Spectral metrics on quantum projective spaces

We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital spectral triple introduced by D'Andrea and Dabrowski. In particular, we establish that the Connes metric metrizes the weak-* topology on the state space of quantum projective space. This generalizes previous work by the second author and Aguilar regarding spectral metrics on the standard Podles spheres.


Introduction
In the last two decades it has become more and more apparent that one of the key open problems in noncommutative geometry is to reconcile this theory with the theory of compact quantum groups, [11,10,54].An important aspect of this problem concerns the construction of spectral triples on q-deformations of classical Lie groups in a way which is compatible with the q-geometry as witnessed by the quantized enveloping algebra, [19,25].We note that there are currently interesting examples of spectral triples available for q-deformations as described in [17,7] for the case of quantum SU (2) and in the general setting in [45].These examples are however less sensible to the underlying q-geometry since the spectra of the corresponding abstract Dirac operators are similar to the spectra of their classical counter parts.Studies of the important example quantum SU(2) do in fact suggest that the current spectral triple framework for noncommutative geometry (with or without twists) is too inflexible to capture all of the rich geometric features of this compact quantum group, [33,30,28,5].
Instead, it appears that some of the quantum flag manifolds are more amenable to q-geometric investigations using spectral triples as the main theoretical framework, see [32,34] where the relationship to the covariant first order differential calculus found by Heckenberger and Kolb is also highlighted, [21,22].The hope is then that these quantum flag manifolds can be used as building blocks for understanding larger and larger parts of the q-deformed classical Lie groups through the lens of noncommutative geometry, see e.g.[33,28] for applications of this idea in the case of quantum SU (2).The correct way of assembling these building blocks is in principle dictated by unbounded KK-theory as pioneered by Mesland in [40] and consolidated in [29,41].It should however be noted that the current version of unbounded KKtheory features a too restrictive focus on unbounded Kasparov modules (bivariant spectral triples), meaning that the incorporation of twisted commutators is currently not allowed in unbounded KK-theory, even though some progress in this direction has been made in [26].
We are in this text studying the noncommutative geometry of the higher dimensional quantum projective spaces which are quantum flag manifolds associated with quantum SU(N).The point of departure is the unital spectral triples introduced by D'Andrea and Dąbrowski in [14] building on the earlier work [15] including Landi as a coauthor.
One of the many fundamental insights of Connes was that any unital spectral triple gives rise to an (extended) metric on the state space of the associated unital C * -algebra, [9].It was then emphasized by Rieffel that it is important to ask whether the Connes metric metrizes the weak- * topology or not.This question leads to the whole theory of compact quantum metric spaces as pioneered by Rieffel in [47,48] and prominently featuring a quantum analogue of the Gromov-Hausdorff distance between compact metric spaces, [49].The quantum Gromov-Hausdorff distance has later on been fine tuned to incorporate more and more structure in the deep and extensive work of Latrémolière, [36,37,38].
The main result of this paper can be formulated as follows: The Connes metric metrizes the weak- * topology on the state space of quantum projective space.In other words, the unital spectral triple introduced by D'Andrea and Dąbrowski turns quantum projective space into a spectral metric space.
As a special case of this theorem we recover the main result from [1] which states that the standard Podleś sphere becomes a spectral metric space when equipped with the Dąbrowski-Sitarz spectral triple introduced in [18].
Before giving some details on our method of proof we pay attention to a certain detail regarding the coordinate algebra appearing in a spectral triple.This choice of coordinate algebra influences the definition of the Connes metric on the state space and the larger this algebra is, the harder it becomes to recover the weak- * topology.We are in this text working with the Lipschitz algebra which is the maximal choice of coordinate algebra (with respect to inclusion) and our result regarding spectral metrics on quantum projective space can therefore not be strengthened.A part from maximality, another feature of the Lipschitz algebra is that it can be reconstructed from the C * -algebra and the abstract Dirac operator of the spectral triple (making it canonical in this sense as well).
In the proof of our main theorem we rely on the structure of quantum SU(N) as a C * -algebraic compact quantum group.More precisely, we apply the coaction of quantum SU(N) on quantum projective space together with a sequence of states converging in the weak- * topology to the counit (restricted to quantum projective space).Each of the states appearing in this sequence provides us with a finite dimensional approximation of the inclusion of quantum projective space into quantum SU(N).It turns out that the precision of this approximation can be estimated by the distance in the Connes metric between the state in question and the restricted counit.
Applying the description of compact quantum metric spaces in terms of finite dimensional approximations appearing in [27], we see that our main result follows, if we can establish that the sequence of states converges to the restricted counit with respect to the Connes metric (and not just in weak- * topology).Establishing this convergence result is still a difficult task, but it can be made substantially less complicated by considering the invariance properties of the Connes metric.We show that the Connes distance between our states is invariant under a conditional expectation from quantum projective space onto a much smaller and commutative unital C * -subalgebra, which is * -isomorphic to the continuous functions on the subset I q = {q 2m | m ∈ N 0 } ∪ {0} of the real line.
We refer to I q as the quantized interval and the above observations ensure that we only need to understand the spectral metric properties of I q (equipped with the unital spectral triple obtained by restriction of the unital spectral triple on quantum projective space).This is a manageable task and it turns out that the relevant metric information can be extracted from difference quotients for continuous functions on I q and the single positive continuous function q −1 x(1 − q 2 x).Notably, this latter function (without the factor q −1 ) also plays a major role for the continuity properties of the Podleś spheres in quantum Gromov-Hausdorff distance investigated in [20,2].
Acknowledgements.The authors gratefully acknowledge the financial support from the Independent Research Fund Denmark through grant no.9040-00107B, 7014-00145B and 1026-00371B.
We would like to thank Marc Rieffel and Walter van Suijlekom for making us aware of the articles [50] and [17].It is also a pleasure to thank our colleague David Kyed for valuable input to our proof of the extension theorem.

Preliminaries
All Hilbert spaces in this text are assumed to be separable and for such a Hilbert space H we let B(H) denote the unital C * -algebra of bounded operators on H.The inner product on the Hilbert space H (which is anti-linear in the first variable) is denoted by •, • and the notation • 2 refers to the corresponding norm on H.The unique C * -norm on a C * -algebra is written as • .
Throughout this article, the deformation parameter q is fixed and belongs to the open unit interval (0, 1).We shall moreover fix a natural number ℓ ∈ N and put N := ℓ + 1.
For integers i, j ∈ Z we apply the notation δ ij for the corresponding Kronecker delta and for a natural number n ∈ N we put n := {1, . . ., n} and 0 := ∅.The unital C * -algebra of matrices of size n is denoted by M n (C) and the corresponding matrix units are written as e ij for i, j in the index set n .

2.1.
Compact quantum metric spaces.In this subsection, we present Rieffel's notion of a compact quantum metric space, see [47,48,49].We notice that the current trend is to describe the theory in the generality of operator systems in line with recent developments in noncommutative geometry, [12,13,51].Keeping in mind the scope of the present paper, we are going to focus on the C * -algebraic framework for compact quantum metric spaces.
Let A be a norm-dense unital * -subalgebra of a unital C * -algebra A and consider a seminorm L : A → [0, ∞).The following terminology can be found in [50]: From now on, we shall assume that L : A → [0, ∞) is a slip-norm.We define an extended metric -referred to as the Monge-Kantorovich metric -on the state space of A by putting for all µ, ν ∈ S(A).The adjective "extended" means that mk L can take the value infinity but otherwise satisfies the usual requirements of a metric.Definition 2.2.We say that the pair (A, L) is a compact quantum metric space if mk L metrizes the weak- * topology on the state space S(A).
If (A, L) is a compact quantum metric space, then it follows from connectedness of S(A) in the weak- * topology that the Monge-Kantorovich metric is actually finite so that (S(A), mk L ) is a metric space in the usual sense.We now quote a result which can be found in [47,Proposition 1.6] and [48,Proposition 2.2].The notation [•] : A → A/C1 refers to the quotient map and the quotient space A/C1 is equipped with the quotient norm • A/C1 (coming from the C * -norm on A).

Proposition 2.3. The following are equivalent:
(1) There exists a constant We say that the pair (A, L) has finite diameter, if the equivalent conditions from Proposition 2.3 are satisfied.Notice that if (A, L) has finite diameter, then the kernel of L agrees with C1.Recently, it was shown in [35] that the pair (A, L) has finite diameter if and only if the Monge-Kantorovich metric is finite (meaning that the extended metric mk L is a metric).As a consequence, we see that every compact quantum metric space has finite diameter.This latter fact can also be deduced from compactness of the state space in the weak- * topology.
We shall now introduce a convenient characterization of compact quantum metric spaces which was obtained in [27,Theorem 3.1].The proof relies to some extent on Rieffel's characterization of compact quantum metric spaces, see [47,Theorem 1.8].We first need a definition.Definition 2.4.Fix a constant ε > 0, a unital C * -algebra B and two unital linear maps ι, Φ : A → B. Suppose that ι is isometric and that Φ is positive.We say that the pair (ι, Φ) is an ε-approximation of (A, L) if (1) The image of Φ is finite dimensional (as a vector space over C); (2) ι(x) − Φ(x) ε • L(x) for all x ∈ A.
As a first example, consider a classical compact metric space (X, d) and let Lip(X) denote the Lipschitz functions on X, sitting as a norm-dense unital * -subalgebra of the unital C * -algebra C(X) of continuous functions on X.We equip the Lipschitz functions with the slip-norm L d defined by so that L d (f ) agrees with the Lipschitz constant of a Lipschitz function f (the above supremum is by convention equal to zero in the case where X is a singleton).It then holds that the pair Lip(X), L d is a compact quantum metric space, see for example [46,Theorem 2.5.4].For a more recent proof using ε-approximations we refer to [27,Lemma 3.4].
An important class of noncommutative examples arises from unital spectral triples, see [9] and [10,Chapter 6].Indeed, suppose that we have a unital spectral triple (A, H, D) and let us identify A with a unital * -subalgebra of the bounded operators on H (via a fixed unital injective * -homomorphism).We define the unital C * -algebra A ⊆ B(H) as the closure of A in operator norm.The unital spectral triple gives rise to a closable derivation d : A → B(H) defined by putting d(x) := cl([D, x]) so that d(x) agrees with the closure of the commutator [D, x] : Dom(D) → H.This construction does in turn yield the slip-norm In this context, the extended metric mk L D on the state space of A is often referred to as the Connes metric.
The terminology "spectral metric space" introduced here below comes from the paper [4].Let us however emphasize that there are many examples of unital spectral triples which are not spectral metric spaces.It can even happen that the kernel of L D agrees with the scalars while some states have distance infinity with respect to the Connes metric, see [35] for an elementary example of this phenomenon.For a more geometric example we refer to [6] where a similar behaviour is exhibited for the Moyal plane (even though this is in the non-compact setting).Definition 2.6.We say that the unital spectral triple (A, H, D) is a spectral metric space if (A, L D ) is a compact quantum metric space.
Let us clarify that the unital * -subalgebra A ⊆ A appearing in the unital spectral triple (A, H, D) can be enlarged in a substantial fashion.Indeed, instead of A we could consider the Lipschitz algebra Lip D (A) ⊆ A defined as follows: An element x ∈ A belongs to Lip D (A) if and only if the supremum is finite, see [8,Theorem 3.8] for a number of equivalent conditions.We record that Lip D (A), H, D is a unital spectral triple and we therefore obtain the slip-norm To avoid confusion we often denote this latter slip-norm by L max D and notice that L max D (x) = L D (x) for all x ∈ A. Likewise, we often apply the notation d max : Lip D (A) → B(H) for the closed derivation coming from the enlarged unital spectral triple.Remark that L max D (x) agrees with the supremum in (2.1) for x ∈ Lip D (A) and that this supremum in turn agrees with the operator norm of d max (x).
It is important to keep in mind that Lip D (A) ⊆ A can very well be different from the domain of the closure d of the derivation d : A → B(H), but it always holds that Dom( d ) ⊆ Lip D (A).In particular we get from Theorem 2.5 that if (Lip D (A), H, D) is a spectral metric space, then (A, H, D) is a spectral metric space.We expect that the converse is not true but at the time of writing we are not aware of any counter examples.
2.2.Quantum SU(N).We are now going to review some of the main concepts pertaining to the quantized version of the special unitary group.Our main reference on these matters is the book [31] and in particular the chapters 6.1 and 9.2.For completeness we also refer the reader to the foundational papers on quantized enveloping algebras, [19,25] and C * -algebraic compact quantum groups, [53,54].
Let us start out by introducing the universal unital C-algebra O(M q (N)) generated by N 2 elements u ij , i, j ∈ N subject to the quadratic relations (2.2) Whenever we have two non-empty subsets I, J ⊆ N with the same number of elements, say n, we may choose 1 i 1 < . . .< i n N and 1 j 1 < . . .< j n N such that I = {i 1 , . . ., i n } and J = {j 1 , . . ., j n }.The quantum n-minor determinant with respect to the subsets I and J is then the element in O(M q (N)) defined by where S n denotes the group of permutations of the set n and ι(σ) is the number of inversions in a permutation σ ∈ S n .For I = J = N we refer to D IJ as the quantum determinant.Moreover, for each i, j ∈ N we define the cofactor The notation O(SL q (N)) refers to the unital C-algebra obtained as the quotient of O(M q (N)) by the ideal generated by D N , N − 1 (so inside O(SL q (N)), the quantum determinant equals one).It follows from [31, Chapter 9.2.3,Proposition 10] that O(SL q (N)) becomes a Hopf algebra with coproduct, counit and antipode given on generators by As in [31,Chapter 9.2.4], the coordinate algebra for quantum SU(N) is defined as the Hopf * -algebra O(SU q (N)) which agrees with O(SL q (N)) as a Hopf algebra, but with involution defined on generators by For later use it is important to record the following further relations inside O(SU q (N)), see [31, Proposition 8, Chapter 9.2.2]: Letting u ∈ M N O(SU q (N)) denote the (N × N)-matrix with entries u ij , i, j ∈ N , we see from (2.3) that u is unitary, and we refer to this unitary matrix as the fundamental unitary.
We also introduce the quantized enveloping algebra of su(N) defined as the universal unital * -algebra U q (su(N)) generated by 3ℓ elements E i , K i , K −1 i , i ∈ ℓ subject to the following relations for all i, j ∈ ℓ : The unital * -algebra U q (su(N)) becomes a Hopf * -algebra with coproduct, counit and antipode determined by the formulae For more details we refer the reader to [31,Chapter 6.1.2]-notice here that Klimyk and Schmüdgen apply the notation Ȗq (su N ) for the Hopf * -algebra we are denoting by U q (su(N)).
Letting e 1 , . . ., e N denote the standard orthonormal basis for C N we obtain the functionals π ij : U q (su(N)) → C given by π ij (η) := e i , π(η)e j .These functionals give rise to a dual pairing of Hopf * -algebras as made explicit in the next theorem.We are presenting an important detail on the proof since it is less easy to find in [31] even though the result is very much related to [31,Theorem 18,Chapter 9.4].
Theorem 2.7.There exists a unique dual pairing of Hopf * -algebras •, • : This identity implies that the functionals π ij : U q (su(N)) → C satisfy the relations in (2.2) inside the dual Hopf * -algebra U q (su(N)) • (with π ij replacing u ij ).We therefore get a unital algebra homomorphism Φ : O(M q (N)) → U q (su(N)) • and, applying the argument from [31, Theorem 18, Chapter 9.4], it follows that Φ vanishes on the element D N , N − 1 ∈ O(M q (N)).It is then not difficult to see that the induced map Φ : O(SU q (N)) → U q (su(N)) • is in fact a homomorphism of Hopf * -algebras and this ends the proof of the theorem.
The dual pairing from Theorem 2.7 gives rise to a left action of U q (su(N)) on O(SU q (N)) (by linear endomorphisms) defined by for x ∈ O(SU q (N)) and η ∈ U q (su(N)).We record that for all r ∈ ℓ and i, j ∈ N .Furthermore, it follows from the defining properties of a dual pairing that we have the identities for all x, y ∈ O(SU q (N)) and η ∈ U q (su(N)) (where we apply the Sweedler notation ).In order to avoid the flip in the first identity in (2.6) one may replace it by the rule d S(η As a consequence of the above considerations we get that . It moreover holds that d E i and d F i are twisted derivations of O(SU q (N)), meaning that they satisfy the twisted Leibniz rules for all x, y ∈ O(SU q (N)).The behaviour of the operations d E i , d F i and d K i with respect to the involution is summarized by the formulae We define quantum SU(N) -denoted C(SU q (N)) -as the universal C *algebra generated by O(SU q (N)).The comultiplication extends to a unital *homomorphism ∆ : C(SU q (N)) → C(SU q (N)) ⊗ min C(SU q (N)) (applying the minimal tensor product), and in this way C(SU q (N)) is turned into a (C * -algebraic) compact quantum group, see [53,54].It also follows by construction that the counit extends to a unital * -homomorphism ǫ : Let us reserve the notation h : C(SU q (N)) → C for the Haar state of the compact quantum group C(SU q (N)) and recall that this state is uniquely determined by the identity h(1) = 1 together with the bi-invariance see [54,Theorem 1.3].As a consequence of the bi-invariance property we get that Let us denote the modular automorphism of the Haar state by θ : O(SU q (N)) → O(SU q (N)), see [54,Theorem 5.6].Record in this respect that θ(x * ) = θ −1 (x) * and that h(xy) = h(yθ(x)) for all x, y ∈ O(SU q (N)).This modular automorphism can be computed explicitly on the generators for O(SU q (N)).By [31, Proposition 34 and Example 9, Chapter 11.3.4] it holds that (2.10) Let L 2 (SU q (N)) denote the Hilbert space associated with the GNS-construction applied to the Haar state.We know from [43, Theorem 1.1] that the Haar state is faithful and it follows that both the GNS-representation π : C(SU q (N)) → B(L 2 (SU q (N))) and the canonical linear map ι : C(SU q (N)) → L 2 (SU q (N)) are injective.We recall that inner products are linear in the second variable so that ι(x), ι(y) = h(x * y) for all x, y ∈ C(SU q (N)).

2.3.
Peter-Weyl decomposition and Schur orthogonality.In this subsection we briefly remind the reader of the Peter-Weyl decomposition of the coordinate algebra O(SU q (N)) and a relevant consequence of the Schur orthogonality relations.These results are in fact available in the more general setting of cosemisimple Hopf algebras, see [31,Chapter 11.2], but we are here focusing on the specific example O(SU q (N)).
Recall that a corepresentation of O(SU q (N)) is a linear map v : A corepresentation v acting on C n (with standard orthonormal basis {e k } n k=1 ) gives rise to the matrix coefficients v ij labelled by i, j ∈ {1, . . ., n} and satisfying that v(e j ) = n i=1 e i ⊗ v ij for all j ∈ n .
Two irreducible corepresentations v and w (both acting on C n ) are said to be equivalent, if there exists an invertible linear map T : Letting O(SU q (N)) denote the set of equivalence classes of irreducible unitary corepresentations we choose a representative v α : The Peter-Weyl decomposition in this setting says that the set of matrix coefficients We emphasize here that the entries u ij for i, j ∈ N of the fundamental unitary u are in fact matrix coefficients of a unitary corepresentation and it therefore follows that O(SU q (N)) is a compact matrix quantum group algebra in the sense of [ For each α ∈ O(SU q (N)) we define the finite dimensional subspace ) and record that C(α) is in fact a coalgebra in its own right since ∆(C(α)) ⊆ C(α) ⊗ C(α).The Schur orthogonality relations then imply that these subspaces are mutually orthogonal meaning that h(x * y) = 0 whenever x ∈ C(α) and y ∈ C(β) for some α = β, see [31, Proposition 15, Chapter 11.2.2].
2.4.Quantum spheres and quantum projective spaces.In this subsection we recall the definition of the Vaksman-Soibelman quantum spheres and the associated quantum projective spaces introduced in [52].Inside the quantum projective space (of complex dimension ℓ) we pay particular attention to a small commutative unital C * -subalgebra which is isomorphic to the continuous functions on the quantized interval I q := q 2m | m ∈ N 0 ∪ {0}.The quantized interval is going to play a pivotal role in our approach to the spectral metric structure of quantum projective spaces.
For each i ∈ N we put z i := u N i so that z i agrees with the i th entry in the last row of the fundamental unitary u.We let O(S 2ℓ+1 q ) denote the smallest unital * -subalgebra of O(SU q (N)) such that z i belongs to O(S 2ℓ+1 q ) for all i in N .This unital * -subalgebra is referred to as the coordinate algebra for the quantum sphere.
The following identities are consequences of the defining relations for the coordinate algebra for quantum SU(N): (2.11) We let O(CP ℓ q ) denote the smallest unital * -subalgebra of O(S 2ℓ+1 q ) such that z i z * j ∈ O(CP ℓ q ) for all i, j ∈ N .This unital * -subalgebra is called the coordinate algebra for the quantum projective space.
We define the unital C * -algebras C(S 2ℓ+1 q ) and C(CP ℓ q ) as the C * -closures of O(S 2ℓ+1 q ) and O(CP ℓ q ) inside C(SU q (N)).We refer to C(S 2ℓ+1 q ) and C(CP ℓ q ) as the quantum sphere and the quantum projective space, respectively.
The coproduct ∆ on the coordinate algebra for quantum SU(N) induces a right coaction on both the coordinate algebra for the quantum sphere and the quantum projective space.These coactions are denoted by and they extend by continuity to right coactions of the compact quantum group C(SU q (N)) on the quantum sphere and the quantum projective space.We introduce the elements x i := z i z * i for i ∈ N together with the sums It is relevant to analyse the ) is * -isomorphic to the continuous functions on the spectrum of y ℓ .An application of [23,Lemma 4.2] (see also [42,Proposition 2.11]) shows that As mentioned earlier, we refer to this subset of R as the quantized interval and apply the notation ) via Gelfand duality.We shall from now on identify C(I q ) with a unital C * -subalgebra of C(CP ℓ q ).To be explicit, this identification sends the inclusion I q → C to the element y ℓ .
The indicator function for the point q 2k ∈ R restricts to a continuous function on Sp(y ℓ ) as soon as k ∈ N 0 .The corresponding projection in C(I q ) (obtained via continuous functional calculus) is denoted by p k .For k = 0, we may describe p 0 as the limit p 0 = lim n→∞ y n ℓ and for k > 0 it can be verified that (2.12) We apply the convention that p k := 0 for k ∈ Z \ N 0 .

Spectral geometry of quantum projective spaces
In this section, we review the spectral geometry of quantum projective spaces as witnessed by the unital spectral triple introduced by D'Andrea and Dąbrowski in [14] building on the earlier work [15] and [18] where the lower dimensional cases are treated.It should be emphasized that related constructions appear in many places, see e.g.[16,34,39], but in the present text we follow the original approach of D'Andrea and Dąbrowski closely.
We start out with a small well-known result relating the adjoint operation on U q (su(N)) to the adjoint operation of closable unbounded operators on the Hilbert space L 2 (SU q (N)).As usual it is convenient to apply Sweedler notation for coproducts.
Lemma 3.1.For every x, y ∈ O(SU q (N)) and ξ ∈ U q (su(N)) it holds that Proof.Applying (2.6) and (2.9) we compute as follows: In order to define a q-deformed analogue of the antiholomorphic forms over quantum projective space we consider the usual exterior algebra Λ(C ℓ ) = ⊕ ℓ k=0 Λ k (C ℓ ).We view Λ(C ℓ ) as a finite dimensional Hilbert space with orthonormal basis {e I } I⊆ ℓ indexed by the power set of ℓ = {1, . . ., ℓ}.
The operation ♯ counts the number of elements in a finite (or empty) set.Hence, for I ⊆ N and j ∈ N we have that ♯(I ∩ j ) is equal to the number of elements in I which are less than or equal to j.We shall also apply the variant over the Kronecker delta such that δ j,I is equal to 1 for j ∈ I and 0 for j / ∈ I.For each j ∈ ℓ we are interested in the q-deformed exterior multiplication operator ε q j : Λ(C ℓ ) → Λ(C ℓ ) determined by ε q j (e I ) := 0 for j ∈ I (−q) −♯(I∩ j ) e I∪{j} for j / ∈ I .
It is important to record the commutation rule ε q i ε q j = −qε q j ε q i for all i < j.Recall that U q (su(ℓ)) ⊆ U q (su(N)) denotes the unital * -subalgebra of U q (su(N)) generated by the elements K r , K −1 r , E r for r < ℓ (for ℓ = 1 this is just the scalars C1).It clearly holds that U q (su(ℓ)) is a Hopf * -subalgebra of U q (su(N)).
Let us clarify the compatibility between the representation σ and the q-deformed exterior multiplication operators as investigated in [14, Proposition 3.9].For each vector v ∈ C ℓ = Λ 1 (C ℓ ) we define ) for all η ∈ U q (su(ℓ)) and v ∈ C ℓ .Let us from now on fix an integer M ∈ Z which is going to index a twist of the antiholomorphic forms by a finitely generated projective module over the coordinate algebra O(CP ℓ q ).Notice that the incorporation of this twist is necessary in order to recover the Dąbrowski-Sitarz spectral triple over the Podleś sphere for ℓ = 1, see Subsection 3.2.
For each I ⊆ ℓ we define the subspace x of the coordinate algebra O(SU q (N)).For each k ∈ {0, 1, . . ., ℓ} we then introduce the subspace Definition 3.3.Let k ∈ {0, 1, . . ., ℓ} and M ∈ Z.The twisted antiholomorphic forms of degree k are defined as the subspace Let us specify that Ω k M carries a left action of the coordinate algebra O(CP ℓ q ) which is inherited from the algebra structure of O(SU q (N)).Indeed, because of (2.13) and (2.14) for every ω ∈ Ω k M and x ∈ O(CP ℓ q ) it holds that (x⊗1)•ω ∈ Ω k M .We shall often view Ω k M as a subspace of the Hilbert space tensor product L 2 (SU q (N)) and the corresponding Hilbert space completion of by taking the direct sum of the representations ρ k on L 2 (Ω k M , h).Notice that the injectivity of ρ for M = 0 relies on the faithfulness of the Haar state (restricted to the quantum projective space C(CP ℓ q )).For M = 0 one also needs the sphere relations N i=1 z i z * i = 1 = N i=1 q 2(N −i) z * i z i together with the extra observation that z j ∈ Ω 0 −1 and z * j ∈ Ω 0 1 for all j ∈ N .The Hilbert space L 2 (Ω M , h) is equipped with the Z/2Z-grading with grading operator γ : For each i ∈ ℓ we introduce the element M i ∈ U q (su(N)) by putting M ℓ := E ℓ and then recursively define We shall also need the element N i ∈ U q (su(N)) given by N With these definitions ready we introduce the linear maps which a priori act on the algebraic tensor product O(SU q (N)) ⊗ Λ(C ℓ ).The next result, which can be found as [14, Proposition 5.6], guarantees that the above linear maps preserve the subspace Ω M ⊆ O(SU q (N)) ⊗ Λ(C ℓ ).We give a few details on the proof.Proof.We focus on showing that the endomorphism ∂ : M it now suffices to fix an r ∈ ℓ − 1 and show that the three commutators vanish.We present the relevant details for the third commutator.
An application of [14,Lemma 3.2] shows that F r commutes with Y i whenever i = r, r + 1 and for the remaining cases we have the identities Similarly, we see from Lemma 3.2 that σ(F r ) commutes with ε q i whenever i = r, r + 1 and the remaining cases satisfy the identities σ(F r )ε q r = ε q r+1 σ(K r ) + q −1/2 ε q r σ(F r ) and σ(F r )ε q r+1 = q 1/2 ε q r+1 σ(F r ).We also record that K r commutes with Y i whenever i = r, r + 1 and that Another application of Lemma 3.2 moreover entails that σ(K r )ε q r = q 1/2 ε q r σ(K r ) and σ(K r )ε q r+1 = q −1/2 ε q r+1 σ(K r ) whereas σ(K r ) commutes with the remaining q-deformed exterior multiplication operators.Combining these formulae, we get that the third commutator in (3.3) is equal to zero.
The above proposition allows us to define the symmetric unbounded operator The closure is denoted by D q := cl(D q ).It is not hard to see that D q anticommutes with the grading operator γ, entailing that D q is odd.The following theorem is part of [14, Theorem 6.2] but D'Andrea and Dąbrowski also treat equivariance, reality and determine the spectral dimension.We present some further discussion on the equivariance condition in the next subsection.
We apply the notation Lip Dq (CP ℓ q ) ⊆ C(CP ℓ q ) for the Lipschitz algebra associated with the above unital spectral triple and the corresponding closed derivation is denoted by d max : Lip Dq (CP ℓ q ) → B L 2 (Ω M , h) .As discussed in Section 2.1, the unital spectral triple from Theorem 3.5 induces two slip-norms which agree on O(CP ℓ q ).Moreover, we have the expression L max Dq (x) := d max (x) for all x ∈ Lip Dq (CP ℓ q ).We are in this text almost exclusively focusing on the slip-norm L max Dq .
3.1.Commutators and equivariance.In this subsection we continue our review of the noncommutative geometry of quantum projective spaces.More precisely, we discuss the equivariance properties and give an explicit computation of the closed derivation d max in the case where the input belongs to the coordinate algebra O(CP ℓ q ).We recall that M ∈ Z is a fixed integer.Let us first consider the restriction of d max to the coordinate algebra: Lemma 3.6.Let x ∈ O(CP ℓ q ).The commutators [∂, x ⊗ 1] and [∂ † , x ⊗ 1] (initially defined on Ω M ) extend to bounded operators on L 2 (Ω M , h) given respectively by the two expressions Proof.By [14, Lemma 5.7] we have that the commutator [∂, x ⊗ 1] extends to the bounded operator ℓ i=1 d N i M * i (x) ⊗ ε q i (which therefore in particular preserves the subspace Ω M ).Hence to prove our formula for We therefore only need to show that . This is straightforward to verify using the recursive definition of M i from (3.2) together with the fact that d Fr (x) = 0 for all r ∈ ℓ − 1 .
The formula for [∂ † , x ⊗ 1] now follows by taking adjoints.Indeed, for ω ∈ Ω M we apply (2.6) and Proposition 3.4 to get that Our next objective is to describe the equivariance properties of the unital spectral triple Lip Dq (CP ℓ q ), L 2 (Ω M , h), D q .Since we are here dealing with the Lipschitz algebra Lip Dq (CP ℓ q ) and not just the coordinate algebra O(CP ℓ q ), the best way of explaining equivariance is to work with the multiplicative unitary which implements the coproduct of quantum SU(N) at the Hilbert space level, see [3] for more details on multiplicative unitaries.
Define the endomorphism W of the algebraic tensor product O(SU q (N)) ⊗ O(SU q (N)) by the formula An application of the Peter-Weyl decomposition and the faithfulness of the Haar state (at the level of the coordinate algebra) shows that W is in fact a linear automorphism of O(SU q (N))⊗O(SU q (N)).The same argument also shows that W induces a unitary operator W on the Hilbert space tensor product L 2 (SU q (N)) ⊗L 2 (SU q (N)).This unitary operator implements the coproduct on the compact quantum group C(SU q (N)) in so far that where the quantities on both sides are interpreted as bounded operators on L 2 (SU q (N)) ⊗L 2 (SU q (N)).
In fact, for the purposes of this text it is relevant to consider a more general situation.Let us look at a unital C * -algebra A and assume that µ : A → C is a fixed faithful state.Let L 2 (A, µ) denote the corresponding Hilbert space coming from the GNS-construction and let π : A → B(L 2 (A, µ)) denote the associated injective unital * -homomorphism.Suppose that Φ : C(SU q (N)) → A is a surjective unital * -homomorphism and put A := Φ O(SU q (N)) so that A ⊆ A is a normdense unital * -subalgebra.Instead of W we may then consider the endomorphism W Φ of O(SU q (N)) ⊗ A defined by the formula It can be verified that W Φ is a linear automorphism of the algebraic tensor product O(SU q (N)) ⊗ A and that this linear automorphism induces a unitary operator W Φ on the Hilbert space tensor product L 2 (SU q (N)) ⊗L 2 (A, µ).
For each k ∈ {0, 1, . . ., ℓ} we define the linear automorphism W k Φ of the algebraic tensor product O(SU q (N)) ⊗ Λ k (C ℓ ) ⊗ A by putting In the statement of the next lemma we apply the notation D q ⊗1 for the closure of the unbounded operator Since the unbounded operator ∂ + ∂ † is essentially selfadjoint (by Theorem 3.5) it holds that D q ⊗1 is selfadjoint.

It moreover holds that the commutator
is equal to zero.In particular, we get that ⊕ ℓ k=0 (W k M ) Φ induces a unitary operator Proof.Let η ∈ U q (su(N)) be given and remark that for every k, m ∈ {0, 1, . . ., ℓ} and every linear operator This in turn implies that both W k Φ and (W k Φ ) −1 preserve the subspace Ω k M ⊗ A and moreover that the commutator in (3.4) is trivial.The remaining claims of the lemma now follow by taking closures.
Remark that we may represent the minimal tensor product C(CP ℓ q ) ⊗ min A faithfully on the Hilbert space tensor product L 2 (Ω M , h) ⊗L 2 (A, µ).It should then be emphasized that the unitary operator (W M ) Φ implements the unital * -homomorphism (1 ⊗ Φ)δ : C(CP ℓ q ) → C(CP ℓ q ) ⊗ min A, where δ denotes the right coaction of C(SU q (N)) on C(CP ℓ q ).In other words, we have that where both sides of the equation are understood to be bounded operators on L 2 (Ω M , h) ⊗L 2 (A, µ).
For a single vector ξ ∈ L 2 (A, µ) we introduce the bounded operator T ξ : given by the formula T ξ (ζ) := ζ ⊗ ξ.We record that the operator norm of T ξ is equal to the Hilbert space norm of ξ.The bounded operator T ξ intertwines the unbounded selfadjoint operators D q and D q ⊗1 in so far that we have the inclusion T ξ D q ⊆ (D q ⊗1)T ξ of unbounded operators.Thus, by taking adjoints we also obtain the inclusion T * ξ (D q ⊗1) ⊆ D q T * ξ .For two vectors ξ, η ∈ L 2 (A, µ) we define the linear functional ϕ ξ,η : A → C by putting ϕ ξ,η (x) := ξ, π(x)η and this linear functional can be promoted to a slice map 1 ⊗ϕ ξ,η on the minimal tensor product C(CP ℓ q ) ⊗ min A. Upon recalling that our minimal tensor product is represented faithfully on the Hilbert space tensor product L 2 (Ω M , h) ⊗L 2 (A, µ) we then obtain the relationship ξ zT η for all z ∈ C(CP ℓ q ) ⊗ min A, where it is understood that both sides are operators on the Hilbert space L 2 (Ω M , h).Lemma 3.8.Let ξ, η ∈ A and let x ∈ Lip Dq (CP ℓ q ).It holds that (1 ⊗ ϕ ξ,η Φ)δ(x) belongs to the Lipschitz algebra Lip Dq (CP ℓ q ) and we have the estimate But this follows in a straightforward fashion from the equivariance properties combined in Lemma 3.7 and (3.5) together with the observations made before the statement of the present lemma.To wit, we use that Relationship with the Dąbrowski-Sitarz spectral triple.Let us spend some time clarifying the relationship between the even unital spectral triple O(CP 1 q ), L 2 (Ω 1 , h), D q treated earlier in this section and the Dąbrowski-Sitarz spectral triple over the Podleś sphere, see [18].We are here focusing on the presentation from [44] since this picture is compatible with the investigations of the spectral metric properties of the Podleś sphere given in [1,2].
We apply the notation u = a * −qb b * a for the entries of the fundamental unitary for quantum SU(2).The standard Podleś sphere C(S 2 q ) is defined as the smallest unital C * -subalgebra of C(SU q (2)) containing A := bb * and B := ab * .The corresponding coordinate algebra is denoted by O(S 2 q ) (and is generated as a unital * -subalgebra by A and B).
The dual pairing described in Theorem 2.7 gives rise to an alternative left action of U q (su(2)) on O(SU q (2)) defined by putting ∂ η (x) := (1 ⊗ η, • )∆(x).This left action provides the coordinate algebra O(SU q (2)) with the structure of a Z-graded algebra with homogeneous subspaces defined by It holds that A 0 = O(S 2 q ) so that A n becomes a left module over A 0 for every n ∈ Z.We let H + and H − denote the Hilbert spaces obtained by taking the closure of respectively A 1 and A −1 inside the Hilbert space L 2 (SU q (2)).The left action of A 0 on the dense subspaces A 1 and A −1 can then be promoted to a pair of injective unital * -homomorphisms ρ ± : C(S 2 q ) → B(H ± ).Taking the direct sum of these two representations we obtain a faithful representation ρ of the Podleś sphere on the Z/2Z-graded Hilbert space H + ⊕ H − .
The linear operators ∂ E : A 1 → A −1 and ∂ F : A −1 → A 1 can be promoted and combined into an odd unbounded operator which turns out to be essentially selfadjoint.We apply the notation / D q for the corresponding selfadjoint closure.
The Dąbrowski-Sitarz spectral triple, in the presentation of Neshveyev and Tuset, agrees with the even unital spectral triple given by O(S 2 q ), H + ⊕ H − , / D q .As discussed after Definition 2.6, the Dąbrowski-Sitarz spectral triple also exists in a Lipschitz version which we denote by Lip / D q (S 2 q ), H + ⊕ H − , / D q .On the other hand, following the constructions of the present paper, we have that O(CP 1 q ) is generated by A = bb * and C := ab as a unital * -subalgebra of O(SU q (2)) whereas x and x .Furthermore, the relevant essentially selfadjoint unbounded operator is given by We now introduce the key ingredient needed for describing the relationship between the two even unital spectral triples reviewed in this subsection.Define the * -automorphism T : C(SU q (2)) → C(SU q (2)) by putting T (u ij ) := q j−i u ji i, j ∈ 2 and record that T 2 agrees with the identity operator.We refer to T as the transpose and the first result regarding this operation is that it leaves the Haar state invariant.Lemma 3.9.We have the identity hT = h.
Proof.This can be viewed as a consequence of the explicit formula for the Haar state on quantum SU(2) obtained by Woronowicz in [53, Appendix A].We recall in this respect that where we use the convention that a k := (a * ) −k for k < 0.
As a consequence of Lemma 3.9, we get that the transpose T induces a selfadjoint unitary operator T : L 2 (SU q (2)) → L 2 (SU q (2)).
Proof.The first step is to verify the relevant identity in the case where η ∈ {E, F, K} and x = u ij for some i, j ∈ 2 (this can be done by a straightforward computation).The more general case where x is arbitrary (and η ∈ {E, F, K}) then follows by noting that T d η T obeys the same algebraic rules as ∂ ν(η) with respect to linear combinations and products in the coordinate algebra O(SU q (2)).The case where both η and x are arbitrary finally follows since both sides of the equation are wellbehaved with respect to products and sums of elements in U q (su(2)).
An application of Lemma 3.9 in combination with Lemma 3.10 shows that the transpose restricts to two selfadjoint unitary operators T + : 1 , h) which intertwine the relevant unbounded selfadjoint operators (up to a factor −q −1/2 ) via the identity It is moreover clear from the definition of the transpose that T restricts to a *isomorphism T : C(S 2 q ) → C(CP 1 q ).These observations yield the main result of this subsection, providing a precise relationship between the slip-norms associated to our two even unital spectral triples.Proposition 3.11.The * -isomorphism T : C(S 2 q ) → C(CP 1 q ) restricts to a *isomorphism between the Lipschitz algebras T : Lip / D q (S 2 q ) → Lip Dq (CP 1 q ) satisfying the identity

L max
Dq T (x) = q −1/2 L max / D q (x) for all x ∈ Lip / D q (S 2 q ).In particular, it holds that Lip Dq (CP 1 q ), L 2 (Ω 1 , h), D q is a spectral metric space if and only if Lip / D q (S 2 q ), H + ⊕ H − , / D q is a spectral metric space.

The extension theorem
Throughout this whole section we let M ∈ Z be fixed since this M appears in the definition of the even unital spectral triple O(CP ℓ q ), L 2 (Ω M , h), D q from Theorem 3.5.
Let us introduce the norm-dense unital * -subalgebra Lip(I q ) ⊆ C(I q ) of q-Lipschitz functions on the quantized interval I q = Sp(y ℓ ) by putting Lip Dq (I q ) := C(I q ) ∩ Lip Dq (CP ℓ q ).Notice here that we are identifying C(I q ) with the unital C * -subalgebra of quantum projective space generated by y ℓ , see the discussion in Subsection 2.4.The q-Lipschitz functions are equipped with the slip-norm L max Dq : Lip Dq (I q ) → [0, ∞) defined as the restriction of the slip-norm L max Dq : Lip Dq (CP ℓ q ) → [0, ∞).Recall here that both L max Dq and the Lipschitz algebra Lip Dq (CP ℓ q ) come from the even unital spectral triple on quantum projective space.
Our aim is now to prove an extension theorem saying that the pair Lip Dq (CP ℓ q ), L max Dq is a compact quantum metric space if and only if the pair Lip Dq (I q ), L max Dq is a compact quantum metric space.The strategy for proving this extension theorem is inspired from the constructions appearing in the paper [2], regarding quantum Gromov-Hausdorff convergence of the standard Podleś spheres with respect to the Dąbrowski-Sitarz spectral triple, [18].In the paper [2], the authors construct a sequence of positive unital endomorphisms of the standard Podleś sphere C(S 2 q ) = C(CP 1 q ) such that all the endomorphisms appearing in this sequence have finite dimensional images.Moreover, the whole sequence converges pointwise to the identity operator in a way which can be estimated by the Monge-Kantorovich metric on the state space of the standard Podleś sphere.
We shall see that similar constructions for the quantum projective space C(CP ℓ q ) will help us verify the conditions in Theorem 2.5 for the pair Lip Dq (CP ℓ q ), L max Dq under the assumption that Lip Dq (I q ), L max Dq is a compact quantum metric space.More precisely, using the techniques from [2] we are able to establish the finite diameter condition and prove the existence of arbitrarily precise ε-approximations.

4.1.
Weak- * convergence to the counit.In this subsection, we shall apply the Haar state h : C(SU q (N)) → C to construct a sequence of states {h k } ∞ k=0 on C(CP ℓ q ) which converges in the weak- * topology to the restriction of the counit ǫ : C(CP ℓ q ) → C. Notice that a related result already appears in [24, Proposition 2.4 and Theorem 2.5], but we are currently unaware of the precise relationship between the statements in [24] and our Proposition 4.3.
For each k ∈ N 0 we put a k := (z * N ) k z k N and introduce the state Notice that h 0 agrees with the restriction of the Haar state to quantum projective space.
In order to prove that the sequence {h k } ∞ k=0 converges to the restricted counit we start out by reducing this problem to the quantized interval I q .This is carried out by means of a conditional expectation of C(CP ℓ q ) onto C(I q ).This conditional expectation is also applied in [43] (and in many other places) but for the sake of completeness we review the construction here.
For N 3 we define the surjective unital * -homomorphism Φ : C(SU q (N)) → C(SU q (N − 1)) by putting For N = 2 we consider the unital C * -algebra of continuous functions on the unit circle C(T) and let w : T → C denote the inclusion.We then define the surjective unital * -homomorphism Φ : C(SU q (2)) → C(T) by the formula In both cases, we record that Φ is compatible with the coalgebra structures so that (Φ ⊗ Φ)∆ = ∆Φ (where the coproduct on C(T) is induced by the group structure of the compact group T).We therefore obtain a conditional expectation where h denotes the Haar state both for N 3 and for N = 2 (for N = 2 the Haar state is just given by Riemann integration with respect to the Haar measure on the unit circle).We remark that the operation (hΦ ⊗ 1)δ agrees with the identity map on C(CP ℓ q ) (in fact this even holds on C(S 2ℓ+1 q ) for N 3), but in the definition of E we are slicing with the state hΦ : C(SU q (N)) → C on the right leg of the tensor product.
Proposition 4.1.The image of the conditional expectation E : C(CP ℓ q ) → C(CP ℓ q ) agrees with the unital C * -subalgebra C(I q ).Proof.For N = 2 we consider the strongly continuous circle action σ on C(CP The statement of the proposition can then be obtained by noting that the image of E agrees with the fixed point algebra for the circle action σ. For N 3, the result is a consequence of [43,Theorem 3.3].Indeed, we may extend the conditional expectation E to a conditional expectation on the quantum sphere, E : C(S 2ℓ+1 q ) → C(S 2ℓ+1 q ) defined by the same formula E(x) = (1 ⊗ hΦ)δ(x) (but now using the right coaction of quantum SU(N) on the quantum sphere).As a consequence of [43,Theorem 3.3] we then get that the image of E coincides with C * (z N , 1) (the smallest unital C * -subalgebra of C(S 2ℓ+1 q ) which contains z N ).The result of the proposition now follows by noting that C * (z N , 1)∩C(CP ℓ q ) = C(I q ).The main point is that the state h k : C(CP ℓ q ) → C for k ∈ N 0 and the restricted counit ǫ : C(CP ℓ q ) → C are invariant under the conditional expectation E: Lemma 4.2.Let x ∈ C(CP ℓ q ) and k ∈ N 0 .It holds that h k (E(x)) = h k (x) and ǫ(E(x)) = ǫ(x).
Proof.We first notice that the bi-invariance property of the Haar state (2.8) implies hE(x) = hΦ h(x) • 1 = h(x) establishing the relevant identity for k = 0.For k ∈ N, the most efficient approach is to apply the modular properties of the Haar state.By the discussions near (2.10) we get that θ(z * N ) = q 2(1−N ) z * N and hence that . Since E is a conditional expectation onto C(I q ) and hE = h we obtain that . The final claim regarding the counit follows by checking that Φ(z i z * j ) = δ iN δ jN = ǫ(z i z * j ) • 1 for all i, j ∈ N so that we also get the identity Φ(x) = ǫ(x)1.Indeed, this latter identity entails that ǫ(E(x)) = hΦ(x) = ǫ(x).
After these preparations we are ready to prove that the sequence of states {h k } ∞ k=0 satisfies the desired convergence property.
Proof.By Lemma 4.2 it is sufficient to show pointwise convergence on C(I q ).Moreover, since {y m ℓ | m ∈ N 0 } is linearly norm-dense in C(I q ), we only need to show that lim k→∞ h k (y m ℓ ) = ǫ(y m ℓ ) for every m ∈ N (so m = 0).For such m it holds that ǫ(y m ℓ ) = 0, and moreover for every k ∈ N 0 we have the estimate • h(a k ) −1 q 2km • y m ℓ , where the third identity follows since both the positive element a k = (z * N ) k z k N and y m ℓ belong to C(I q ) (which is a commutative C * -algebra).4.2.Monge-Kantorovich-convergence to the counit.Let us for a little while assume that the pair Lip Dq (I q ), L max Dq is a compact quantum metric space (this will be established later on in Theorem 5.In this subsection, we shall prove that the Monge-Kantorovich distance between h k and the restricted counit ǫ : C(CP ℓ q ) → C actually agrees with the Monge-Kantorovich distance between h k | C(Iq) and ǫ| C(Iq) .As a consequence we get that the sequence of states {h k } ∞ k=0 converges to the restricted counit ǫ : C(CP ℓ q ) → C with respect to the Monge-Kantorovich distance on the state space S C(CP ℓ q ) .From now on we no longer assume that Lip Dq (I q ), L max Dq is a compact quantum metric space.
We first establish that the conditional expectation E : C(CP ℓ q ) → C(CP ℓ q ) is a contraction for the slip-norm L max Dq : Lip Dq (CP ℓ q ) → [0, ∞).Lemma 4.4.Let x ∈ Lip Dq (CP ℓ q ).It holds that E(x) ∈ Lip Dq (I q ) and we have the inequality L max Dq (E(x)) L max Dq (x).Proof.We put A = C(SU q (ℓ)) for N 3 and A = C(T) for N = 2 and in both cases we equip A with the Haar state h (which is indeed faithful, see [43,Theorem 1.1] for the case where N 3).We then have the surjective unital * -homomorphism Φ : C(SU q (N)) → A from (4.1) and (4.2).The result of the lemma therefore follows from Lemma 3.8 by noting that E = (1 ⊗ ϕ 1,1 Φ)δ.
We may now prove the main result of this subsection.Proposition 4.5.For all k ∈ N 0 we have the identity

Proof. The inequality mk L max
Dq ) is satisfied since the q-Lipschitz functions Lip Dq (I q ) are contained in the Lipschitz algebra Lip Dq (CP ℓ q ) and the relevant slip-norms agree on the q-Lipschitz functions.To show the reverse inequality, we let x ∈ Lip Dq (CP ℓ q ) with L max Dq (x) 1 be given.Applying Lemma 4.2 and Lemma 4.4, we obtain the estimate 4.3.Finite dimensional approximations.In this subsection we are going to prove the extension theorem.The main step is to show that if Lip Dq (I q ), L max Dq is a compact quantum metric space, then there exist arbitrarily precise finite dimensional approximations for the pair Lip Dq (CP ℓ q ), L max Dq (cf.Theorem 2.5).For each k ∈ N 0 our prospective finite dimensional approximation is defined as the positive unital map β k : C(CP ℓ q ) → C(SU q (N)) β k (x) := (h k ⊗ 1)δ(x), where we recall that δ denotes the right coaction of quantum SU(N) on quantum projective space, see Subsection 2.4.Record that β 0 (x) = h(x) • 1 since h 0 agrees with the restriction of the Haar state to C(CP ℓ q ).Our first result is that β k has finite dimensional image.The proof relies on the Peter-Weyl decomposition of O(SU q (N)) together with the Schur orthogonality relations, see Subsection 2.3.Lemma 4.6.For all k ∈ N 0 it holds that β k : C(CP ℓ q ) → C(SU q (N)) has finite dimensional image.
Proof.We are in fact going to prove a stronger result, namely that a certain extension of β k has finite dimensional image.Recall from Subsection 4.1 that a k := (z * N ) k z k N and define the state We may then introduce the positive unital map Since the state h k agrees with the restriction of H k to C(CP ℓ q ) we obtain that B k (x) = β k (x) for all x ∈ C(CP ℓ q ).Hence, to establish the result of the lemma it suffices to show that B k has finite dimensional image.
Recall that θ : O(SU q (N)) → O(SU q (N)) denotes the modular automorphism of the Haar state.Putting w k := z k N θ((z k N ) * ) we then obtain that Using the Peter-Weyl decomposition we may choose a finite subset F ⊆ O(SU q (N)) such that w k ∈ α∈F C(α).For every β ∈ O(SU q (N)) \ F and x ∈ C(β) it holds that ∆(x) ∈ C(β) ⊗C(β) and the Schur orthogonality relations then imply that Applying the Peter-Weyl decomposition again we can now conclude that This shows that B k O(SU q (N)) is finite dimensional and by continuity and density we infer that the image of B k : C(SU q (N)) → C(SU q (N)) must be finite dimensional too.
Proposition 4.7.Let x ∈ Lip Dq (CP ℓ q ) and let k ∈ N 0 .We have the inequality Proof.Let ξ, η ∈ O(SU q (N)) be two vectors with ξ 2 and η 2 dominated by one (inside the Hilbert space L 2 (SU q (N))).Since x = (ǫ| C(CP ℓ q ) ⊗ 1)δ(x), we have the identities We may now apply Lemma 3.8 to the case where Φ is the identity operator on C(SU q (N)) (and µ is the Haar state h).Indeed, an application of this lemma and the definition of the Monge-Kantorovich metric yield the estimates: The result of the proposition then follows since C(SU q (N)) is represented faithfully on L 2 (SU q (N)), see the discussion towards the end of Subsection 2.2.Indeed, for every y ∈ C(SU q (N)) we get that We are now prepared for the proof of the extension theorem.Theorem 4.8.It holds that Lip Dq (CP ℓ q ), L max Dq is a compact quantum metric space if and only if Lip Dq (I q ), L max Dq is a compact quantum metric space.
Proof.Since Lip Dq (I q ) is a unital * -subalgebra of Lip Dq (CP ℓ q ) and the two slipnorms agree on Lip Dq (I q ), it follows from Theorem 2.5 that if Lip Dq (CP ℓ q ), L max Dq is a compact quantum metric space, then this holds for Lip Dq (I q ), L max Dq as well.Suppose that Lip Dq (I q ), L max Dq is a compact quantum metric space.Then we get from Proposition 4.3 that lim k→∞ mk L max Dq h k | C(Iq) , ǫ| C(Iq) = 0 and hence by Proposition 4.5 that also lim k→∞ mk L max Dq h k , ǫ| C(CP ℓ q ) = 0. Let ε > 0 be given.Choose a k 0 ∈ N 0 such that mk L max Dq (h k 0 , ǫ| C(CP ℓ q ) ) ε.If we let i : C(CP ℓ q ) → C(SU q (N)) denote the inclusion, it then follows from Lemma 4.6 and Proposition 4.7 that i, β k 0 is an ε-approximation of Lip Dq (CP ℓ q ), L max Dq .By Theorem 2.5, it now only remains to show that Lip Dq (CP ℓ q ), L max Dq has finite diameter.To this end, we first notice that mk L max Dq (h 0 , ǫ| C(CP ℓ q ) ) < ∞.This is a consequence of Proposition 4.5 together with our assumption that Lip Dq (I q ), L max Dq is a compact quantum metric space.The finite diameter condition for Lip Dq (CP ℓ q ), L max Dq then follows from Proposition 4.7.Indeed, we get the estimates

The quantum metric structure of the quantized interval
Let us again fix an integer M ∈ Z and recall that M is relevant in the construction of the unital spectral triple Lip Dq (CP ℓ q ), L 2 (Ω M , h), D q .Remark that this unital spectral triple restricts to a unital spectral triple over the q-Lipschitz functions namely Lip Dq (I q ), L 2 (Ω M , h), D q .
In view of Theorem 4.8, our aim is now to prove that the pair Lip Dq (I q ), L max Dq is a compact quantum metric space (so that the unital spectral triple over the q-Lipschitz functions becomes a spectral metric space).As a first step in this direction, we compute derivatives of the indicator functions associated to isolated points in I q , see (2.12).5.1.Derivatives of indicator functions.For each k ∈ {0, 1, . . ., ℓ} we recall from Section 3 that L 2 (Ω k M , h) denotes the closure of the twisted antiholomorphic forms Ω k M inside the Hilbert space tensor product L 2 (SU q (N)) ⊗ Λ k (C ℓ ).
In this subsection, we shall prove that the projection p m ∈ C(I q ) belongs to the domain of ∇ for all m ∈ N 0 and compute the derivative ∇(p m ).We recall here that the projection p m was introduced in Subsection 2.4 and corresponds to the isolated point q 2m in the spectrum of y ℓ .
For each i ∈ ℓ we start out by defining the operation It is then relevant to notice that ∇ i is a twisted derivation in the sense that for all x, y ∈ O(S 2ℓ+1 q ), see (2.13) and (2.7).Moreover, using (2.6) we may specify how ∇ i behaves with respect to the adjoint operation: For every r ∈ N we also record the following formulae which are consequences of (2.5): ∇ i (z r ) = 0 and ∇ i (z * r ) = d E i •...•E ℓ (u N r ) * = (−q) i−N u * ir .
(5.1)By Lemma 3.6 the relationship between the derivation ∇ and the above twisted derivations can be clarified.Indeed, we have that ∇(x)(ξ) = ℓ i=1 ∇ i (x)(ξ) ⊗ e i for all ξ ∈ L 2 (Ω 0 M , h) and x ∈ O(CP ℓ q ).(5.2) In the next three short lemmas we present a couple of commutation relations involving the twisted derivation ∇ i and the generators for the coordinate algebra of the quantum sphere.We recall that x s := z s z * s for all s ∈ N and that y r := r s=1 x s for all r ∈ N .Lemma 5.1.For i ∈ ℓ and s, r ∈ N with s = r we have the identity z s ∇ i (z * r ) = ∇ i (z * r )z s .

5.3.
Estimates on Lipschitz functions.In order to show that the pair Lip Dq (I q ), L max Dq is a compact quantum metric space, our strategy is to apply the slip-norm L max Dq to obtain estimates on an arbitrary q-Lipschitz function f .In fact, we need to control how fast the corresponding sequence f (q 2m )} ∞ m=0 converges to f (0).Since the quantity L max Dq (f ) dominates the operator norm of the gradient ∇(f ) applied to an arbitrary q-Lipschitz function it is relevant to relate the bounded operator ∇(f ) * ∇(f ) to the values of the function f .The next proposition in combination with Proposition 5.7 provide us with a formula for ∇(f ) * ∇(f ) in terms of the difference quotient D(f ) and a single explicit positive continuous function on the quantized interval I q .
Proof.We first recall that the identities in (2.11) imply that y ℓ = q −2 (1 − z * N z N ) and z * N (1 − z N z * N )z N = q 2 y ℓ (1 − q 2 y ℓ ).For every i ∈ ℓ we then obtain from the considerations in the beginning of Subsection 5.1 that Combining these observations with the identities from (2.3) and (5.2) we obtain the desired result: We apply the notation G := q −1 y ℓ (1 − q 2 y ℓ ) for the positive continuous function on the quantized interval I q appearing in Proposition 5.8.Proposition 5.9.Let f ∈ Lip Dq (I q ) and let x, y ∈ I q .We have the estimate Proof.Let m ∈ N 0 and notice first of all that it suffices to establish the result for x = q 2m and y = q 2(m+1) .Using Proposition 5.7 and Proposition 5.8 together with the fact that G(q 2m ) q 2m (1 − q 2 ) we arrive at the estimates: q m 1 − q 2 = f (q 2m ) − f (q 2(m+1) ) • √ 1 − q (q m − q m+1 ) √ 1 + q .
Theorem 5.10.The pair Lip Dq (I q ), L max Dq is a compact quantum metric space.
Proof.To ease the notation, put C q := √ 1+q √ 1−q .The finite diameter condition is a consequence of Proposition 5.9.Indeed, for every f ∈ Lip Dq (I q ) we get that f − f (0) C q • L max Dq (f ).Let ε > 0 be given and choose N ∈ N 0 such that C q • q N ε.Let id : C(I q ) → C(I q ) denote the identity operator and define the positive unital map Ψ : C(I q ) → C(I q ) by putting Ψ(f )(x) := f (x) for x q 2N f (q 2N ) for x q 2N .Since the quantized interval I q only contains finitely many points which are larger than or equal to q 2N we get that Ψ has finite dimensional image.For each f ∈ Lip Dq (I q ) we moreover obtain from Proposition 5.9 that Dq (f ).This shows that (id, Ψ) is an ε-approximation of the pair Lip Dq (I q ), L max Dq .It therefore follows from Theorem 2.5 that Lip Dq (I q ), L max Dq is a compact quantum metric space.
Proof.This follows immediately by combining Theorem 4.8 and Theorem 5.10.
As a corollary to our main theorem we recover the main result from [1] as a special case, see the discussion in Subsection 3.2 and in particular Proposition 3.11.
Corollary 5.12.The Lipschitz algebra version of the Dąbrowski-Sitarz spectral triple Lip / D q (S 2 q ), H + ⊕ H − , / D q is a spectral metric space.
10).It then follows from Proposition 4.3 that the restricted sequence of states h k | C(Iq) ∞ k=0 converges to the restricted counit ǫ| C(Iq) with respect to the Monge-Kantorovich distance mk L max Dq on the state space S C(I q ) .