N · h N + 1 i for m N .
An application of Lemma 3.2 then proves the result of the present lemma. (cid:3)
Quantum fuzzy spheres.
Our second ingredient is a quantum analogue of the fuzzyspheres, which we will introduce in this section, and afterwards equip each of them with anoperator system spectral triple. These operator system spectral triples provide each of thequantum fuzzy spheres with the structure of a compact quantum metric space. Moreover,we are going to link the quantum fuzzy spheres to the Podleś spheres by showing thatthe image of the Berezin transform in degree N agrees with the quantum fuzzy sphere indegree N , thus obtaining natural quantum analogues of classical results, see [12, 45, 47]. Definition 3.4.
We define the quantum fuzzy sphere in degree N ∈ N as the C -linear span F Nq := span C (cid:8) A i B j , A i ( B ∗ ) j | i, j ∈ N , i + j N (cid:9) ⊆ C ( S q ) . We immediately remark that the vector space dimension of F Nq agrees with ( N + 1) which in turn is the dimension of the classical fuzzy sphere M N +1 ( C ) . Since F Nq ⊆ C ( S q ) is closed, unital and stable under the adjoint operation we may think of the quantumfuzzy sphere as a concrete operator system. Moreover, since ( O ( S q ) , H + ⊕ H − , D q ) is aneven unital spectral triple we immediately obtain an even operator system spectral triple ( F Nq , H + ⊕ H − , D q ) for the quantum fuzzy spheres. In particular, we may equip F Nq withthe seminorm L D q : F Nq → [0 , ∞ ) defined by L D q ( x ) := max {k ∂ ( x ) k , k ∂ ( x ) k} . Thus, L D q on the quantum fuzzy sphere is just the restriction of the seminorm on C ( S q ) arising from the Dąbrowski-Sitarz spectral triple. Since we already know that L D q is a Lip-norm on C ( S q ) we immediately obtain that L D q is also a Lip-norm on F Nq . We summarisethis observation in a lemma: Lemma 3.5.
The pair ( F Nq , L D q ) is a compact quantum metric space. We shall now see that the quantum fuzzy sphere in degree N agrees with the image ofthe Berezin transform β N : C ( S q ) → C ( S q ) . Lemma 3.6.
It holds that F Nq = Im ( β N ) .Proof. Since the vector space dimension of F Nq agrees with the vector space dimension ofIm ( β N ) it suffices to show that F Nq ⊆ Im ( β N ) (see Lemma 3.3). Moreover, since β N ( x ∗ ) = β N ( x ) ∗ we only need to show that A k B l ∈ Im ( β N ) for all k, l ∈ N with k + l N . However,using that A = bb ∗ and B = ab ∗ we see from (2.4) that A k B l = ( bb ∗ ) k ( ab ∗ ) l ∈ span C (cid:8) u nij | n k + l ) , i, j ∈ { , , . . . , n } (cid:9) . Moreover, since A k B l ∈ O ( S q ) we must in fact have that A k B l ∈ span C (cid:8) u mim | m k + l , i ∈ { , , . . . , m } (cid:9) , see (2.12). Since k + l N we now obtain the result of the present lemma by applyingLemma 3.3. (cid:3) HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 17
When q = 1 the classical fuzzy sphere in degree N is, by definition, given as M N +1 ( C ) and the classical Berezin transform agrees with the composition b N = σ N ◦ ˘ σ N , where σ N : M N +1 ( C ) → C ( S ) is the so-called covariant Berezin symbol and ˘ σ N is its adjoint (seeeg. [47, Section 3.3.1] or [45, Section 2]). In the quantised setting we have only definedthe composition thus leaving out a treatment of the covariant Berezin symbol. However,since the quantum Berezin transform at q = 1 agrees with the classical Berezin transformwe obtain that F N = σ N (cid:0) M N +1 ( C ) (cid:1) . Using our seminorm L D : F N → [0 , ∞ ) we thereforealso obtain a seminorm on the classical fuzzy sphere M N +1 ( C ) by deeming the covariantBerezin symbol to be a Lip-norm isometry. At least a priori, this seminorm is differentfrom the one considered by Rieffel in [45] and the one arising from the Grosse-PrešnajderDirac operator considered in [5, 19, 47].3.3. Derivatives of the Berezin transform.
Our aim in this section is to show that thequantum Berezin transform is a Lip-norm contraction and we are going to achieve this forelements in the coordinate algebra O ( S q ) . In Proposition 3.12 here below we shall thus seethat L D q ( β N ( x )) L D q ( x ) for all x ∈ O ( S q ) . It turns out that the derivation ∂ : O ( S q ) → M (cid:0) O ( SU q (2)) (cid:1) coming from the Dąbrowski-Sitarz spectral triple does not have any goodequivariance properties with respect to the Berezin transform and this makes the proofof the inequality L D q ( β N ( x )) L D q ( x ) a delicate matter. Our strategy is to conjugatethe derivation ∂ with the fundamental corepresentation unitary u ∈ M ( O ( SU q (2))) andthereby obtain an operation which is equivariant with respect to the Berezin transform.It is in fact possible to describe the conjugated derivation u∂u ∗ entirely in terms of the right action of the quantum enveloping algebra, even though the derivation ∂ comes fromthe left action of the quantum enveloping algebra. To this end, recall that the rightaction on O ( SU q (2)) of an element g ∈ U q ( su (2)) is given by the linear endomorphism δ g : O ( SU q (2)) → O ( SU q (2)) defined by δ g ( x ) := ( h g, ·i ⊗ x ) . Remark that δ g ( x ) ∈O ( S q ) for all x ∈ O ( S q ) since ∆( O ( S q )) ⊆ O ( SU q (2)) ⊗ O ( S q ) . Lemma 3.7.
Let g ∈ U q ( su (2)) . It holds that δ g β N ( x ) = β N δ g ( x ) for all x ∈ O ( S q ) .Proof. Remark first that Im ( β N ) ⊆ O ( S q ) so that it makes sense to look at the compo-sition δ g β N , see Lemma 3.3. Then by coassociativity of the coproduct ∆ : O ( SU q (2)) →O ( SU q (2)) ⊗ O ( SU q (2)) we have that δ g β N ( x ) = ( h g, ·i ⊗ ⊗ h N )∆( x ) = ( h g, ·i ⊗ ⊗ ⊗ h N )(1 ⊗ ∆)∆( x )= (1 ⊗ h N )∆( h g, ·i ⊗ x ) = β N δ g ( x ) for all x ∈ O ( S q ) . This proves the lemma. (cid:3) We are particularly interested in the three linear maps δ e , δ f and δ k : O ( SU q (2)) →O ( SU q (2)) . We record that δ k is an algebra automorphism whereas δ e and δ f are twisted derivations, meaning that δ e ( x · y ) = δ e ( x ) · δ k ( y ) + δ − k ( x ) · δ e ( y ) and δ f ( x · y ) = δ f ( x ) · δ k ( y ) + δ − k ( x ) · δ f ( y ) (3.2)for all x, y ∈ O ( SU q (2)) . The compatibility between our three operations and the involutionon O ( SU q (2)) is described by the identities δ e ( x ∗ ) = − q − · δ f ( x ) ∗ , δ f ( x ∗ ) = − q · δ e ( x ) ∗ and δ k ( x ∗ ) = δ − k ( x ) ∗ . (3.3)In the case where q = 1 , we emphasise that our conventions imply that δ k agrees with theidentity automorphism of O ( SU (2)) , and hence we obtain that δ e and δ f are derivations on O ( SU (2)) in the usual sense of the word. However, for q = 1 we also have the interestingderivation δ h : O ( SU (2)) → O ( SU (2)) coming from the third generator h ∈ U ( su (2)) . Thisextra derivation relates to the adjoint operation via the formula δ h ( x ∗ ) = − δ h ( x ) ∗ .It is convenient to record the formulae: δ k ( a ) = q / · a δ e ( a ) = 0 δ f ( a ) = − q · bδ k ( a ∗ ) = q − / · a ∗ δ e ( a ∗ ) = b ∗ δ f ( a ∗ ) = 0 (3.4) δ k ( b ) = q − / · b δ e ( b ) = − q − · a δ f ( b ) = 0 δ k ( b ∗ ) = q / · b ∗ δ e ( b ∗ ) = 0 δ f ( b ∗ ) = a ∗ Moreover, for q = 1 we in addition have that δ h ( a ) = a , δ h ( b ) = − b , δ h ( a ∗ ) = − a ∗ and δ h ( b ∗ ) = b ∗ . We define the algebra automorphism τ := δ k ∂ k : O ( SU q (2)) → O ( SU q (2)) and notice thatit follows from the defining commutation relations in O ( SU q (2)) that bx = τ ( x ) b and b ∗ x = τ ( x ) b ∗ for all x ∈ O ( SU q (2)) . (3.5)Clearly, τ agrees with δ k when restricted to O ( S q ) . In general, when θ : O ( SU q (2)) →O ( SU q (2)) is an algebra automorphism, we shall apply the notation [ y, x ] θ := yx − θ ( x ) y for the twisted commutator between two elements x and y ∈ O ( SU q (2)) . With this notationwe now obtain: Lemma 3.8.
We have the identities [ a ∗ , x ] δ k = (1 − q ) q / b∂ e ( x ) and [ a, x ] δ k = (1 − q ) q − / b ∗ ∂ f ( x ) for all x ∈ O ( S q ) .Proof. The operation x [ a ∗ , x ] δ k satisfies a twisted Leibniz rule, meaning that [ a ∗ , xy ] δ k = [ a ∗ , x ] δ k y + δ k ( x )[ a ∗ , y ] δ k , HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 19 for all x, y ∈ O ( S q ) . It moreover follows from (3.5) that the operation x b∂ e ( x ) satisfiesthe same twisted Leibniz rule so that b∂ ( xy ) = b∂ ( x ) y + δ k ( x ) b∂ e ( y ) , for all x, y ∈ O ( S q ) . In order to prove the first identity of the lemma, it thus suffices tocheck that a ∗ x − δ k ( x ) a ∗ = (1 − q ) q / b∂ e ( x ) for x ∈ { A, B, B ∗ } . This can be done in a straightforward fashion using that ∂ e ( A ) = − q − / b ∗ a ∗ ∂ e ( B ) = q − / ( b ∗ ) (3.6) ∂ e ( B ∗ ) = − q − / ( a ∗ ) , which can be seen from (2.11). The second identity of the lemma follows by a similarargument, or, alternatively, from the first identity by applying the involution. (cid:3) We recall that u := u = (cid:18) a ∗ − qbb ∗ a (cid:19) ∈ M (cid:0) O ( SU q (2)) (cid:1) denotes the fundamental corepresentation unitary and that ∂ : O ( S q ) → M ( O ( SU q (2))) denotes the derivation ∂ = (cid:18) ∂ ∂ (cid:19) = (cid:18) q − / ∂ f q / ∂ e (cid:19) . Lemma 3.9.
We have the identities [ u, x ] δ k = (1 − q ) (cid:18) bq − b ∗ (cid:19) ∂ ( x ) and [ u ∗ , δ k − ( x )] δ k = (1 − q ) ∂ ( x ) (cid:18) q − bb ∗ (cid:19) for all x ∈ O ( S q ) .Proof. Let x ∈ O ( S q ) be given. By definition of the fundamental corepresentation unitary u , the identities in (3.5), and Lemma 3.8 we see that ux − δ k ( x ) u = (cid:18) [ a ∗ , x ] δ k − q [ b, x ] δ k [ b ∗ , x ] δ k [ a, x ] δ k (cid:19) = (1 − q ) (cid:18) b∂ ( x ) 00 q − b ∗ ∂ ( x ) (cid:19) = (1 − q ) (cid:18) bq − b ∗ (cid:19) ∂ ( x ) . This proves the first identity of the lemma. The remaining identity then follows from thecomputation (1 − q ) ∂ ( x ) (cid:18) q − bb ∗ (cid:19) = ( q − (cid:16) (cid:18) bq − b ∗ (cid:19) ∂ ( x ∗ ) (cid:17) ∗ = ( δ k ( x ∗ ) u − ux ∗ ) ∗ = u ∗ δ − k ( x ) − xu ∗ = [ u ∗ , δ − k ( x )] δ k . (cid:3) We are now ready to show that the operation x u∂ ( x ) u ∗ is a twisted derivation on O ( S q ) . Proposition 3.10.
It holds that u∂ ( xy ) u ∗ = u∂ ( x ) u ∗ δ k ( y ) + δ k − ( x ) u∂ ( y ) u ∗ for all x, y ∈ O ( S q ) .Proof. Let x, y ∈ O ( S q ) be given. We compute that u∂ ( xy ) u ∗ = u∂ ( x ) yu ∗ + ux∂ ( y ) u ∗ = u∂ ( x ) u ∗ δ k ( y ) + δ k − ( x ) u∂ ( y ) u ∗ − u∂ ( x )[ u ∗ , δ k ( y )] δ k − + [ u, x ] δ k − ∂ ( y ) u ∗ , so (upon conjugating with u ) we have to show that u ∗ [ u, x ] δ k − ∂ ( y ) = ∂ ( x )[ u ∗ , δ k ( y )] δ k − u. Notice now that (cid:18) q − bb ∗ (cid:19) u = (cid:18) q − bb ∗ abq − a ∗ b ∗ − qb ∗ b (cid:19) = u ∗ (cid:18) bq − b ∗ (cid:19) . Thus, applying Lemma 3.9 we obtain that u ∗ [ u, x ] δ k − ∂ ( y ) = ( x − u ∗ δ k − ( x ) u ) ∂ ( y ) = − [ u ∗ , δ k − ( x )] δ k u∂ ( y )= ( q − ∂ ( x ) (cid:18) q − bb ∗ (cid:19) u∂ ( y ) = ( q − ∂ ( x ) u ∗ (cid:18) bq − b ∗ (cid:19) ∂ ( y )= − ∂ ( x ) u ∗ [ u, y ] δ k = ∂ ( x )[ u ∗ , δ k ( y )] δ k − u. This proves the proposition. (cid:3)
In order to finish our computation of the conjugated derivation x u∂ ( x ) u ∗ we intro-duce two twisted derivations δ , δ : O ( S q ) → O ( S q ) by setting δ := q / δ e and δ := q − / δ f . For q = 1 , we furthermore define the twisted derivation δ := δ k − δ − k q − q − : O ( S q ) → O ( S q ) and for q = 1 we simply put δ := δ h : O ( S ) → O ( S ) . Here, and below, the adjective“twisted” is to be understood in the sense of the following Leibniz type rule: δ i ( xy ) = δ i ( x ) δ k ( y ) + δ k − ( x ) δ i ( y ) , for x, y ∈ O ( S q ) and i ∈ { , , } ; that this holds follows from (3.2). For q ∈ (0 , we nowassemble this data into a twisted derivation δ : O ( S q ) → M ( O ( S q )) by the formula δ ( x ) := (cid:18) − δ ( x ) δ ( x ) δ ( x ) δ ( x ) (cid:19) . We record that δ ( x ∗ ) = − δ ( x ) ∗ , see (3.3) for this. Proposition 3.11.
It holds that u∂ ( x ) u ∗ = δ ( x ) for all x ∈ O ( S q ) . HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 21
Proof.
Using Proposition 3.10 we see that the operations x u∂ ( x ) u ∗ and x δ ( x ) satisfy the same twisted Leibniz rule and they also behave in the same way with respect tothe adjoint operation. It therefore suffices to verify the required identity on the generators A, B ∈ O ( S q ) . This can be carried out by a straightforward computation using (3.6) and(3.4) together with the defining relations for O ( SU q (2)) , indeed: u∂ ( A ) u ∗ = (cid:18) a ∗ − qbb ∗ a (cid:19) (cid:18) ab − b ∗ a ∗ (cid:19) (cid:18) a b − qb ∗ a ∗ (cid:19) = (cid:18) a ∗ aba ∗ + qbb ∗ a ∗ b − qb ∗ abb ∗ − ab ∗ a ∗ a (cid:19) = (cid:18) ba ∗ − ab ∗ (cid:19) = δ ( A ) , and u∂ ( B ) u ∗ = (cid:18) a ∗ − qbb ∗ a (cid:19) (cid:18) q − a ( b ∗ ) (cid:19) (cid:18) a b − qb ∗ a ∗ (cid:19) = (cid:18) − a ∗ a b ∗ − qb ( b ∗ ) a q − a ∗ a a ∗ − qb ( b ∗ ) q − b ∗ a a ∗ + a ( b ∗ ) b (cid:19) = (cid:18) − ab ∗ q − aa ∗ − qbb ∗ ab ∗ (cid:19) = δ ( B ) . (cid:3) We may now show that the Berezin transform is a Lip-norm contraction:
Proposition 3.12.
Let q ∈ (0 , and N ∈ N . The Berezin transform β N : C ( S q ) → C ( S q ) is a Lip-norm contraction in the sense that k ∂β N ( x ) k k ∂ ( x ) k for all x ∈ O ( S q ) .Proof. Since u ∈ M (cid:0) O ( SU q (2)) (cid:1) is unitary we obtain from Proposition 3.11 that k ∂β N ( x ) k = k u · ∂β N ( x ) · u ∗ k = k δβ N ( x ) k . Then, since β N : C ( S q ) → C ( S q ) is a complete contraction we get from Lemma 3.7 that k δβ N ( x ) k = k β N δ ( x ) k k δ ( x ) k = k ∂ ( x ) k . This proves the result of the proposition. (cid:3) Quantum Gromov-Hausdorff convergence
The aim of the present section is to prove our main convergence results, namely that thequantum fuzzy spheres F Nq converge to the Podleś sphere S q as the matrix size N grows,and that the Podleś spheres S q converge to the classical 2-sphere (with its round metric)as q tends to 1. However, before approaching the actual convergence results, quite a bit ofpreparatory analysis is needed. As it turns out, the key to the convergence results is thatthe quantum Berezin transform β N provides a good approximation of the identity map onthe Lip unit balls, and we prove this in the sections to follow. In the following section we provide the essential upper bound on k β N ( x ) − x k for x in the Lip unit ball, and in thenext section we prove that this upper bound indeed goes to zero as N tends to infinity.4.1. Approximation of the identity.
Throughout this section we let N ∈ N and q ∈ (0 , be fixed. We let ǫ : C ( S q ) → C denote the restriction of the counit to the Podleśsphere and we recall that h N : C ( S q ) → C is the state given by h N ( x ) := h (cid:0) ( a ∗ ) N xa N (cid:1) ·h N +1 i . We start out by proving an equivariance property for our derivations ∂ , ∂ : O ( S q ) →O ( SU q (2)) . Lemma 4.1.
Let x ∈ O ( S q ) . It holds that (1 ⊗ ∂ )∆( x ) = ∆ ∂ ( x ) and (1 ⊗ ∂ )∆( x ) = ∆ ∂ ( x ) . Proof.
First note that since O ( S q ) is a left O ( SU q (2)) -comodule the formulae in the lemmaare well defined. As explained in Section 2, the derivations ∂ , ∂ : O ( S q ) → O ( SU q (2)) aregiven by ∂ ( y ) = q / (1 ⊗ h e, ·i )∆( y ) and ∂ ( y ) = q − / (1 ⊗ h f, ·i )∆( y ) for all y ∈ O ( S q ) . Using the coassociativity of ∆ : O ( SU q (2)) → O ( SU q (2)) ⊗ O ( SU q (2)) we then obtain that (1 ⊗ ∂ )∆( x ) = q / (1 ⊗ ⊗ h e, ·i )(1 ⊗ ∆)∆( x ) = q / (∆ ⊗ h e, ·i )∆( x ) = ∆( ∂ ( x )) . A similar proof applies when ∂ is replaced by ∂ . (cid:3) The next lemma is a consequence of the above Lemma 4.1, and standard properties ofthe minimal tensor product:
Lemma 4.2.
Let φ : C ( SU q (2)) → C be a bounded linear functional. It holds that L D q (cid:0) ( φ ⊗ x ) (cid:1) k φ k · L D q ( x ) for all x ∈ O ( S q ) . We let d q ( h N , ǫ ) ∈ [0 , ∞ ) denote the Monge-Kantorovič distance between the two states h N , ǫ : C ( S q ) → C . Recall that this is defined as d q ( h N , ǫ ) := sup (cid:8) | h N ( x ) − ε ( x ) | | x ∈ O ( S q ) , L D q ( x ) (cid:9) . Proposition 4.3.
We have the inequality k x − β N ( x ) k d q ( h N , ǫ ) · L D q ( x ) for all x ∈ O ( S q ) .Proof. Let x ∈ O ( S q ) be given. We record that x = (1 ⊗ ǫ )∆( x ) and hence x − β N ( x ) = (cid:0) ⊗ ( ǫ − h N ) (cid:1) ∆( x ) . Notice now that x − β N ( x ) ∈ C ( S q ) ⊆ B (cid:0) L ( S q ) (cid:1) and let η, ζ ∈ L ( SU q (2)) with k η k , k ζ k be given. We let φ η,ζ : C ( SU q (2)) → C , φ η,ζ ( y ) := h η, ρ ( y ) ζ i , denote the associatedcontractive linear functional, and compute that φ η,ζ ( x − β N ( x )) = ( ǫ − h N )( φ η,ζ ⊗ x ) . HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 23
Using the definition of the Monge-Kantorovič distance together with Lemma 4.2 we thenobtain that (cid:12)(cid:12) φ η,ζ ( x − β N ( x )) (cid:12)(cid:12) d q ( h N , ǫ ) · L D q (cid:0) ( φ η,ζ ⊗ x ) (cid:1) d q ( h N , ǫ ) · k φ η,ζ k · L D q ( x ) d q ( h N , ǫ ) · L D q ( x ) . Taking the supremum over vectors η and ζ ∈ L ( SU q (2)) with k η k , k ζ k we obtain that k x − β N ( x ) k d q ( h N , ǫ ) · L D q ( x ) . (cid:3) Convergence to the counit.
Throughout this section we let q ∈ (0 , be fixed. Wesee from Proposition 4.3 that in order to establish that the Berezin transform provides agood approximation of the identity map on the Lip unit ball, we only need to verify that lim N →∞ d q ( h N , ǫ ) = 0 . Since we already know that ( C ( S q ) , L D q ) is a compact quantummetric space, the convergence in the Monge-Kantorovič metric is equivalent to convergencein the weak ∗ -topology. We apply the notation O ( A, ⊆ C ( S q ) for the smallest unital ∗ -subalgebra containing the element A ∈ C ( S q ) . Proposition 4.4.
Let q ∈ (0 , . The sequence of states { h N } ∞ N =0 converges to ǫ : C ( S q ) → C in the weak ∗ -topology and hence lim N →∞ d q ( h N , ǫ ) = 0 .Proof. It suffices to show that lim N →∞ h N ( x ) = ǫ ( x ) , (4.1)for all x in the norm-dense unital ∗ -subalgebra O ( S q ) ⊆ C ( S q ) . Notice next that h N ( x ) =0 = ǫ ( x ) whenever x = yB i or x = y ( B ∗ ) i for some i ∈ N and some y ∈ O ( A, . Indeed,this is clear for the counit and for the state h N this follows since the Haar state is invariantunder the modular automorphism ν : O ( SU q (2)) → O ( SU q (2)) . When proving (4.1) wemay thus restrict our attention to the case where x ∈ O ( A, . Using next that O ( A,
1) = span C (cid:8) ( a ∗ ) k a k | k ∈ N (cid:9) we only need to show that h N (cid:0) ( a ∗ ) k a k (cid:1) converges to ǫ (( a ∗ ) k a k (cid:1) = 1 for all k ∈ N . But thisfollows from the computation here below, where we utilize the fact that ( a ∗ ) N + k = u N + k together with (2.8): h N (cid:0) ( a ∗ ) k a k (cid:1) = h N + 1 i h (cid:0) ( a ∗ ) N + k a N + k (cid:1) = h N +1 ih N + k +1 i = 1 − q N +1) h k ih N + k +1 i −→ N →∞ . (cid:3) Corollary 4.5.
Let q ∈ (0 , . For each ε > there exists an N ∈ N such that k x − β N ( x ) k ε · L D q ( x ) for all N > N and all x ∈ O ( S q ) .Proof. By Proposition 4.3 we have the inequality k x − β N ( x ) k d q ( h N , ǫ ) · L D q ( x ) , and by Proposition 4.4 we have that lim N →∞ d q ( h N , ǫ ) = 0 . (cid:3) The above corollary in combination with Proposition 3.12 now makes it an easy task toshow that that the quantum fuzzy spheres F Nq converge to the Podleś sphere C ( S q ) as N approaches infinity, and we carry out the details of this argument in Section 4.5. However,in order to show that C ( S q ) converges to C ( S ) as q tends to , we need to estimate thedistance d q ( h N , ǫ ) in a suitably uniform manner with respect to the deformation parameter q . As a first step in this direction we show that the states h N and ǫ : C ( S q ) → C may berestricted to the unital C ∗ -subalgebra C ∗ ( A, ⊆ C ( S q ) without changing their Monge-Kantorovič distance. This will play out to our advantage since we already carried outa careful analysis of the compact quantum metric space (cid:0) C ∗ ( A, , L D q (cid:1) in [16]. In fact, C ∗ ( A, is a commutative unital C ∗ -algebra and the Lip-norm L D q : C ∗ ( A, → [0 , ∞ ] comes from an explicit metric on the spectrum of the selfadjoint positive operator A . Weshall give more details on these matters in the next section.Letting i : C ∗ ( A, → C ( S q ) denote the inclusion we specify that d q ( h N ◦ i, ǫ ◦ i ) := (cid:8) | h N ( x ) − ǫ ( x ) | | x ∈ O ( A, , L D q ( x ) (cid:9) and d q ( h N , ǫ ) := (cid:8) | h N ( x ) − ǫ ( x ) | | x ∈ O ( S q ) , L D q ( x ) (cid:9) . (4.2)In order to prove that these two quantities agree, we define the strongly continuous circleaction σ R : S × C ( SU q (2)) → C ( SU q (2)) by the formulae σ R ( z, a ) := za and σ R ( z, b ) := z − b This circle action gives rise to the operation Φ : C ( SU q (2)) → C ( SU q (2)) given by Φ ( x ) := 12 π Z π σ R ( e it , x ) dt. For z ∈ S and x ∈ O ( S q ) we record the relation σ R ( z, ∂ ( x )) = ∂ ( σ R ( z, x )) , which can be proved by a direct computation on the generators A, B and B ∗ and anapplication of the Leibniz rule. Using that ∂ : O ( S q ) → M (cid:0) C ( SU q (2)) (cid:1) is closable (seethe discussion after Theorem 2.1) we then obtain the identity ∂ (Φ ( x )) = Φ ( ∂ ( x )) for all x ∈ O ( S q ) . Remark that the restriction of Φ to C ( S q ) yields a conditional ex-pectation onto C ∗ ( A, which maps the coordinate algebra O ( S q ) onto O ( A, . Theseobservations immediately yield the next result: Lemma 4.6.
Let q ∈ (0 , . We have the inequality L D q (Φ ( x )) L D q ( x ) for all x ∈ O ( S q ) . As alluded to above, we have the following:
Lemma 4.7.
Let q ∈ (0 , . It holds that d q ( h N , ǫ ) = d q ( h N ◦ i, ǫ ◦ i ) for all N ∈ N . HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 25
Proof.
Let N ∈ N be given. The inequality d q ( h N ◦ i, ǫ ◦ i ) d q ( h N , ǫ ) is clearly satisfied.To prove the remaining inequality, let x ∈ O ( S q ) with L D q ( x ) be given. Using Lemma4.6 we then obtain that L D q (Φ ( x )) . Next, since h N (cid:0) σ R ( z, x ) (cid:1) = h N ( x ) and ǫ (cid:0) σ R ( z, x ) (cid:1) it follows that h N ( x ) = h N (Φ ( x )) and ǫ ( x ) = ǫ (Φ ( x )) and hence that (cid:12)(cid:12) h N ( x ) − ǫ ( x ) (cid:12)(cid:12) = (cid:12)(cid:12) ( h N ◦ i )(Φ ( x )) − ( ǫ ◦ i )(Φ ( x )) (cid:12)(cid:12) d q ( h N ◦ i, ǫ ◦ i ) . This proves the present lemma. (cid:3)
Uniform approximation of the identity.
In this section we are no longer consid-ering q ∈ (0 , to be fixed but rather as a variable deformation parameter. For this reasonwe decorate our generators a, b ∈ O ( SU q (2)) and A, B ∈ O ( S q ) with an extra subscript,e.g. writing A q instead of A . Likewise, we put h n i q := h n i = n − X m =0 q n . We are, however, fixing a δ ∈ (0 , and restrict our attention to the case where q ∈ [ δ, .We start out by shortly reviewing some of the results obtained in [16]. The unital C ∗ -subalgebra C ∗ ( A q , ⊆ C ( S q ) is commutative and is therefore isomorphic to the continuousfunctions on the spectrum of A q . For q = 1 this spectrum is given by X q := { q m | m ∈ N } ∪ { } ⊆ [0 , and for q = 1 the spectrum X q agrees with the whole closed unit interval [0 , . We are from now on tacitly identifying the C ∗ -algebra C ∗ ( A q , with the C ∗ -algebra C ( X q ) . It was proved in [16] that the restriction of the seminorm L D q : C ( S q ) → [0 , ∞ ] to C ∗ ( A q , agrees with the Lipschitz constant seminorm arising from a metric ρ q : X q × X q → [0 , ∞ ) . For q = 1 this metric is given by the explicit formula ρ q ( q m , q l ) = (cid:12)(cid:12)(cid:12) ∞ X k = m (1 − q ) q k p − q k +1) − ∞ X k = l (1 − q ) q k p − q k +1) (cid:12)(cid:12)(cid:12) (4.3)for all m, l ∈ N . We let h q : C ( X q ) → C denote the restriction of the Haar state to the C ∗ -subalgebra C ∗ ( A q , ⊆ C ( SU q (2)) . For q = 1 we remark that h agrees with the usualRiemann integral on C ([0 , .Let N ∈ N . As mentioned earlier we are interested in estimating the Monge-Kantoroviçdistance between the two states h N and ǫ : C ( S q ) → C in a suitably uniform mannerwith respect to the deformation parameter q ∈ [ δ, . We now present an estimate on thisquantity which involves the metric ρ q : X q × X q → [0 , ∞ ) . Lemma 4.8.
Let q ∈ [ δ, and N ∈ N . We have the estimate d q ( ǫ, h N ) h N + 1 i q q N · h q (cid:0) a Nq ( a ∗ q ) N · ρ q ( − , (cid:1) . Proof.
By Lemma 4.7 we have that d q ( ǫ, h N ) = d q ( ǫ ◦ i, h N ◦ i ) , where i : C ∗ ( A q , → C ( S q ) denotes the inclusion. The composition ǫ ◦ i : C ∗ ( A q , → C agrees with the pure state on C ( X q ) given by evaluation at ∈ X q . Let now ξ ∈ C ( X q ) be given. Since the restriction L D q : C ( X q ) → [0 , ∞ ] agrees with the Lipschitz constant seminorm coming from the metric ρ q : X q × X q → [0 , ∞ ) we obtain that (cid:12)(cid:12) ǫ ( ξ ) − h N ( ξ ) (cid:12)(cid:12) = (cid:12)(cid:12) h N (cid:0) ξ (0) − ξ (cid:1)(cid:12)(cid:12) h N (cid:0) | ξ (0) − ξ | (cid:1) L D q ( ξ ) · h N (cid:0) ρ q ( − , (cid:1) = L D q ( ξ ) · h N + 1 i q q N · h q (cid:0) a Nq ( a ∗ q ) N · ρ q ( − , (cid:1) , where the last identity uses the definition of state h N : C ( S q ) → C together with thetwisted tracial property of the Haar state, see (3.1) and (2.6). The result of the lemmanow follows from the definition of the quantity d q ( ǫ ◦ i, h N ◦ i ) = d q ( ǫ, h N ) as recalled in(4.2). (cid:3) We now carry out a more detailed analysis of the right hand side of the estimate ap-pearing in Lemma 4.8. First of all we treat the dependency of the (restriction of the) Haarstate h q : C ( X q ) → C on the deformation parameter q ∈ [ δ, . Our next lemma can also bededuced from [8, Page 195], but for the convenience of the reader we here provide a shortself contained argument. Consider the polynomial algebra C [ x, y ] as a unital ∗ -subalgebraof C (cid:0) [ δ, × [0 , (cid:1) and define the linear map H : C [ x, y ] → C ([ δ, given by H ( x j y k )( q ) := q j h k + 1 i q . Lemma 4.9.
The linear map H : C [ x, y ] → C ([ δ, extends to a norm-contraction H : C ([ δ, × [0 , → C ([ δ, such that H ( ξ )( q ) = h q ( ξ ( q, − ) | X q ) for all q ∈ [ δ, .Proof. Let first q ∈ [ δ, be fixed. By definition of H : C [ x, y ] → C ([ δ, we have that H ( x j y k )( q ) = q j h k + 1 i q = q j · h q ( A kq ) = h q (cid:0) ( x j · y k )( q, − ) | X q (cid:1) for all j, k ∈ N . Remark that the formula for h q ( A kq ) can be found in [27, Chapter 4,Equation (51)]. Then, using that h q : C ( X q ) → C is a state for every q ∈ [ δ, , we obtainthat k H ( ξ ) k ∞ = sup q ∈ [ δ, (cid:12)(cid:12) H ( ξ )( q ) (cid:12)(cid:12) = sup q ∈ [ δ, (cid:12)(cid:12) h q ( ξ ( q, · ) | X q ) (cid:12)(cid:12) sup ( q,s ) ∈ [ δ, × X q (cid:12)(cid:12) ξ ( q, s ) (cid:12)(cid:12) k ξ k ∞ for all ξ ∈ C [ x, y ] . The result of the lemma now follows since C [ x, y ] ⊆ C ([ δ, × [0 , isdense in supremum norm. (cid:3) Continuing our treatment of the right hand side of the estimate in Lemma 4.8 we nowanalyze how the functions ρ q ( − ,
0) : X q → [0 , ∞ ) depend on the deformation parameter q ∈ [ δ, . We record that ρ q ( q m ,
0) = ∞ X k = m (1 − q ) q k p − q k +1) = ∞ X k =0 (1 − q ) q k + m p − q k + m +1) (4.4) HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 27 whenever q = 1 and m ∈ N , see (4.3). In order to deal with these expressions we let ( q, s ) ∈ [ δ, × [0 , and define the continuous decreasing function ζ q,s : ( − , ∞ ) → [0 , ∞ ) by the formula ζ q,s ( x ) := (1 − q ) q x · √ s p − sq x +1) Each of these functions can be estimated from above as follows: ζ q,s ( x ) q x p − q x +1) q x p − q for all x > . We may thus introduce the function f : [ δ, × [0 , → [0 , ∞ ) by putting f ( q, s ) := (cid:26) P ∞ k =0 ζ q,s ( k ) for q = 12 arcsin( √ s ) for q = 1 . (4.5)Comparing with the formula for the metric in (4.4) we immediately see that f ( q, q m ) = ∞ X k =0 ζ q,q m ( k ) = ∞ X k =0 (1 − q ) q k + m p − q k + m +1) = ρ q ( q m , (4.6)whenever q = 1 and m ∈ N . Lemma 4.10.
The function f : [ δ, × [0 , → [0 , ∞ ) is continuous.Proof. We focus on proving continuity at a point of the form (1 , s ) for a fixed s ∈ [0 , ,since the continuity of the restriction f | [ δ, × [0 , follows as the sequence of partial sums { P mk =0 ζ − , − ( k ) } ∞ m =0 converges in supremum norm to f on compact subsets of [ δ, × [0 , .We remark that for each fixed ( q, s ) ∈ [ δ, × [0 , it holds that the function γ q,s : ( − , ∞ ) → R given by γ q,s : x − q ln( q ) · q · arcsin( √ s · q x +1 ) is an antiderivative to ζ q,s : ( − , ∞ ) → R . Moreover, since ζ q,s : ( − , ∞ ) → R is positiveand decreasing we obtain the estimates Z ∞ ζ q,s ( x ) dx ∞ X k =0 ζ q,s ( k ) Z ∞− ζ q,s ( x ) dx. In order to compute the above integrals we record the following formulae: lim x →∞ γ q,s ( x ) = 0 γ q,s (0) = 1 − q ln( q ) · q arcsin( √ s · q )lim x →− γ q,s ( x ) = 1 − q ln( q ) · q arcsin( √ s ) . We thereby obtain the estimates − − q ln( q ) · q arcsin( √ s · q ) f ( q, s ) − − q ln( q ) · q arcsin( √ s ) for all ( q, s ) ∈ [ δ, × [0 , . The continuity of the function f : [ δ, × [0 , → [0 , ∞ ) at thefixed point (1 , s ) ∈ [ δ, × [0 , now follows by noting that lim q → − q ln( q ) · q = − . (cid:3) We are now ready for the final step regarding the continuity properties of the right handside of the estimate in Lemma 4.8. For each q ∈ [ δ, and N ∈ N , we compute that a Nq ( a ∗ q ) N = (1 − q − N − A q ) · (1 − q − N − A q ) · . . . · (1 − A q ) . We then define the continuous function g N : [ δ, × [0 , → [0 , ∞ ) g N ( q, s ) := h N + 1 i q N · (1 − q − N − · s ) · (1 − q − N − s ) · . . . · (1 − s ) (4.7)and note that g N ( q, − ) | X q = h N +1 i q N · a Nq ( a ∗ q ) N .The next result summarizes what we have obtained so far and is thus an immediate con-sequence of Lemma 4.8, Lemma 4.9 and Lemma 4.10 together with (4.6) and (4.7): Lemma 4.11.
For each N ∈ N and each q ∈ [ δ, we have the estimate d q ( ǫ, h N ) H ( f · g N )( q ) . We finish this section by proving three lemmas culminating in a uniform estimate on thedistance between the Berezin transform and the identity map.
Lemma 4.12.
For each q ∈ [ δ, , it holds that lim N →∞ H ( f · g N )( q ) = 0 .Proof. Let q ∈ [ δ, be given. We first remark that it follows from the definition in (4.5)(see also (4.6)) that f ( q,
0) = 0 . Since the restriction of the counit to C ∗ ( A q , ∼ = C ( X q ) isgiven by evaluation at ∈ X q this translates into the identity ǫ ( f ( q, · ) | X q ) = 0 . Moreover,we notice that H ( f · g N )( q ) = h q (cid:0) g N ( q, − ) | X q · f ( q, − ) | X q (cid:1) = h N (cid:0) f ( q, − ) | X q (cid:1) . The result of the lemma now follows since the sequence of states { h N } ∞ N =0 converges to therestriction of the counit ǫ : C ( S q ) → C in the weak ∗ -topology by Proposition 4.4. (cid:3) Lemma 4.13.
For each ε > , each q ∈ [ δ, and each N ∈ N there exists an N > N and an open interval I ⊆ R with q ∈ I such that k x − β N ( x ) k ε · L D q ( x ) for all q ∈ I ∩ [ δ, and all x ∈ O ( S q ) .Proof. Let ε > , q ∈ [ δ, and N ∈ N be given. By Lemma 4.12 we may choose an N > N such that H ( f · g N )( q ) < ε/ . Then, using Lemma 4.10 and Lemma 4.9, we seethat H ( f · g N ) ∈ C ([ δ, . We may therefore choose our open interval I ⊆ R such that (cid:12)(cid:12) H ( f · g N )( q ) − H ( f · g N )( q ) (cid:12)(cid:12) < ε/ for all q ∈ I ∩ [ δ, . Combining this result with Lemma4.11 and Proposition 4.3 we then obtain that k x − β N ( x ) k d q ( h N , ǫ ) · L D q ( x ) (cid:12)(cid:12) H ( f · g N )( q ) (cid:12)(cid:12) · L D q ( x ) ε · L D q ( x ) HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 29 for all q ∈ I ∩ [ δ, and all x ∈ O ( S q ) . This ends the proof of the lemma. (cid:3) Lemma 4.14.
For each ε > and each q ∈ [ δ, there exists an N ∈ N and an openinterval I ⊆ R with q ∈ I such that k x − β N ( x ) k ε · L D q ( x ) for all q ∈ I ∩ [ δ, and all x ∈ O ( S q ) .Proof. If q = 1 the statement follows directly from Lemma 4.13. If q = 1 , we first choosean N ∈ N such that d ( h N , ǫ ) < ε , and hence k x − β N ( x ) k ε · L D ( x ) , for all N > N .By applying Lemma 4.13 we then obtain an N > N and an open interval I which satisfyour claim. (cid:3) Quantum fuzzy spheres converge to the fuzzy sphere.
In this section we provethat the quantum fuzzy spheres converge to the classical fuzzy sphere as the deformationparameter q tends to . As remarked in Section 3.2, rather than thinking of the fuzzysphere as a matrix algebra we will consider its image F N := σ N ( M N +1 ( C )) under thecovariant Berezin symbol σ N : M N +1 ( C ) → C ( S ) , see Lemma 3.6 and the discussion afterthis lemma. We recall that each of the quantum fuzzy spheres F Nq ⊆ C ( S q ) is equippedwith the Lip-norm arising from restricting the Lip-norm L D q : C ( S q ) → [0 , ∞ ] , and in thisway each F Nq becomes a compact quantum metric space. Proposition 4.15.
Let N ∈ N . The quantum fuzzy spheres (cid:0) F Nq (cid:1) q ∈ (0 , vary continuouslyin the parameter q with respect to the quantum Gromov Hausdorff distance.Proof. Let δ ∈ (0 , be given. By [8, Proposition 7.1], there exists a continuous fieldof C ∗ -algebras over [ δ, with total space C ( SU • (2)) such that C ( SU q (2)) agrees withthe fibre at q ∈ [ δ, . There exist continuous sections of this field A • , B • ∈ C ( SU • (2)) mapping to the generators A q and B q ∈ C ( S q ) ⊆ C ( SU q (2)) under the quotient mapev q : C ( SU • (2)) → C ( SU q (2)) for each q ∈ [ δ, . Let us define the subspace V := span C (cid:8) A i • B j • , A i • ( B ∗• ) j | i, j ∈ N , i + j N (cid:9) ⊆ C ( SU • (2)) and let V sa ⊆ V denote the real part V sa := (cid:8) x + x ∗ | x ∈ V (cid:9) so that V sa becomes a realvector space of dimension ( N + 1) containing the unit from C ( SU • (2)) .We remark that it follows from Definition 3.4 that the image ev q ( V sa ) agrees with theorder unit space ( F Nq ) sa for each q ∈ [ δ, . Therefore, upon defining k y k q := k ev q ( y ) k foreach q ∈ [ δ, we obtain a continuous field {k · k q } q ∈ [ δ, of order unit norms on ( V sa , .For each q ∈ [ δ, we now record the formulae ∂ ( A q ) = − b ∗ q a ∗ q , ∂ ( B q ) = ( b ∗ q ) and ∂ ( B ∗ q ) = − q − ( a ∗ q ) . Since we have continuous sections a • , b • ∈ C ( SU • (2)) (mapping tothe generators of C ( SU q (2)) under the quotient map ev q ) we then obtain a continuousfamily of Lip-norms { L q } q ∈ [ δ, on V sa defined by L q ( y ) := L D q ( ev q ( y )) for each q ∈ [ δ, .The assumptions in [44, Theorem 11.2] are thereby fulfilled, and since δ ∈ (0 , wasarbitrary this implies the claimed continuity result. (cid:3) Quantum fuzzy spheres converge to the Podleś sphere.
In this section we fixa q ∈ (0 , and show that the quantum fuzzy spheres F Nq converge to C ( S q ) as N tendsto infinity. This follows directly from our analysis of the quantum Berezin transform andthe following lemma, which is certainly part of the folklore knowledge, but seems not to bedirectly available in the literature. We remark that when the quantum Gromov-Hausdorffdistance is replaced by Latrémolière’s quantum propinquity, the statement can be found in[33, Theorem 6.3], but for the benefit of the reader, we include a proof for the correspondingstatement in our setting. Lemma 4.16.
Let ( X, L ) be a compact quantum metric space and let Y ⊆ X be a sub-operator system such that ( Y, L | Y ) is again a compact quantum metric space. Let ε > .If for every x ∈ X there exists y ∈ Y such that L ( y ) L ( x ) and k x − y k εL ( x ) , then dist Q (cid:0) ( X, L ); (
Y, L | Y ) (cid:1) ε .Proof. As explained in Section 2.3, the compact quantum metric spaces ( X, L ) and ( Y, L | Y ) give rise to order unit compact quantum metric space denoted by A := { x ∈ X sa | L ( x ) < ∞} and B := { y ∈ Y sa | L ( y ) < ∞} . And in fact, since dist Q (cid:0) ( X, L ); (
Y, L | Y ) (cid:1) = dist Q (cid:0) ( A, L ); (
B, L | B ) (cid:1) one may as well pass to the order unit setting when considering matters related to thequantum Gromov-Hausdorff distance.We define a seminorm L ε on A ⊕ B by setting L ε ( a, b ) := max (cid:8) L ( a ) , L ( b ) , ε k a − b k (cid:9) . By construction, L ε ( a, b ) = 0 if and only if ( a, b ) = t (1 , for some t ∈ R . By assumption,for each a ∈ A there exists b ∈ B such that L ( b ) L ( a ) and k a − b k εL ( a ) and hence L ε ( a, b ) L ( a ) . Conversely, if b ∈ B we trivially have that L ε ( b, b ) L ( b ) . This proves that the assumptionsin [44, Theorem 5.2] are fulfilled, and from this we obtain that L ε : A ⊕ B → [0 , ∞ ) is anadmissible Lip-norm.Next, denote by π : A ⊕ B → A and π : A ⊕ B → B the natural projections, and note thatby [44, Proposition 3.1], the dual maps π ∗ : S ( A ) → S ( A ⊕ B ) and π ∗ : S ( B ) → S ( A ⊕ B ) are isometries for the associated Monge-Kantorovič metrics. Given ψ ∈ S ( B ) , we extend ψ to a state ˜ ψ on A using the Hahn-Banach theorem, see [4, Chapter II (1.10)]. For ( a, b ) ∈ A ⊕ B with L ε ( a, b ) , we have k b − a k ε and thus (cid:12)(cid:12) π ∗ ( ψ )( a, b ) − π ∗ ( ˜ ψ )( a, b ) (cid:12)(cid:12) = (cid:12)(cid:12) ψ ( b ) − ˜ ψ ( a ) (cid:12)(cid:12) = (cid:12)(cid:12) ˜ ψ ( b − a ) (cid:12)(cid:12) k b − a k ε. Hence ρ L ε (cid:0) π ∗ ( ψ ) , π ∗ ( ˜ ψ ) (cid:1) ε . Conversely, given ϕ ∈ S ( A ) we have ϕ | B ∈ S ( B ) and ananalogous computation proves the inequality ρ L ε (cid:0) π ∗ ( ϕ | B ) , π ∗ ( ϕ ) (cid:1) ε . Hence, we haveshown that dist ρ Lε H (cid:0) π ∗ ( S ( A )) , π ∗ ( S ( B )) (cid:1) ε, and therefore dist Q (cid:0) ( A, L ); (
B, L | B ) (cid:1) ε as desired. (cid:3) HE PODLEŚ SPHERES CONVERGE TO THE SPHERE 31
Our result regarding convergence of the quantum fuzzy spheres towards the Podleśsphere can now be stated and proved. We emphasize that the domain of our Lip-norm L D q : C ( S q ) → [0 , ∞ ] is given by the coordinate algebra O ( S q ) . Theorem 4.17.
For each q ∈ (0 , it holds that lim N →∞ dist Q (cid:0) F Nq ; C ( S q ) (cid:1) = 0 .Proof. This follows from Proposition 3.12, Corollary 4.5 and Lemma 4.16. (cid:3)
Note that when q = 1 , Theorem 4.17 gives a variation of Rieffel’s original result [45,Theorem 3.2], but since our Lip-norm on F N is a priori different from the one consideredin [45], we do not recover the classical result verbatim.4.6. The Podleś spheres converge to the sphere.
In this section we prove the mainresult of the paper, which at this point follows rather easily from the analysis carried outin the previous sections.
Theorem 4.18.
For any q ∈ (0 , one has lim q → q dist Q (cid:0) C ( S q ); C ( S q ) (cid:1) = 0 .Proof. Let ε > be given. By Proposition 3.12, Proposition 4.14 and Lemma 4.16 thereexists an open interval I containing q and an N ∈ N such that dist Q ( F Nq ; C ( S q )) < ε/ for all q ∈ I ∩ (0 , . Upon shrinking I if needed, Proposition 4.15 shows that wemay assume dist Q ( F Nq ; F Nq ) < ε/ for all q ∈ I ∩ (0 , . Given q ∈ I , the inequal-ity dist Q (cid:0) C ( S q ); C ( S q ) (cid:1) < ε now follows from the triangle inequality for the quantumGromov-Hausdorff distance, see [44, Theorem 4.3]. (cid:3) Remark . The above theorem is in reality a result about the coordinate algebras O ( S q ) , q ∈ (0 , , in so far that these are exactly the domains of the Lip-norms L D q : C ( S q ) → [0 , ∞ ] . We do now know whether the same convergence result holds when L D q is replacedby L max D q as defined in Remark 2.7. However, we conjecture that the quantum Gromov-Hausdorff distance between ( C ( S q ) , L D q ) and ( C ( S q ) , L max D q ) is equal to zero for each q ∈ (0 , and this would of course imply such a convergence result. References [1] Konrad Aguilar.
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Department of Mathematics and Computer Science, The University of Southern Den-mark, Campusvej 55, DK-5230 Odense M, Denmark
Email address : [email protected] Department of Mathematics and Computer Science, The University of Southern Den-mark, Campusvej 55, DK-5230 Odense M, Denmark
Email address : [email protected] Department of Mathematics and Computer Science, The University of Southern Den-mark, Campusvej 55, DK-5230 Odense M, Denmark
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