Spectral Metric Spaces on Extensions of C*-Algebras

We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2- and 3-spheres giving a new perspective on these algebras in noncommutative geometry.

1. Introduction 1.1. Background. Spectral triples, a central concept of noncommutative geometry, provide an analytical language for geometric objects. A prototype is given by the triple (C 1 (M), L 2 (M, S), D) which is a spectral triple on the algebra C(M) of continuous functions on M, where M is a compact Riemannian manifold equipped with a spin C (or spin) structure, C 1 (M) a dense "smooth" subalgebra of C(M) and D is the corresponding Dirac operator acting on L 2 (M, S).
Connes [13], [12] introduced spectral triples as a potential means of describing the homology and index theoretic aspects in the more general language of (locally) compact topological spaces, as well as to develop a theory of cyclic cohomology mimicking the de-Rham cohomology theory of manifolds. Further, Connes shows that geometric information about a Riemannian manifold M, such as the geodesic distance and dimension, can all be recovered from the Dirac triple on C(M).
Spectral triples are motivated by Kasparov theory and can be regarded as "Dirac-type" or elliptic operators on general C * -algebras (usually assumed separable). In particular a spectral triple defines a K-homology class. Spectral triples with good properties can be used to encode geometric information on a C * -algebra. Besides the link between summability and dimension which is well understood in the commutative case, we mention two examples of current areas of research.
The first is Connes' reconstruction programme, the aim of which is to find conditions or axioms under which a spectral triple on a commutative C * -algebra can provide the spectrum of the algebra with the structure of a manifold. Several reconstruction theorems have been suggested in what has become a very prominent area of research (see for example [27], [15]). Besides the noncommutative tori, there do not seem to be many examples of noncommutative C * -algebras at present for which this sort of analysis can be extended to.
The second one is the idea to regard spectral triples as noncommutative (quantum) metric spaces, beginning with Connes' observation [12] that the Dirac triple on a Riemannian spin C manifold M recovers the geodesic distance between two points on the manifold. In fact Connes' expression for the geodesic distance extends immediately to a metric on the space of probability measures on M. In more recent and general language, a spectral triple on a C *algebra determines a Lipschitz seminorm on the self-adjoint part of the smooth subalgebra, an analogue of the classical notion of Lipschitz continuous functions. In e.g. [34], [35] and [36] Rieffel studies Lipschitz seminorms of this kind extensively. Under mild conditions such a seminorm defines a metric on the state space of the algebra by a formula analogous to the manifold case. However, in general, Lipschitz seminorms and corresponding metrics may be quite arbitrary. A natural condition one would expect this metric to satisfy is that it induces the weak- * -topology on the state space and Rieffel makes this the defining condition of his notion of a quantum metric space. Rieffel found a very useful characterisation of this metric condition for unital C * -algebras ( [34], cf. Prop.2.5 below for the statement). We will refer to this condition as Rieffel's metric condition. Latrémolière later extended much of this work to non-unital C * -algebras in [25] and [26]. A C * -algebra equipped with a spectral triple satisfying this metric condition is sometimes called a spectral metric space.
Despite the longevity of spectral triples as a subject of study, general methods of constructing spectral triples on C * -algebras are not well understood, much less still those satisfying the metric condition. There have been successful constructions of so-called spectral metric spaces on certain noncommutative C*-algebras, such as approximately finite dimensional algebras ( [10]), group C*-algebras of discrete hyperbolic groups ( [29]) and algebras arising as q-deformations of the function algebras of simply connected simple compact Lie groups ( [28]).
Building on previous authors' works, we are particularly interested in 'building block' constructions i.e. constructing new spectral triples from old ones, which is also in the sprit of permanence properties. This point of view has been used by various authors to attempt to construct spectral triples on crossed products of C*-algebras by certain discrete groups ( [2], [22]). More specifically, the authors of those two references study C * -dynamical systems (A, G, α) in which the algebra A is equipped with the structure of a spectral triple with good metric properties and consider under what conditions it is possible to write down a spectral triple on A⋊ r,α G using a natural implementation of the external product in Kasparov theory. It turns out that a necessary and sufficient condition is the requirement that the action of G essentially implements an isometric action on the underlying spectral metric space. This is satisfied for a variety of group actions and, via this construction, the authors in collaboration with A. Skalski and S. White ( [22]) were able to write down spectral triples with good metric properties on both the irrational rotation algebras and the Bunce-Deddens algebras and some of their generalisations.
Spectral triples define Baaj-Julg cycles, the unbounded analogue of a Kasparov bimodule in KK-theory ( [1]). This perspective is increasingly being examined by various authors to write down spectral triples on C * -algebras by means of an unbounded version of Kasparov's internal product, which is defined for C * -algebras A, D, B and p, q ∈ {0, 1} as a map ⊗ B : KK p (A, D) × KK q (D, B) → KK p+q (A, B). There are a couple of important recent developments in this area: Gabriel and Grensing ([20]) consider the possibility of writing down spectral triples on certain Cuntz-Pimsner algebras, generalising the setting of ordinary crossed products by Z but with the same property that the triples they construct represent the image of a given triple under the boundary map in the resulting six-term exact sequence. They succeed in implementing these techniques to construct a variety of spectral triples on certain quantum Heisenberg manifolds. Goffeng and Mesland ([21]) investigate how the Kasparov product can, under suggested modifications, be used to write down spectral triples on Cuntz-Krieger algebras, beginning with the spectral triple on the underlying subshift space. It is anticipated therefore that there will be a considerable interest in the interplay between spectral triples and the Kasparov product in the near future.
In this paper we construct spectral triples on extensions of C*-algebras out of given ones on the ideal and the quotient algebra. We are, however, primarily concerned with those which satisfy Rieffel's metric condition, thus implementing the structure of a quantum metric space on the extension, beginning with related structures on both the quotient and ideal. Techniques in Kasparov theory will be important to us too, but certain technical difficulties will prevent us from being able to give a full description of the resulting triples in terms of their representatives in K-homology. We remark that the ideas in this paper are closely linked to those of Christensen and Ivan ( [11]) and are to some extent a generalisation of their results.
1.2. Outline of the paper. We assume throughout the paper that all C * -algebras and Hilbert spaces are separable. Given a C * -algebra E and an essential ideal I ⊆ E, and given spectral triples on both I and E/I, is there any way of constructing a spectral triple on E out of the given spectral triples? In this paper we will be looking at the situation in which the quotient is a unital C * -algebra A and the ideal is the tensor product of a unital C * -algebra by the algebra of compact operators, that is, we consider extensions of the form, This is a generalisation of the situation considered by Christensen and Ivan [11], who looked at short exact sequences of the form, They exploited the fact that a certain class of C * -extensions by compacts (those which are semisplit) can be spatially represented over a Hilbert space: as outlined in Section 2.7 of [23], we can regard E as a subalgebra of the bounded operators on an infinite dimensional Hilbert space H generated by compacts on P H and the operators {P π A (a)P ∈ B(H) : a ∈ A}, where π A : A → B(H) is a faithful representation and P ∈ B(H) is an orthogonal projection with infinite dimensional range. The algebra acts degenerately, only on the subspace P H.
There is a certain generalisation of this picture for semisplit extensions by general stable ideals of the form (3) which is due to Kasparov ([24], see also [3]). For such extensions, E can always be regarded as a subalgebra of L B (ℓ 2 (B)) = L B , the C * -algebra of bounded Blinear and adjointable operators on the Hilbert module ℓ 2 (B). In fact, using semisplitness, there is a representation π : A → L B (ℓ 2 (B) ⊕ ℓ 2 (B)) ∼ = L B (ℓ 2 (B)) and a projection P ∈ L B such that E is generated by P π(A)P and P (K ⊗ B)P = P (K B )P (cf. Section 3 for more details). However, to construct spectral triples on E we need a representation on a Hilbert space, not Hilbert module. Our given spectral triples come with concrete representations π A : A → B(H A ) and π B : B → B(H B ) on Hilbert spaces. It seems reasonable to study those extensions which act naturally on the tensor product H A ⊗ H B , possibly degenerately i.e. only on a subspace of this tensor product. More precisely, we consider representations of the form π : Not all extensions can be brought into this form. In Section 3 we show that it is possible if the Busby invariant satisfies a certain factorisation property. In order to describe it in somewhat more detail recall that a short exact sequence of C * -algebras, is characterised by a * -homomorphism ψ : A → Q B , the Busby invariant, where Q B := L B /K B is sometimes called the generalised Calkin algebra with respect to the C * -algebra B.
Since L B ∼ = M(K ⊗ B) there is an embedding of the ordinary Calkin algebra Q = M(K)/K into Q B and the condition which characterises the extensions we consider is that there exists a semisplit extension of A by K of the type (2) with Busby invariant ψ 0 : A → Q such that ψ factors through ψ 0 and the natural inclusion of Q into Q B = M(K ⊗ B)/K ⊗ B. In KK-theoretic language, we need the class of ψ in KK 1 (A, B) = Ext(A, B) −1 to factor into the class of an extension ψ 0 in K 1 (A) = Ext −1 (A) and the K 0 (B)-class of 1 B ∈ B, i.e.
In this situation we can view the algebra E as a concrete subalgebra of B(H A ⊗ H B ) generated by elements of the form P aP ⊗ 1 B and k ⊗ b, where P ∈ B(H A ) is an orthogonal infinite dimensional projection, a ∈ A, b ∈ B and k is a compact operator in P H (cf. Corollary 3.2). Throughout the paper we will assume that our extension is of this form. To avoid technicalities we will assume that P aP ∩K = {0} which is true for essential extensions and can always be arranged by replacing π A by an infinite ampliation π ∞ A . Starting from a pair of spectral triples (A, H A , D A ) on A and (B, H B , D B ) on B (cf. Section 2.1 for notation), Kasparov theory can be used to write down a spectral triple on A ⊗ B whose representative in K-homology is the external Kasparov product of the representatives of (A, H A , D A ) and (B, H B , D B ). When the spectral triple on A is odd and the spectral triple on B is even, i.e. there is a direct sum representation π + B ⊕ π − B and D B , acting on H B ⊗ C 2 , decomposes as the matrix, then the spectral triple can be defined on the spatial tensor product A ⊗ B acting on the Hilbert space (which can be interpreted as sum two graded tensor products), whereas the product of two ungraded triples is represented by the matrix acting on H A ⊗C 2 (see for example [13] pp. 433-434). We mention these formulae since they serve as an inspiration for the Dirac operator we are going to write down for the extension. In fact our operator will be a combination of these two formulae which makes it difficult to interpret our construction in K-homological terms. Returning to our set-up, our assumptions imply that we can, omitting representations, write down an isomorphism, where [P, a] is a compact operator on H A . E can be regarded as a concrete subalgebra of B(H A ⊗ H B ) acting degenerately (effectively only on P H A ⊗ H B ). The corresponding representation is denoted by π. There is another representation π σ : E → B(H A ⊗ H B ) given as the composition of the quotient map σ : E → A and the natural representation . π σ is non-degenerate but not faithful. The information coming from both representations is essential to writing down a spectral triple on E which encodes the metric behaviour of both the ideal and quotient parts of the extension. We will use this information, the presence of Dirac operators D A on H A and D B on H B together with the aforementioned ideas in Kasparov theory to build spectral triples on E. The representation of this triple will be a suitable combination of the two representations of E.
In order to build a spectral triple we need the further requirement that P commutes with D A which then decomposes into the direct sum where C(H A ) is the dense * -subalgebra of elements x ∈ K(H A ) such that xdom(D A ) ⊆ dom(D A ) and both xD A and D A x extend to bounded operators (P -regularity cf. Defs.4.1, 4.4). One may think of C(H A ) as the dense * -subalgebra of 'differentiable compacts', hence the notation used. We summarise our assumptions on the extension and spectral triples in the following definition which is consistent with the article [11].
is said to be of Toeplitz type if there exists an infinite dimensional projection P ∈ B(H A ) such that [P, π A (a)] ∈ K(H A ), is then referred to as a Toeplitz triple for the extension.
If moreover (A, H A , D A ) is a spectral triple such that D A and P commute and [P, π A (a)] ∈ C(H A ) for all a ∈ A, then the quadruple (A, H A , D A , P ) (or just P ) is said to be of Toeplitz type.
A Toeplitz type quadruple is said to be P -injective if ker(D p A ∩ P H A ) = {0}. When P coincides with the orthogonal projection into the closed span of the positive eigenspace for D A , then the smoothness assumption turns out to be equivalent to saying that not only [D A , π A (a)] but also [|D A |, π A (a)] is a bounded operator for each a ∈ A, which is related to the concept of regularity for spectral triples in Riemannian geometry due to Connes and Moscovici ( [16]).
We state here the two main results of our paper asserting the existence of spectral triples with good summability properties on Toeplitz-type extensions under the assumption that (A, H A , D A , P ) is of Toeplitz type (Theorem 4.7) and that Rieffel's metric condition is preserved under the mild extra assumption that D A is P -injective (Theorem 5.7).
Before we can do so we need to introduce further notation. Let Π 1 , Π 2 : E → B(H A ⊗ H B ⊗ C 2 ) be the representations given by Π 1 = π σ ⊕ π σ and Π 2 = π ⊕ π σ (4) and consider operators D 1 , D 2 , D 3 on H A ⊗ H B ⊗ C 2 given by and Theorem 4.7. Let A and B be unital C * -algebras and suppose that E arises as the short exact sequence (3) and that there exist spectral triples (A, H A , D A ) on A and (B, H B , D B ) on B, represented via π A and π B respectively, and an orthogonal projection P ∈ B(H A ) such that (A, H A , D A , P ) is of Toeplitz type. Let Then (E, H, D), represented via Π, defines a spectral triple on E. Moreover, the spectral dimension of this spectral triple is computed by the identity The Dirac operator D of the spectral triple defines a Lipschitz seminorm L D which in turn defines in good cases a metric on the state space of the C * -algebra. We address the question of whether our spectral triples satisfy Rieffel's metric condition, which is a necessary and sufficient condition for the metric on the state space to induce the weak- * -topology (cf. Prop.2.5 and the definition thereafter). In this case the spectral triple together with the Lipschitz seminorm is called a spectral metric space. We show that under our natural assumptions this is always the case. We go on to show that there are numerous examples of C * -algebra extensions which can be given the structure of a spectral metric space. Our main focus is the single-parameter noncommutative (quantum) spheres C(S n q ) for n ≥ 2, which can be iteratively defined as C * -algebra extensions of smaller noncommutative spheres. We shall specifically study the cases n = 2 (the equatorial Podleś spheres) and n = 3 (the quantum SU q (2) group) and merely comment on how these two examples can be used to study their higher dimensional counterparts.
The noncommutative spheres have garnered a lot of attention in the literature as examples of noncommutative manifolds and many spectral triples have been suggested (e.g. [5], [7], [17]), though most of these from a very different perspective to ours, for example by looking at the representation theory of the ordinary SU(2) group and focusing on those triples which behave equivariantly with respect to the group co-action.
We remark that Chakraborty and Pal also considered the question of finding Lip-metrics starting from given ones on the ideal and quotient for extensions of the same type as ours [4]. However, our goal is to construct spectral triples, rather than compact quantum metric spaces. Our spectral triples give rise to Lip-metrics with properties similar to theirs but we have existence results for Dirac operators and our constructions seem to be fairly different.

2.
A review of spectral triples and quantum metric spaces 2.1. Spectral Triples. We begin with a short exposition of spectral triples. For more information and context, we recommend the articles [32], [39] and [27] which provide an excellent exposition of the theory and motivation behind spectral triples. We remind the reader that C * -algebras and Hilbert spaces are assumed to be separable throughout this article.
Definition 2.1. Let A be a separable C * -algebra. A spectral triple (A, H, D) on A is given by a faithful * -representation π : A → B(H) on a Hilbert space H, a dense * -subalgebra A ⊆ A and a linear densely defined unbounded self-adjoint operator D on H such that for each a ∈ A and (2) π(a)(I + D 2 ) −1 is a compact operator for each a ∈ A.
Unlike the majority of definitions provided in the literature, we do not make the assumption that the C * -algebra A is unital, or indeed that the representation over H is nondegenerate. Using faithfulness on the other hand we may identify A with π(A) and therefore omit the representation from notation, in particular writing aξ for π(a)ξ.
For a spectral triple on A, it is sometimes convenient to study the maximal Lipschitz subalgebra, ](η, ξ)) * ensures that C 1 (A) is closed under involution. It is well known that C 1 (A) becomes an operator algebra when equipped with the norm a 1 := π(a) + [D, π(a)] and viewed as a concrete subalgebra of the bounded operators on the first Sobolev space, H 1 := dom(D), of H, the latter being a Hilbert space with respect to the inner product η 1 , η 2 1 := η 1 , η 2 + Dη 1 , Dη 2 . Depending on the context, it can be useful to think of C 1 (A) as either a dense * -subalgebra of A or as a Banach algebra in its own right.
Definition 2.2. The spectral dimension of (A, H, D), defined on a unital C * -algebra A, is given by It can be shown that Here Tr is the usual unbounded trace on B(H).
We will often write s 0 (D) instead of s 0 (A, H, D) and employ this notation also for the summability of an unbounded essentially self-adjoint operator.
Summability can also be defined for spectral triples of non-unital C * -algebras, as advocated by Rennie in [31].
Because of the relationship between spectral triples and Fredholm modules in K-homology, spectral triples are often distinguished into odd and even varieties: A spectral triple on A is called graded or even if there exists an operator γ ∈ B(H) such that γ 2 = id, γπ(a) = π(a)γ for each a ∈ A and γD = −Dγ. Otherwise it will be called ungraded or odd.
Stated in a different way, an even spectral triple is one which can be formally represented via a direct sum representation over an orthogonal direct sum of Hilbert spaces H = H + ⊕ H − over which π and D decompose as

2.2.
Compact quantum metric spaces. One of the most interesting aspects of spectral triples in differential geometry is the possibility to recover the metric information of the manifold from the spectral triple. Connes [12] generalises this observation by showing that a spectral triple (A, H, D) on a C * -algebra A defines an extended metric (i.e. allowing the metric to take the value ∞) d C : Connes' metric d C in general depends on the algebra A, so it is often better to write d A to stress this dependence. The motivating example is prescribed by the Dirac triple on a connected spin c manifold M defined on the dense subalgebra of "C ∞ -functions" for which [D, f ] = df . The restriction of Connes' metric to the point evaluation measures d C (p x , p y ) then coincides with the geodesic metric d γ (x, y) along M.
In [34], [35] and [36], Rieffel considered the more general setting of Lipschitz seminorms, which can be viewed as a generalisation of metric spaces, or Lipschitz functions, to orderunit spaces. The theory is based on the observation of Kantorovich and Rubinstein, who demonstrated that a metric on a compact topological space can be extended naturally to the set of probability measures on that space. Recent work by Latrémolière in [25] and [26] has extended much of this work to the setting of non-unital C * -algebras.
In the context of this paper, a Lipschitz seminorm on a separable C * -algebra A is a seminorm L : A → R + defined on a dense subalgebra A which is closed under involution with the property L(a * ) = L(a) for each a ∈ A and also L(1) = 0 whenever A is unital. We say that a Lipschitz seminorm L is nondegenerate if the set {a ∈ A : L(a) = 0} is trivial or contains only multiples of the identity when A is unital. As pointed out in [22], nondegeneracy of L is independent of the choice of the dense subalgebra A, but it should be stressed that many of the properties of the Lipschitz seminorm will depend on A.
A Lipschitz seminorm on A determines an extended metric d A,L on S(A) (occasionally written d A , or d L ) in a way which provides a noncommutative analogue of the Monge-Kantorovich distance when A is commutative. The metric is given by If L is a Lipschitz seminorm on A, so is L d A,L . When the Lipschitz seminorm L is lower semicontinuous, so that the set {a ∈ A : L(a) ≤ r} is closed in A for any and hence all r > 0, then L = L d A,L . We shall further call L closed if L is lower semicontinuous and A = dom(L d A,L ). Hence, starting from a lower semicontinuous seminorm, the above procedure can be used to extend L to a closed seminorm. All these observations are well known in the case when A is unital and the procedure of replacing A with its unitisation A = A ⊕ CI and introducing the new seminorm L(a, λ) := L(a) can easily be used to generalise these results to the non-unital case. Rieffel addresses the question of whether a metric induced by a Lipschitz seminorm on a unital separable C * -algebra (or order-unit space) has finite diameter and whether it induces the weak- * -topology of S(A) which is a compact metrisable Hausdorff space in this situation. To state his result we introduce some notation: for a given Lipschitz seminorm L on A, define where B A is the closed unit ball in A. We will refer to the conditions 2.(a) and 2.(b) in Prop.2.5 as Rieffel's metric conditions or just metric conditions. The situation when A is non-unital is rather more complicated, but Latrémolière ([25]) shows that, provided one works only with Lipschitz seminorms which give a metric on S(A) with finite diameter, things are not too complicated. For this case he provides conditions similar to the ones in the preceeding Proposition 2.5 which characterise those seminorms that induce the weak- * -topology on S(A). Definition 2.6. Let A be a separable C * -algebra equipped with a Lipschitz seminorm L on a suitable dense subalgebra A with the property that d A,L determines a metric of finite diameter inducing the weak- * -topology of S(A). Then the pair (A, L) is called a quantum metric space (or compact quantum metric space when A is unital).
Thus (A, L) with A unital will be compact quantum metric space if and only if (A, d A,L ) satisfies Rieffel's metric conditions.
The final definition is motivated by a similar definition in [2] which we will follow in this paper.

Extensions and Kasparov's KK-Theory.
In this section we recall and develop some background in KK-theory related to extensions. Further information can be found in Kasparov's seminal paper [24] and in [3], [23].
For a separable C * -algebra A, we write ℓ 2 (A) to mean the Hilbert bimodule A consisting of sequences of the form (a n ) n∈N such that n a * n a n converges in norm, equipped with the inner product (a n ), (b n ) := n a * n b n . We write L A to mean the set of adjointable right A-linear operators on ℓ 2 (A), which becomes a C * -algebra when equipped with the operator norm. We denote by K A the C * -subalgebra of L A consisting of the closed linear span of operators of the form θ x,y (z) = x y, z , x, y, z ∈ ℓ 2 (A). Then K A is an ideal in L A and is isomorphic to the spatial tensor product, K ⊗ A, of A by the compact operators on a separable infinite dimensional Hilbert space. The algebra L A is isomorphic to M(K ⊗ A). We denote the quotient L A /K A by Q A and will also use the notation L for B(ℓ 2 ) and Q for the quotient B(ℓ 2 )/K.

3.1.
Background and set-up. The extensions in this article are unital short exact sequences of separable C * -algebras of the form, Recall that this means ι is an injective * -homomorphism and regarded as an inclusion map, σ is a surjective * -homomorphism and im(ι) = ker(σ). We will always assume that the C *algebras A and B are unital and that K ⊗ B is the stabilisation of B by compact operators on a separable, infinite dimensional Hilbert space. Additionally we will always require the ideal K ⊗ B to be essential, i.e. it has non-zero intersection with any other ideal I ⊆ E.
The Busby invariant of (10) is a * -homomorphism ψ : A → L B /K B =: Q B . The *homomorphism ψ can be regarded as a characteristic of the extension itself, since the extension can be recovered from ψ, up to isomorphism, as the pullback C * -algebra, (here, q B : L B → Q B is the quotient map). The assumptions above imply that ψ and consequently the map π : E → L B , π(x, a) = x is injective. The maps fit together in the commuting diagram We do not consider all such extensions, but restrict our attention to the situation in which the Busby invariant ψ admits a unital completely positive lift, i.e., there is a unital completely positive map s : A → L B such that q B •s = ψ. This is equivalent to the existence of a ucp lift of σ. Such extensions are called semisplit. A well known application of the Kasparov-Stinespring Theorem shows that, in this setting, there is a faithful representation ρ : The existence of such a map is not automatic, unless A is a nuclear C * -algebra in which case the existence of a completely positive lift follows from the Choi-Effros lifting theorem. The Stinespring dilation (P, ρ) can be used to define a Kasparov cycle ψ * which is the element of KK 1 (A, B) represented by the triple (ℓ 2 (B) ⊕ ℓ 2 (B), ρ, 2P − 1) = (ℓ 2 (B), ρ, 2P − 1). A well known result of Kasparov ([24]) says that there is a six-term exact sequence in both K-theory and K-homology. In the case of K-homology the sequence has the form where the boundary maps are defined by taking the internal Kasparov product with ψ * .

3.2.
Toeplitz type extensions and KK-theory. In this section we discuss a characterisation of Toeplitz type extensions showing that they form a large class. Moreover, we will introduce the representations of the extension algebra which are relevant in order to define our spectral triple on the extension algebra.
In what follows, we will assume B is unital and we shall let j : L → L B , : Q → Q B , q : L → Q and q B : L B → Q B be the natural maps, so that q B • j = • q. The main result of this section is contained in the following: (1) There is a * -homomorphism ψ 0 : A → Q such that • ψ 0 = ψ.
(1) =⇒ (2): Starting from a homomorphism ψ 0 : A → Q as above, we can define E 0 as the pullback C * -algebra There are natural maps π 0 : E 0 → B(ℓ 2 ) and σ 0 : E 0 → A. Similarly, as pointed out in (11) E is given as a pullback E = L B ⊕ (q B ,ψ) A, so that we also have natural maps π : E → L B and σ : E → A. Notice that ker(σ 0 ) = ker(q • π 0 ) is isomorphic to the algebra of compact operators. The map r : E 0 → E can be defined by r ((x, a)) := (j • π 0 (x), a), which is injective and thus establishes the first part of the proof.
It is clear that in the setting of the last Proposition 3.1, the map ψ 0 is injective if and only if ψ is injective, so that we can assume the extension corresponding to the top row of (12) is essential. If s : A → L is a completely positive lift for ψ 0 then j • s is a completely positive lift for ψ.
To apply this to our Toeplitz type extensions recall (Def.1.1) that an extension (3) is of Toeplitz type if π A : A → B(H A ) and π B : B → B(H B ) are faithful representations with [P, π A (a)] ∈ K(H A ) for each a ∈ A, P π A (a)P ∩ K(P H A ) = {0}, and E is isomorphic to the subalgebra of B(H A ⊗ H B ) generated by K(P H A ) ⊗ π B (B) and P π A (A)P ⊗ CI B . Thus, omitting the representations, the extension is of the form (π A , π B , P ) is called the corresponding Toeplitz triple. From this extension we obtain the extension There is a natural inclusion map r : E 0 ֒→ E. Moreover, E 0 embedds naturally into L ∼ = B(H A ) defining a degenerate (i.e. non-unital) but faithful representation π 0 : E 0 → L.
Similarly, there is a degenerate but faithful representation π : E → B(H A ⊗ H B ) given by its very definition as a subalgebra. Since H A is separable and infinite dimensional we . We can therefore think of π as a representation π : E → L B . There is a natural inclusion r : E 0 → E (using that B is unital) such that the diagrams in Prop.3.1.(2) commute.
Note that the Busby invariants for the extension E and E 0 are given by ψ(a) = q B ((P ⊗ I)(π A (a) ⊗ I)(P ⊗ I)) and ψ 0 (a) = q(P π A (a)P ), which implies that • ψ 0 = ψ. Thus starting from a Toeplitz type extension E of A by K ⊗ B we found an extension E 0 of A by K satisfying the conditions (1) and, hence, (2) of Prop.3.1.
We mention that with this interpretation s(a) := P π A (a)P ⊗ 1 can be regarded as a completely positive map s : is the Busby invariant of the extension. Hence s is a cp-lift of the extension and (P ⊗1, π A ⊗1) can be regarded as Stinespring dilation of the semisplit extension (16).
Conversely, given an essential semisplit extension then there exists faithful representations π A : A → B(H A ) and π B : B → B(H B ) and P ∈ B(H A ) an infinite dimensional projection such that E 0 ∼ = P π A (A)P + K(P H A ) and if we define E by E = K(P H A ) ⊗ B + P AP ⊗ CI B and π, π 0 , r, ψ, ψ 0 as before then (1) and (2)  Proof. The first part follows from the discussion preceeding the Corollary. For the second part it is known ([23], 2.7.10) that the required representation π A and projections can be found for every semisplit extension of the form 0 → K → E 0 → A → 0 (it is given by the Stinespring dilation we decribed.) Once we have that we can use any faithful representation π B : B → B(H B ) and define E and π, π 0 , r, ψ, ψ 0 as before satisfying the required identities.
When a Toeplitz triple exists, we have our * -homomorphism π : E → B(H A ⊗ H B ) given on the generators via where a ∈ A and k ∈ K(P H A ) which is faithful but degenerate (i.e. not unital). We also have another representation π σ : E → B(H A ⊗ H B ) given by where σ : E → A is the quotient map in the extension, given by where a ∈ A, b ∈ B and k ∈ K(P H A ). The representation π σ is non-degenerate (unital) but not faithful.

Construction of the spectral triple.
We will now begin to describe the steps needed to construct a spectral triple on an extension (10) of Toeplitz type with Toeplitz triple (π A , π B , P ) introduced earlier in Def.1.1. To discuss the second condition we formally introduce the notion of differentiability for compact operators in the next definition. Remark 4.2. Our motivation for choosing the term D A -differentiable compacts is based on the following observation: the same information as was given above can be used to write down an even spectral triple on the algebra of compact operators. It is given by the triple We note that C(H A ) can be viewed as a Banach * -algebra when equipped with the norm y 1 := y + max{ D A y , yD A }. Thus C(H A ) plays the role of the 'differentiable' elements with respect to this choice of spectral triple. Proof. If (1) holds, then the operators where the third identity follows from the first two. Thus (2) holds.
That (2)  Let us add the following comments on the three equivalent conditions in Proposition 4.3. The first condition can be compared to a smoothness criterion proposed by Wang in [40] whilst the second was studied by Christensen and Ivan in [11]. In the special situation in which P is the orthogonal projection onto the span of the positive eigenspaces of D A the third condition can be rephrased as requiring that the commutator [|D A |, a] is bounded for all a ∈ A. This condition is the first part of Connes and Moscovici's regularity (called smoothness by some authors) which requires for all a ∈ A that [D A , a] and δ(a) := [|D A |, a] are bounded but, moreover, that a and [D A , a] both lie in ∞ n=1 dom(δ n ) ( [16]).
The first condition of Proposition 4.3 has already been mentioned in the introduction as a smoothness assumption. We formalise this in the following definition. 4.2. The spectral triple on E. In this section we define the Dirac operator on the extension algebra and establish some of its basic properties. We suppose that we have C * -algebras A, B, E, a short exact sequence (10), spectral triples (A, H A , D A ) on A and (B, H B , D B ) on B represented via π A and π B respectively and an orthogonal projection P ∈ B(H A ) such that (π A , π B , P ) is a Toeplitz triple and the quadruple (A, H A , D A , P ) is of Toeplitz type (cf. Defs.1.1 and 4.4).
As pointed out in the introduction, the Dirac operator for the extension algebra E will be a combination of two formulae for Kasparov products. Recall the definition of the following representations and consider unbounded operators D 1 , D 2 , D 3 on H A ⊗ H B ⊗ C 2 given by and and finally To show that all these operators are essentially self-adjoint we only need to show that each of them possesses a complete orthonormal basis of eigenvectors. Indeed note that if T : H → H is an unbounded linear operator on a complex separable Hilbert space H with a complete set of orthonormal eigenvectors (ξ n ) ⊆ H and corresponding sequence of real eigenvalues (λ n ) so that T ξ n = λ n ξ n and thus lin{ξ n : n ∈ N} ⊆ dom(T ), then it is easy to see that T ⊆ T * = T * * = cl(T ) so that T is essentially self-adjoint. Such an operator will be called diagonalisable (with real eigenvalues).
Not all of the operators defined in (22) -(25) have compact resolvent but D 1 and D I do have which can be shown by finding their eigenvalues. This also allows to prove summability results.
(2) By assumption |λ m |, |µ n | → ∞ and therefore also |λ ′ m |, |µ ′ n | → ∞ as m, n → ∞. Since we have shown in the proof of (1) that the eigenvalues of D 1 and D I are given by ± λ 2 m + µ 2 n and ± λ ′2 m + µ ′2 n respectively it is easy to see that the sequences of absolute values of them tend to infinity. Thus D 1 and D I have compact resolvent.
Next we need to show boundedness of commutators with our operators D 1 and D I . In order to do so let us point out the following elementary identity for commutators of matrices. Proof. Let e = x + P π A (a)P ⊗ 1 be an element in E, where x ∈ C(P H A ) ⊙ B. Then direct calculations using (26) reveal that which is bounded. Next we determine the following commutators, again using (26). (We will omit the A and B indices of D A and D B as well as the representation π A .) where we have used that D q ⊗ 1 ⊥ π(e). Next Using these identities we obtain The claim is now evident since all entries in all operator matrices of the commutators are indeed bounded.
We are now ready to state and prove the first of our main results.
Theorem 4.7. Let A and B be unital C * -algebras and suppose that E arises as the short exact sequence (3) and that there exist spectral triples (A, H A , D A ) on A and (B, H B , D B ) on B, represented via π A and π B respectively, and an orthogonal projection P ∈ B(H A ) such that (A, H A , D A , P ) is of Toeplitz type. Let Then (E, H, D), represented via Π, defines a spectral triple on E. Moreover, the spectral dimension of this spectral triple is computed by the identity Proof. Note first that the representation Π is faithful and that D = D 1 ⊕ D I . Since 4.3. The algebra C 1 (E). Theorem 4.7 only provides the existence of a spectral triple for the dense subalgebra E of E. Given the Dirac operator we defined, it is natural to ask how large we can allow the dense subalgebra to be. More specifically, we ask: what is the largest 'smooth' subalgebra of E in which the construction in Theorem 4.7 defines a spectral triple? There seems to be a natural such algebra, the maximal Lipschitz algebra C 1 (E) associated to our Dirac operator mentioned after Def.2.1. It is also an extension fitting into the short exact sequence (27). We think of E as being represented via π on H A ⊗H B so that E ⊆ K(P H A )⊗B+P AP ⊗1, where the sum is algebraically direct. Then C 1 (E) is the * -subalgebra of E comprising all e ∈ E such that (i) π A (a)(dom(D A )) ⊆ dom(D A ) and [D A , π A (a)] is closable and bounded.
A , π A (a)] are closable and bounded. We now define C 1 (K B ) ⊆ C 1 (E) to be the dense * -subalgebra of K B consisting of all Finally, let C 1,P (A) be the * -subalgebra of A consisting of all a ∈ A such that (i) π A (a)(dom(D A )) ⊆ dom(D A ) and [D A , π A (a)] is closable and bounded.
(ii) π A (a)(dom(D p A )) ⊆ dom(D A ) and π A (a)(dom(D q A )) ⊆ dom(D A ) and [D p A , π A (a)], [D q A , π A (a)] are closable and bounded.
Remark 4.8. Our definitions imply that we obtain the following short exact sequence where ι 1 and σ 1 are the natural inclusion and quotient map respectively.

The metric condition for extensions.
We are interested in the construction of spectral metric spaces and, as such, we turn now to the question of whether the spectral triple on E satisfies Rieffel's metric condition (cf. Prop.2.5), giving E the structure of a spectral metric space. There is an abundance of Lipschitz seminorms on each of A, B and E which we could study, depending on the choice of smooth subalgebras. In this section we will focus on the situation in which the smooth subalgebras (cf. Definition 2.1) are A = C 1,P (A) and B = C 1 (B) and show that it is possible to construct a Lipschitz seminorm on C 1 (E) coming from a spectral triple with the desired properties. Our results can be adjusted to fit the setting of dense subalgebras possibly smaller than C 1,P (A) or C 1 (B).
To this end, we assume that the spectral triples ( are bounded subsets of A/CI A and B/CI B respectively and that the sets are norm totally bounded. From Rem.4.8 and the standing assumption of essentialness of our extension we conclude that for every elementẽ ∈ C 1 (E)/CI E there is a unique x ∈ C 1 (K B ) andã ∈ C 1,P (A)/CI A such thatẽ = (x + P aP ⊗ I) ∼ . In this sense we have: Remark 5.1. The equality C 1 (E)/CI E = C 1 (K B )+P (C 1,P (A)/CI A )P ⊗CI B holds, where the sum is direct.
Notice that I E = P I A P ⊗ I B = P ⊗ I B .
We now introduce the following spaces, where X and Y play the role of subscripts and do not refer to other objects.
Recall from Def.1.1 that a Toeplitz type quadruple (A, H A , D A , P ) is called P -injective if ker(D p A ) ∩ P H A = {0} (i.e. the operator D p A | P H A has an inverse in K (P H A ).) It will be necessary for us to impose this condition throughout this section.
We remark that necessarily ker(D p A ∩ P H A ) is finite rank, so that if P -injectivity fails then we can merely replace P with P − P ker(D A ) , where P ker(D A ) is the orthogonal projection onto ker(D A ). This procedure does not affect any other aspects of the extension. Using Lemma 4.5 we have the following observation.
Remark 5.2. One checks that the expressions for [D 1 , Π 1 (e)] and [D I , Π 2 (e) ⊕ Π 2 (e)] in the proof of Lemma 4.6 define bounded operators for all e ∈ C 1 (E) ⊇ E. Indeed, for any e = x + P aP ⊗ I B ∈ C 1 (E) such that x ∈ C 1 (K B ) and a ∈ C 1,P (A) we have The spectral triple (E, Π, D) on E in Theorem 4.7 determines a seminorm L = L D on C 1 (E) given by L(e) = max{ [D 1 , Π 1 (e)] , [D I , Π 2 (e) ⊕ Π 2 (e)] }. Our first objective is to show that L is a nondegenerate Lipschitz seminorm: Proof. The proof consists in showing C 1 (E) ∩ L −1 ({0}) = CI E . To this end, let e = x + P aP ⊗ I B , where x ∈ C 1 (K B ) and a ∈ C 1,P (A). If L(e) = 0 then [D 1 , Π 1 (e)] = 0 so that [D A , π A (a)] = 0 from eq. (28). Since D A implements a nondegenerate spectral triple on A, necessarily a = λI A for some λ ∈ C, so that we can write e = x + λI E . Moreover, we have [D I , Π 2 (x) ⊕ Π 2 (x)] = 0, so by (29) and (30) this means that (D p A ⊗ 1)π(x) = (D p A ⊗ 1)x = 0. By P -injectivity, x = 0, completing the proof. Then Proof. To show the first inclusion, let e ∈ C 1 (E) be such that L(e) ≤ 1 and write e = x + P aP ⊗ I B for unique x ∈ K B and a ∈ C 1,P (A). We need to show that [D A , π A (a)] ≤ 1 and Now L(e) ≤ 1 is equivalent to [D 1 , Π 1 (e)] ≤ 1 and [D I , Π 2 (e) ⊕ Π 2 (e)] ≤ 1. This means that the norms of all entries in the matrix expressions in Rem.5.2 are bounded by 1 from which we obtain the following inequalities: , we must have [D p A , a] ≤ 2, using (i) and (v). Then (iii) and (iv) imply Now (ii) and (31) imply This shows that x ∈ 7 U X and the result follows. The second inclusion can be shown in a similar way.
The next Lemma is immediate from our definitions.
Lemma 5.5. Let σ : E → A be the quotient map with induced mapσ : E/CI → A/CI. Then the maps are isometric bijections. Therefore, since (C 1,P (A), H A , D A ) satisfies Rieffel's metric condition, U Y ⊆ E/CI E is bounded and U Y,1 ⊆ E is totally bounded.
The uniform norm estimates in the next result is of key importance to establish Rieffel's metric condition. It uses a norm estimate from the proof of Lemma 5.4.
, let {P k } k∈N be the spectral projections of Y and write Q n := n k=1 P k . Then for each ǫ > 0 there exists an N ∈ N such that, for each x ∈ U X and for all n ≥ N , Moreover, for each x ∈ U X and for each n ∈ N, x n ≤ 3 Y , where x n := (Q n ⊗ 1)x(Q n ⊗ 1).
Proof. Since Y is a compact operator, it quickly follows that for each ǫ > 0 there exists an N ∈ N such that Y − Y Q n ≤ ǫ 6 and Y − Q n Y ≤ ǫ 6 whenever n ≥ N . For x ∈ U X , using P D A Y = P , we obtain where the last inequality follows from (31) in the proof of Lemma 5.4. This proves the second statement. To prove the first statement note that for all x ∈ U X and n ≥ N and similarly We can now prove our second main result. Proof. According to Rieffel's criteria (Prop.2.5) we need to show that U E is bounded and U E,1 is totally bounded. By Lemma 5.5 we know that U Y and U Y,1 are bounded, respectively totally bounded. By Lemma 5.4 we know that U E ⊆ 7U X + U Y and U E,1 ⊆ 7U X + U Y,1 . So we have only to show that the set U X ⊆ K B is totally norm bounded. Using Lemma 5.6, it will suffice for us to show that the sets are totally bounded for each n ∈ N. Since we may regard (Q n ⊗ 1)U X (Q n ⊗ 1) as a subset of M mn (B), where m n = dim(Q n ), any given element in this set can be expressed in the form is an orthonormal basis for the finite dimensional Hilbert space Q n H A . We shall denote the corresponding projections in . Since these commute with 1 ⊗ D B , we have that for x ∈ U X and n ∈ N, These estimates tell us that the sets Q n U X Q n are contained in the sets Now we recall our assumption that the spectral triple on B satisfies Rieffel's metric condition, so that U B,1 is totally bounded and consequently the sets S n are totally bounded as well. This concludes the proof of the Theorem.
If In this case of split extensions other constructions are possible. For instance we can use the following representation and Dirac operator which seems more natural from the point of view of K-homology.
6.2. Extensions by compacts. An extension by compacts is a short exact sequence of the form, which we have mentioned before. From our point of view, these extensions correspond to the instance B = C, the continuous functions on a single point. The canonical spectral triple on this space is the 'one-point' triple (C, C, 0). A second re-statement of Theorem 4.7 is as follows: Proposition 6.2. Let A be a unital C * -algebra, endowed with a spectral triple (A, H A , D A ). Let E ∼ = P AP +K(P H A ) be a unital extension of A by compact operators such that [P, a] is a compact operator for each a ∈ A, P AP ∩ K(P H A ) = {0} and the quadruple (A, H A , D A , P ) is of Toeplitz type. Then (E, H, D), represented via Π, defines a spectral triple on E. Here, The spectral dimension of this triple is the same as the spectral dimension of (A, H A , D A ). Moreover, if D p A is invertible and the spectral triple (A, H A , D A ) satisfies the Rieffel metric condition then so does the spectral triple on E.
It is worth comparing our spectral triples with those considered by Christensen and Ivan [11]. They make the same assumptions that we do, but the difference is that their triple acts on the Hilbert space P H A ⊕ P H A ⊕ QH A , rather than the larger Hilbert space H A ⊗ C 6 . Their Dirac operator, like ours, is designed to obtain a spectral triple with good metric properties. In the spirit of Rieffel-Gromov-Hausdorff theory, Christensen-Ivan introduce extra parameters α, β ∈ (0, 1) which can be used to study the effects of "recovering" metric data on either the quotient algebra or the compacts itself, coming from the extension. 6.3. Noncommutative spheres. Example 6.3. The quantum group SU q (2) was introduced by Woronowicz as a 1-parameter deformation of the ordinary SU (2) group [41]. When one considers the isomorphism SU (2) ∼ = S 3 of topological Lie groups, we can identify its C * -algebra with a 1-parameter deformation of the continuous functions on the 3-sphere, C(S 3 q ), for each q ∈ [0, 1]. It can be formally defined as the universal C * -algebra for generators α and β subject to the relations α * α + β * β = I, αα * + q 2 ββ * = I, αβ = qβα, αβ * = qβ * α, β * β = ββ * .
Woronowicz shows that the C * -algebras C(S 3 q ) are all isomorphic for q ∈ [0, 1). For q ∈ (0, 1), there is an alternative description of C(S 3 q ) as a symplectic foliation (see [4], [7], [8]): write H := ℓ 2 (N 0 ) ⊗ ℓ 2 (Z) and let S and T be respectively the unilateral shift on ℓ 2 (N 0 ) and the bilateral shift on ℓ 2 (Z), i.e Se k := e k+1 for each k ≥ 0 and T e k := e k+1 for each k ∈ Z. Let N q ∈ K(ℓ 2 (N 0 )) be defined by N q e k := q k e k . There exists a representation of C(S 3 q ) over H defined by: and this representation is faithful. By considering the map σ : C(S 3 q ) → C(T) sending β to 0 and α to the generator T * of C(T), we soon obtain a short exact sequence, 0 → K ⊗ C(T) → C(S 3 q ) → C(T) → 0. We obtain an isomorphism, where P ∈ B(ℓ 2 (Z)) is the usual Toeplitz projection, with the property that [P, x] is a compact operator for each x ∈ C(T) and P xP ⊗ I ∈ C(S 3 q ) for each x ∈ C(T). Note that we can write π(α) = −P T * P (1 − 1 − N 2 q ) ⊗ I + P T * P ⊗ I whilst π(β) ∈ K ⊗ C(T).
There are numerous other constructions of spectral triples on the algebra C(SU q (2)) in the literature, mostly with different spectral dimensions and no information about Rieffel's metric condition. The precise relation between those and our construction is unclear. The first spectral triples on C(SU q (2)) were constructed by Chakraborty and Pal in [5] and [6], whose focus was very different to ours. The named authors show that any spectral triple on C(SU q (2)) which is of a certain natural form and which is equivariant for the quantum group co-action of SU q (2) must have spectral dimension at least 3, which is in contrast to our spectral triple of dimension 2. In [7] the same authors construct spectral triples on C(SU q (2)) using an altogether different approach, focusing on those triples which are equivariant for the action of T 2 on C(SU q (2)), which might be closer to our spectral triple. The construction in [6] was used and further developed by Connes [14]. A different construction of a 3 + -summable spectral triple on C(SU q (2)) was developped in [18] using the classical Dirac operator. In another paper [19] the same authors give a construction of this triple via an extension using the cosphere bundle defined in [14] which appears somewhat similar to our construction.
As in Example 6.3, we can formulate the existence of spectral triples for the algebras C(S 2 q ) as follows: first, on B we introduce the two-point triple, which turns B into a spectral metric space C 2 , C 2 , γ := 0 1 1 0 .
The noncommutative n-spheres for higher dimensions can be defined inductively on n. The spheres of odd dimension arise as short exact sequences of the form 0 → K ⊗ C(S 1 ) → C(S 2n+1 q ) → C(S 2n−1 q ) → 0, n ≥ 1. and the spheres of even dimension as short exact sequences of the form 0 → K ⊗ C 2 → C(S 2n q ) → C(S 2n−1 q ) → 0, n ≥ 1.
We suspect that the same process that was used to construct spectral metric spaces on C(S 2 q ) and C(S 3 q ) can, via this procedure, lead to the construction of spectral metric spaces for 1-parameter quantum spheres of any integer dimension. We can then relate these to similar constructions in the literature, e.g. [7], [8].

Outlook.
In addition to the questions mentioned at the end of Example 6.3 and 6.5 we briefly raise a number of questions related to this article which seem interesting.
1. The spectral triple we construct on the extension (3) behaves well with respect to summability and induces metrics on the state space. This was our main goal. However, the following question still remains: Question 7.1. What is the KK-theoretical meaning of the spectral triple we construct on the extension? 2. Our construction of spectral triples is restricted to a special class of extensions (Toeplitz type extensions) but is applicable to several concrete examples as demonstrated in the last section. As discussed, there are similarities between a general semisplit extension by a stable ideal the Toeplitz type extensions we consider. 3. Rieffel proposed a notion of distance between compact quantum metric spaces, modelled on the Gromov-Hausdorff distance ( [34,35,36]). It has since been used in a number of questions relating to C * -algebras endowed with seminorms. Some of the results are quite surprising: Rieffel ([37]) shows how the common observation in quantum physics that 'matrices converge to the 2-sphere' can be illustrated quite well using Rieffel-Gromov-Hausdorff convergence.
There are various perspectives that we could take with respect to convergence for extensions in this chapter, especially for algebras arising as q-deformations. One is to try to mimic the convergence studied by Christensen and Ivan in their approach. They construct a two-parameter family of spectral triples (T , H A , D α,β ) for extensions of the form 0 → K → T → A → 0, and for α, β > 0, for which the quantum metric spaces converge to those on A and K as α → 0 and β → 0. However, for example in the case of the Podleś spheres this turns out not to be sufficient to study the Gromov-Hausdorff convergence aspects of varying the parameter q. The following two questions seem interesting, though we point out that the situation addressed in those questions is quite different from the matrix algebra convergence in [37] since the parameter q does not change the algebras of the Podles spheres. We remark that classically q can be regarded as a label for Poisson structures on S 2 ([38]). Question 7.3. Suppose that (q n ) n∈N ⊆ (0, 1) is a sequence converging to q ∈ (0, 1) and let (A(C(S 2 qn )), L) be one of the compact quantum metrics on the Podleś sphere C(S 2 qn ) for n ∈ N as defined in Thm.6.6. Is it true that (A(C(S 2 qn )), L) converges to (A(C(S 2 q )), L) for Rieffel-Gromov-Hausdorff convergence? Question 7.4. Suppose now that (q n ) n∈N ⊆ (0, 1) converges to 1. Let (C 1 (S 2 ), L D ) be the usual Lipschitz seminorm on the algebra C(S 2 ) ∼ = C(S 2 1 ) for which the restriction of the metric to S 2 is the geodesic metric. Is it true that (A(C(S 2 qn )), L) converges to (C 1 (S 2 )), L), or any equivalent Lipschitz pair on the two-sphere, for Rieffel-Gromov-Hausdorff convergence?