aa r X i v : . [ m a t h . OA ] S e p METRIC ASPECTS OF NONCOMMUTATIVEHOMOGENEOUS SPACES
HANFENG LI
Dedicated to Marc A. Rieffelin honor of his seventieth birthday
Abstract.
For a closed cocompact subgroup Γ of a locally com-pact group G , given a compact abelian subgroup K of G and ahomomorphism ρ : ˆ K → G satisfying certain conditions, Land-stad and Raeburn constructed equivariant noncommutative defor-mations C ∗ ( ˆ G/ Γ , ρ ) of the homogeneous space G/ Γ, generalizingRieffel’s construction of quantum Heisenberg manifolds. We showthat when G is a Lie group and G/ Γ is connected, given any normon the Lie algebra of G , the seminorm on C ∗ ( ˆ G/ Γ , ρ ) inducedby the derivation map of the canonical G -action defines a com-pact quantum metric. Furthermore, it is shown that this compactquantum metric space depends on ρ continuously, with respect toquantum Gromov-Hausdorff distances. Introduction
In recent years, the quantum Heisenberg manifolds have receivedquite some attention. These interesting C ∗ -algebras were constructedby Rieffel [28] as deformation quantizations of the Heisenberg mani-folds, and carry natural actions of the Heisenberg group. The classi-fication of these C ∗ -algebras up to isomorphism (in most cases) andMorita equivalence (in all cases) has been achieved by Abadie and hercollaborators [1, 2, 3, 4]. These C ∗ -algebras also appear in the work ofConnes and Dubois-Violette on noncommutative 3-spheres [10, 11].Aiming partly at giving a mathematical foundation for various ap-proximations in the string theory, such as the fuzzy spheres, namelythe matrix algebras M n ( C ), converging to the 2-sphere S , Rieffel de-veloped a theory of compact quantum metric spaces and quantum Date : May 21, 2009.2000
Mathematics Subject Classification.
Primary 46L87; Secondary 53C23,46L57.Partially supported by NSF Grant DMS-0701414.
Gromov-Hausdorff distance between them [31, 32, 33]. As the infor-mation of the metric on a compact metric space X is encoded in theLipschitz seminorm on the algebra of continuous functions on X , aquantum metric on (the compact quantum space represented by) aunital C ∗ -algebra A is a (possibly + ∞ -valued) seminorm on A satisfy-ing suitable conditions (see Section 5 below for detail).One important class of examples of compact quantum metric spacescomes from ergodic actions of a compact group G on a unital C ∗ -algebra A , which should be thought of as the translation action of G on a noncommutative homogeneous space of G . Given any lengthfunction on G , such an ergodic action induces a quantum metric on A [30] (see [25] for a generalization to ergodic actions of co-amenablecompact quantum groups). This class of examples includes the (fuzzy)spheres above and the noncommutative tori. When G is a compactconnected Lie group and the length function comes from the geodesicdistance associated to some bi-invariant Riemannian metric on G , thisseminorm can also be defined in terms of the derivation map on thespace of once differentiable elements of A with respect to the G -action[31, Proposition 8.6]. Explicitly, denote by σ X ( b ) the derivation of aonce differentiable element b of A with respect to an element X of theLie algebra g of G (see Section 3 below for detail). Then the seminorm L ( b ) is defined as the norm of the linear map g → A sending X to σ X ( b ) when b is once differentiable, or ∞ otherwise.It is natural to ask what conditions are needed to guarantee that L defined above gives rise to a quantum metric when G is not compact.Rieffel raised the question about the quantum Heisenberg manifoldsin [33]. In [38] Weaver studied some sub-Riemannian metric on thequantum Heisenberg manifolds, which does not quite fit into the aboveframework. In [9] Chakraborty showed that certain seminorm associ-ated to some ℓ -norm does define a quantum metric on the quantumHeisenberg manifolds. Since the ℓ -norm is bigger than the C ∗ -norm,this seminorm is bigger than the seminorm L defined above. Thus theresult in [9] is weaker than what Riffel’s question asks for.Our first main result in this article is an affirmative answer to Rief-fel’s question. In fact, we shall deal more generally with Landstad andRaeburn’s noncommutative homogeneous spaces. In [22] Landstad andRaeburn generalized Rieffel’s construction to obtain equivariant defor-mations of compact homogeneous spaces G/ Γ, starting from a locallycompact group G , a closed cocompact subgroup Γ of G , a compactabelian subgroup K of G , and a homomorphism ρ : ˆ K → G satisfyingcertain conditions. These C ∗ -algebras were denoted by C ∗ r ( ˆ G/ Γ , ρ ) and ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 3 were further studied in [20]. We shall see in Proposition 2.7 below thatthese algebras coincide with certain universal C ∗ -algebras, which wedenote by C ∗ ( ˆ G/ Γ , ρ ). For our result to be valid for these algebras,we shall assume conditions (S1)-(S5) (see Sections 2, 3, and 4 below).Among these conditions, (S1)-(S3) are essentially the same but slightlyweaker than the conditions of Landstad and Raeburn. The conditions(S4) and (S5) are just that G is a Lie group and G/ Γ is connected.
Theorem 1.1.
Let G, Γ , K and ρ satisfy the conditions (S1)-(S5). Fixa norm on the Lie algebra g of G . Denote by L ρ the seminorm on C ∗ ( ˆ G/ Γ , ρ ) defined above for the canonical action α of G on C ∗ ( ˆ G/ Γ , ρ ) .Then ( C ∗ ( ˆ G/ Γ , ρ ) , L ρ ) is a C ∗ -algebraic compact quantum metric space. Since Rieffel introduced his quantum Gromov-Hausdorff distance in[31], several variations have appeared [18, 19, 24, 25, 26, 35, 39]. Amongthese quantum distances, probably the most suitable one in our cur-rent situation is the distance dist nu discussed in [19, Section 5], whichis the unital version of the quantum distance introduced in [26, Re-mark 5.5]. As pointed out in [19, Section 5], this distance is no lessthan the distances introduced in [18, 31]. It is also no less than thedistances in [35] (see Appendix below). Our second main result saysthat the compact quantum metric spaces ( C ∗ ( ˆ G/ Γ , ρ ) , L ρ ) depend on ρ continuously. Let us mention that among the conditions (S1)-(5), onlythe conditions (S1) and (S2) involve ρ . Theorem 1.2.
Fix G , Γ , and K so that there exists ρ satisfying theconditions (S1)-(S5). Denote by Ω the set of all ρ satisfying the con-ditions (S1) and (S2), equipped with the weakest topology making themaps Ω → G sending ρ to ρ ( s ) to be continuous for each s ∈ ˆ K . Then Ω is a locally compact metrizable space. Fix a norm on the Lie algebra g of G . Then for any ρ ′ ∈ Ω , dist nu ( C ∗ ( ˆ G/ Γ , ρ ) , C ∗ ( ˆ G/ Γ , ρ ′ )) → as ρ → ρ ′ . This paper is organized as follows. In Section 2 we recall Landstadand Raeburn’s construction of noncommutative homogeneous spaces,and establish some general properties of these noncommutative spaces.The relation between the derivations coming from two canonical groupactions on C ∗ ( ˆ G/ Γ , ρ ) is established in Section 3. In Section 4 we showthat in the nondeformed case L ρ is essentially the Lipschitz seminormcorresponding to some metric on G/ Γ. A general result of establishingcertain seminorm being a quantum metric by the help of a compactgroup action is proved in Section 5. Theorems 1.1 and 1.2 are proved inSections 6 and 7 respectively. In an appendix we compare the distancedist nu and the proximity Rieffel introduced in [35]. HANFENG LI
Acknowledgements.
I am grateful to Wei Wu for comments.2.
Noncommutative homogeneous spaces
In this section we recall Landstad and Raeburn’s construction ofnoncommutative deformations of homogeneous spaces, discuss someexamples, and establish some general properties of these noncommuta-tive homogeneous spaces. These properties are of independent interestthemselves.Let G be a locally compact group. Throughout this paper, we makethe following standard assumptions:(S1) K is a compact abelian subgroup of G , and ρ : ˆ K → G is agroup homomorphism from its Pontryagin dual ˆ K into G suchthat ρ ( ˆ K ) commutes with K .(S2) Γ is a closed subgroup of G commuting with K and satisfiesΩ γ ( s ) := γρ ( s ) γ − ρ ( − s ) is in K for all s ∈ ˆ K, γ ∈ Γ and h Ω γ ( s ) , t i = h Ω γ ( t ) , s i for all s, t ∈ ˆ K, γ ∈ Γ , where h· , ·i denotes the canonical pairing between K and ˆ K .Denote by C b ( G ) the Banach algebra of bounded continuous C -valued functions on G , equipped with the pointwise multiplication andthe supremum norm. Endow K with its normalized Haar measure.Consider the action of K on C b ( G ) induced by the right multiplicationof K on G . For f ∈ C b ( G ), let f s ∈ C b ( G ) for s ∈ ˆ K be the partialFourier transform defined by f s ( x ) := R K < k, s >f ( xk ) dk for x ∈ G (this is denoted by ˆ f ( x, s ) in (1.3) of [22]). Note that although theaction of K on C b ( G ) may not be strongly continuous, we do have f s ∈ C b ( G ). Then C b , ( G ) := { f ∈ C b ( G ) | k f k ∞ , := X s ∈ ˆ K k f s k < ∞} is a Banach ∗ -algebra [21, Proposition 5.2] with norm k · k ∞ , andoperations f ∗ g ( x ) = X s,t f s ( xρ ( t )) g t ( xρ ( − s )) , (1) f ∗ ( x ) = f ( x ) . (2)Fix a left invariant Haar measure on G . For each s ∈ ˆ K denote by P s the projection on L ( G ) corresponding to the restriction of the left ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 5 regular representation L | K of K in L ( G ), i.e., P s = Z K h k, s i L k dk, where L y ξ ( x ) = ξ ( y − x ) for ξ ∈ L ( G ), x, y ∈ G . Then C b , ( G ) has afaithful ∗ -representation V on L ( G ) [22, Proposition 1.3] given by V ( f ) = X s,t P t L ρ ( s ) M ( f ) L ρ ( − t ) P s , (3)where M is the representation of C b ( G ) on L ( G ) given by M ( f ) ξ ( x ) = f ( x − ) ξ ( x ). Denote by C ( G/ Γ) the C ∗ -algebra of continuous C -valuedfunctions on G/ Γ vanishing at ∞ , and think of it as a C ∗ -subalgebraof C b ( G ) via the quotient map G → G/ Γ. The space C , ( G/ Γ , ρ ) := C ( G/ Γ) ∩ C b , ( G, ρ ) is a closed ∗ -subalgebra of C b , ( G, ρ ), and the non-commutative homogeneous space C ∗ r ( ˆ G/ Γ , ρ ) of Landstad and Raeburnis defined as the closure of V ( C , ( G/ Γ , ρ )) [22, Theorem 4.3].Clearly the left translations α y defined by α y ( f )( x ) = f ( y − x ) for y ∈ G extend to isometric ∗ -automorphisms of C , ( G/ Γ , ρ ). They alsoextend to ∗ -automorphisms of C ∗ r ( ˆ G/ Γ , ρ ) [22, Theorem 4.3]. We shallsee later that this action of G on C ∗ r ( ˆ G/ Γ , ρ ) is strongly continuous.Before discussing properties of these noncommutative homogeneousspaces, let us look at some examples. Example 2.1.
Let H be the 3-dimensional Heisenberg group consist-ing of matrices of the form y z x as a subgroup of GL(3 , R ). Denote by Z the subgroup consisting ofelements with x = y = 0 and z ∈ Z . Then we can write the elementsof G := H /Z as ( x, y, e πiz ) for x, y, z ∈ R . Fix a positive integer c .TakeΓ = { ( x, y, e πiz ) ∈ G | x, y, cz ∈ Z } , K = { (0 , , e πiz ) ∈ G | z ∈ R } . Take µ, ν ∈ R and define ρ : Z = ˆ K → G by ρ ( s ) = ( sµ, sν, e πis µ · ν ) . The C ∗ -algebra C ∗ r ( ˆ G/ Γ , ρ ) is isomorphic to Rieffel’s quantum Heisen-berg manifold D in [28, Theorem 5.5] (see [22, page 493]). HANFENG LI
Example 2.2. (cf. [22, Example 4.17]) Let H n be the 2 n +1-dimensionalHeisenberg group consisting of matrices of the form y y · · · y n z · · · x · · · x · · · · · · · · · · · · · · · · · · · · · as a subgroup of GL( n + 2 , R ). Denote by Z the subgroup consistingof elements with x = · · · = x n = y = · · · = y n = 0 and z ∈ Z . Thenwe can write the elements of G := H n /Z as ( x, y, e πiz ) for x, y ∈ R n and z ∈ R . Fix positive integers b , . . . , b n , d , . . . , d n and c such that b j d j | c for all j . Set b = ( b , . . . , b n ) and d = ( d , . . . , d n ) ∈ Z n . TakeΓ = { ( x, y, e πiz ) ∈ G | b · x, d · y, cz ∈ Z } , K = { (0 , , e πiz ) ∈ G | z ∈ R } . Take µ, ν ∈ R n and define ρ : Z = ˆ K → G by ρ ( s ) = ( sµ, sν, e πis µ · ν ) . The C ∗ -algebra C ∗ r ( ˆ G/ Γ , ρ ) is a higher-dimensional generalization ofExample 2.1. Example 2.3.
Let n ≥
3. Let W be the subgroup of GL( n, Z ) consist-ing of upper triangular matrices ( a j,l ) with diagonal entries all being 1.Denote by Z the subgroup consisting of matrices whose entries are all0 except diagonal ones being 1 and a ,n being an integer. Then we canwrite the elements of G := W/Z as ( a j,l ) with a ,n ∈ T . Fix a positiveinteger c . TakeΓ = { ( a j,l ) ∈ G | a c ,n = 1 and a j,l ∈ Z if ( j, l ) = (1 , n ) } ,K = { ( a j,l ) ∈ G | a j,l = 0 if j < l and ( j, l ) = (1 , n ) } . Take µ, ν ∈ R and define ρ : Z = ˆ K → G by( ρ ( s )) j,l = sµ, if ( j, l ) = (2 , n ) ,sν, if ( j, l ) = (1 , n − ,e πis µ · ν , if ( j, l ) = (1 , n ) , , for other j < l. For n = 3 we get the quantum Heisenberg manifold in Example 2.1again.In the rest of this section we establish some properties of C ∗ r ( ˆ G/ Γ , ρ ).Denote by C ∗ ( ˆ G/ Γ , ρ ) the enveloping C ∗ -algebra of the Banach ∗ -algebra C , ( G/ Γ , ρ ) [36, page 42]. By the universality of C ∗ ( ˆ G/ Γ , ρ ) ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 7 there is a canonical surjective ∗ -homomorphism C ∗ ( ˆ G/ Γ , ρ ) → C ∗ r ( ˆ G/ Γ , ρ )such that the diagram C , ( G/ Γ , ρ ) ' ' OOOOOOOOOOOO / / C ∗ ( ˆ G/ Γ , ρ ) (cid:15) (cid:15) C ∗ r ( ˆ G/ Γ , ρ )commutes.Clearly the right translations β k ( f )( x ) = f ( xk ) for k ∈ K extendto isometric ∗ -automorphisms of C , ( G/ Γ , ρ ). Recall the action α of G on C , ( G/ Γ , ρ ) defined before Example 2.1. Then α and β induceactions of G and K on C ∗ ( ˆ G/ Γ , ρ ) respectively, which we still denoteby α and β respectively. For each s ∈ ˆ K , set B s := { f ∈ C ( G/ Γ) | f = f s } . (4) Lemma 2.4.
The actions α and β of G and K on C , ( G/ Γ , ρ ) ( C ∗ ( ˆ G/ Γ , ρ ) resp.) commute with each other and are strongly continuous. Thespectral spaces { f ∈ C , ( G/ Γ , ρ ) | β k ( f ) = h k, s i f for all k ∈ K } and { a ∈ C ∗ ( ˆ G/ Γ , ρ ) | β k ( a ) = h k, s i a for all k ∈ K } of β correspondingto s ∈ ˆ K are exactly B s , and the norm of B s in C , ( G/ Γ , ρ ) and C ∗ ( ˆ G/ Γ , ρ ) is exactly the supremum norm.Proof. Clearly α and β commute with each other. It is also clear that B s = { f ∈ C , ( G/ Γ , ρ ) | β k ( f ) = h k, s i f for all k ∈ K } and that thenorm of B s in C , ( G/ Γ , ρ ) is exactly the supremum norm. It followsthat the restrictions of the actions α and β on B s ⊆ C , ( G/ Γ , ρ ) arestrongly continuous for each s ∈ ˆ K . For any f ∈ C , ( G/ Γ , ρ ), one has f s ∈ B s for each s ∈ ˆ K . For any ε > F ⊆ ˆ K such that P s ∈ ˆ K \ F k f s k < ε . Then k f − P s ∈ F f s k ∞ , = P s ∈ ˆ K \ F k f s k <ε . Therefore ⊕ s ∈ ˆ K B s is dense in C , ( G/ Γ , ρ ). It follows that theactions α and β are strongly continuous on C , ( G/ Γ , ρ ). Note thatthe canonical homomorphism C , ( G/ Γ , ρ ) → C ∗ ( ˆ G/ Γ , ρ ) is contractive[36, Proposition 5.2]. Consequently, the induced actions of α and β on C ∗ ( ˆ G/ Γ , ρ ) are also strongly continuous.Note that the subalgebra B of C , ( G/ Γ , ρ ) is a C ∗ -algebra, whichcan be identified with C ( G/K
Γ). Since the natural homomorphism C , ( G/ Γ , ρ ) → C ∗ r ( ˆ G/ Γ , ρ ) is injective, so is the canonical homomor-phism C , ( G/ Γ , ρ ) → C ∗ ( ˆ G/ Γ , ρ ). As injective ∗ -homomorphisms be-tween C ∗ -algebras are isometric, we conclude that the homomorphismof B into C ∗ ( ˆ G/ Γ , ρ ) is isometric. For any f ∈ B s one has f ∗ ∗ f ∈ B HANFENG LI and the supremum norm of f ∗ ∗ f is equal to the square of the supre-mum norm of f . It follows that the homomorphism C , ( G/ Γ , ρ ) → C ∗ ( ˆ G/ Γ , ρ ) is isometric on B s . In particular, the image of B s in C ∗ ( ˆ G/ Γ , ρ ) is closed.Since the action β of K on C ∗ ( ˆ G/ Γ , ρ ) is strongly continuous, thespectral space { a ∈ C ∗ ( ˆ G/ Γ , ρ ) | β k ( a ) = h k, s i a for all k ∈ K } is theimage of the continuous linear operator C ∗ ( ˆ G/ Γ , ρ ) → C ∗ ( ˆ G/ Γ , ρ )sending a to R K h k, s i β k ( a ) dk . It follows that the image of B s = { f ∈ C , ( G/ Γ , ρ ) | β k ( f ) = h k, s i f for all k ∈ K } in C ∗ ( ˆ G/ Γ , ρ ) isdense in { a ∈ C ∗ ( ˆ G/ Γ , ρ ) | β k ( a ) = h k, s i a for all k ∈ K } . Thereforethe image of B s in C ∗ ( ˆ G/ Γ , ρ ) is exactly { a ∈ C ∗ ( ˆ G/ Γ , ρ ) | β k ( a ) = h k, s i a for all k ∈ K } . (cid:3) We refer the reader to [8, Chapter 2] for the basics of nuclear C ∗ -algebras. Proposition 2.5.
The C ∗ -algebra C ∗ ( ˆ G/ Γ , ρ ) is nuclear.Proof. By Lemma 2.4 the action β of K on C ∗ ( ˆ G/ Γ , ρ ) is stronglycontinuous, and its fixed-point subalgebra is B , a commutative C ∗ -algebra, and hence is nuclear [8, Proposition 2.4.2]. For any C ∗ -algebracarrying a strongly continuous action of a compact group, the algebrais nuclear if and only if the fixed-point subalgebra is nuclear [14, Propo-sition 3.1]. Consequently, C ∗ ( ˆ G/ Γ , ρ ) is nuclear. (cid:3) We shall need the following well-known fact a few times (see forexample [8, Proposition 4.5.1]).
Lemma 2.6.
Let H be a compact group, and let σ j be a strongly con-tinuous action of H on a C ∗ -algebra A j for j = 1 , . Let ϕ : A → A be an H -equivariant ∗ -homomorphism. Then ϕ is injective if and onlyif the restriction of ϕ on the fixed-point subalgebra A H is injective. Inparticular, if ϕ is surjective and ϕ | A H is injective, then ϕ is an isomor-phism. Proposition 2.7.
The canonical ∗ -homomorphism C ∗ ( ˆ G/ Γ , ρ ) → C ∗ r ( ˆ G/ Γ , ρ ) is an isomorphism.Proof. We shall apply Lemma 2.6 to show that the canonical ∗ -homomorphism ϕ : C ∗ ( ˆ G/ Γ , ρ ) → C ∗ r ( ˆ G/ Γ , ρ ) is an isomorphism. By [22, Lemma 4.4]the action β on C , ( G/ Γ , ρ ) extends to an action of K on C ∗ r ( ˆ G/ Γ , ρ ),which we denote by β ′ . Clearly ϕ is K -equivariant. By Lemma 2.4 β is strongly continuous on C ∗ ( ˆ G/ Γ , ρ ). Since ϕ is contractive, it fol-lows that β ′ is strongly continuous on C ∗ r ( ˆ G/ Γ , ρ ). By Lemma 2.4 the ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 9 fixed-point subalgebra ( C ∗ ( ˆ G/ Γ , ρ )) K is B . Since the homomorphism C , ( G/ Γ , ρ ) → C ∗ r ( ˆ G/ Γ , ρ ) is injective, we see that the restriction of ϕ on ( C ∗ ( ˆ G/ Γ , ρ )) K is injective. Therefore the conditions of Lemma 2.6are satisfied and we conclude that ϕ is an isomorphism. (cid:3) We refer the reader to [15] for a comprehensive treatment of C ∗ -algebraic bundles , which are usually called Fell bundles now. Noticethat for f s ∈ B s and g t ∈ B t the product f s ∗ g t is in B s + t and f ∗ s is in B − s . Also || f ∗ s ∗ f s || = || f s || . Therefore we have a Fell bundle B ρ = { B s } s ∈ ˆ K over ˆ K with operations given by (1) and (2). It is easyto see that C , ( G/ Γ , ρ ) is exactly the L -algebra of B ρ (cf. the proofof [21, Proposition 5.2]). Thus the C ∗ -algebra C ∗ ( ˆ G/ Γ , ρ ) is also theenveloping C ∗ -algebra C ∗ ( B ρ ) of the Fell bundle B ρ .Next we discuss what happens if we let ρ vary continuously. Werefer the reader to [13, Chapter 10] for the basics of continuous fieldsof Banach spaces and C ∗ -algebras. On page 505 of [22] Landstad andRaeburn pointed out that it seems reasonable that we shall get a con-tinuous field of C ∗ -algebras, but no proof was given there. This isindeed true, and we give a proof here. To be precise, fix G , Γ and K ,let W be a locally compact Hausdorff space and for each w ∈ W weassign a ρ w satisfying (S1) and (S2) such that the map w ρ w ( s ) iscontinuous for each s ∈ ˆ K . Notice that B ρ as a Banach space bundleover ˆ K do not depend on ρ . For clarity we denote the product and ∗ -operation in (1) and (2) by f s ∗ w g t and f ∗ w s . For any f s ∈ B s and g t ∈ B t , clearly the maps w f s ∗ w g t and w f ∗ w s are both contin-uous. This leads to the next lemma, which is a slight generalizationof [5, Proposition 3.3, Theorem 3.5]. The proof of [5, Proposition 3.3,Theorem 3.5], which in turn follows the lines of [29], is easily seen tohold also in our case. Lemma 2.8.
Let H be a discrete group and A h be a vector space foreach h ∈ H . Let W be a locally compact Hausdorff space and for each w ∈ W assign norms and algebra operations making A w = { A h } h ∈ H into a Fell bundle in such a way that for any f s ∈ A s and g t ∈ A t the map w
7→ || f s || w ∈ R is continuous (then we have a continuousfield of Banach spaces ( A s , || · || w ) w ∈ W over W for each s ∈ H ) andthe sections w f s ∗ w g t ∈ B st and w f ∗ w s ∈ B s − are continuousin the above continuous fields of Banach spaces ( B st , || · || w ) w ∈ W and ( B s − , || · || w ) w ∈ W respectively . Then the map w
7→ || f || w is uppersemi-continuous for each f ∈ ⊕ s ∈ H A s , where || · || w is the norm on theenveloping C ∗ -algebra C ∗ ( A w ) and extends the norm of A s as part of A w for each s ∈ H . Moreover, if H is amenable, then { C ∗ ( A w ) } w ∈ W is a continuous field of C ∗ -algebras with the field structure determinedby the continuous sections w f for all f ∈ ⊕ s ∈ H A s . Since every discrete abelian group is amenable [27, page 14], fromProposition 2.7 we get
Proposition 2.9.
Fix G, Γ and K . Let W by a locally compact Haus-dorff space and for each w ∈ W let ρ w satisfy (S1) and (S2) such thatthe map w ρ w ( s ) is continuous for each s ∈ ˆ K . Then { C ∗ ( ˆ G/ Γ , ρ w ) } w ∈ W is a continuous field of C ∗ -algebras with the field structure determinedby the continuous sections w f for all f ∈ ⊕ s ∈ ˆ K B s . Derivations
In this section we prove Proposition 3.3, to establish the relationbetween derivations coming from α and β .Throughout the rest of this paper, we assume:(S3) G/ Γ is compact.(S4) G is a Lie group.The examples in Section 2 all satisfy these conditions.We refer the reader to [17, Section 1.3] for the discussion about dif-ferentiable maps into Fr´echet spaces. We just recall that a continuousmap ψ from a smooth manifold M into a Fr´echet space A is contin-uously differentiable if for any chart ( U, φ ) of G , where U is an opensubset of some Euclidean space R n and φ is a diffeomorphism from U onto an open set of M , the derivative D ( ψ ◦ φ )( x, h ) = lim R ∋ ν → ψ ◦ φ ( x + νh ) − ψ ◦ φ ( x ) ν exists for all ( x, h ) ∈ ( U, R n ) and is a jointly continuous map from( U, R n ) into A . In such case, D ( ψ ◦ φ )( x, h ) is linear on h , and dependsonly on ψ and the tangent vector u := φ ∗ ( v x,h ) of M at φ ( x ), where v x,h denotes the tangent vector h at x . Thus we may denote D ( ψ ◦ φ )( x, h )by ∂ u ψ . Then ∂ u ψ is linear on u .Denote by g and k the Lie algebras of G and K respectively. For astrongly continuous action σ of G on a Banach space A as isometricautomorphisms, we say that an element a ∈ A is once differentiable with respect to σ if the orbit map ψ a from G into A sending x to σ x ( a )is continuously differentiable. Then the set A of once differentiableelements is a linear subspace of A . For any a ∈ A and any compactlysupported smooth C -valued function ϕ on G , it is easily checked that R G ϕ ( x ) σ x ( a ) dx is in A . As a can be approximated by such elements,we see that A is dense in A . Thinking of g as the tangent space of ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 11 G at the identity element, for each X ∈ g we have the linear map σ X : A → A sending a to ∂ X ψ a . Fix a norm on g . We define aseminorm L on A by setting L ( a ) to be the norm of the linear map g → A sending X to σ X a . Lemma 3.1.
Let σ be a strongly continuous action of G on a Banachspace A as isometric automorphisms. For any a ∈ A , one has L ( a ) = sup = X ∈ g k σ e X ( a ) − a kk X k . Proof.
The proof is similar to that of [31, Proposition 8.6]. Let X ∈ g with k X k = 1. One hassup ν> k σ e νX ( a ) − a k ν ≥ lim ν → + k σ e νX ( a ) − a k ν = k σ X ( a ) k . For any ν >
0, one also has k σ e νX ( a ) − a k = k Z ν σ e zX ( σ X ( a )) dz k ≤ Z ν k σ e zX ( σ X ( a )) k dz = Z ν k σ X ( a ) k dz = ν k σ X ( a ) k . Therefore sup ν> k σ e νX ( a ) − a k ν = k σ X ( a ) k . Thus sup = X ∈ g k σ e X ( a ) − a kk X k = sup X ∈ g , k X k =1 sup ν> k σ e νX ( a ) − a k ν = sup X ∈ g , k X k =1 k σ X ( a ) k = L ( a ) . (cid:3) Lemma 3.2.
Let σ be a strongly continuous action of G on a Banachspace A as isometric automorphisms. Then A is a Banach space withthe norm p ( a ) := L ( a ) + k a k . Suppose that σ ′ is a strongly continuousisometric action of a topological group H on A , commuting with σ .Then H preserves A , and the restriction of σ ′ on A preserves thenorm p and is strongly continuous with respect to p .Proof. Let { a n } n ∈ N be a Cauchy sequence in A under the norm p .Then as n goes to infinity, a n converges to some a ∈ A , and σ X ( a n )converge to some b X in A uniformly on X in bounded subsets of g . Let ̺ : [0 , → G be a continuously differentiable curve in G . Thenlim z → σ ̺ν + z ( a n ) − σ ̺ν ( a n ) z = σ ̺ ν ( σ ̺ ′ ν ( a n )) for all ν ∈ [0 , σ ̺ ν ( a n ) − σ ̺ ( a n ) = Z ν σ ̺ z ( σ ̺ ′ z ( a n )) dz. Letting n → ∞ we get σ ̺ ν ( a ) − σ ̺ ( a ) = Z ν σ ̺ z ( b ̺ ′ z ) dz. Therefore lim z → σ ̺z ( a ) − σ ̺ ( a ) z = σ ̺ ( b ̺ ′ ). It follows easily that a ∈ A and σ X ( a ) = b X for all X ∈ g . Consequently, a n converges to a in A under the norm g , and hence A is a Banach space under the norm g .Clearly σ ′ preserves A and the norm p . For any a ∈ A , the set of σ X ( a ) for X in the unit ball of g is compact. Then for any h ∈ H and ε >
0, when h ′ ∈ H is close enough to h , one has k σ ′ h ( a ) − σ ′ h ′ ( a ) k < ε and k σ X ( σ ′ h ( a )) − σ X ( σ ′ h ′ ( a )) k = k σ ′ h ( σ X ( a )) − σ ′ h ′ ( σ X ( a )) k < ε for all X in the unit ball of g . Consequently, p ( σ ′ h ( a ) − σ ′ h ′ ( a )) = L ( σ ′ h ( a ) − σ ′ h ′ ( a )) + k σ ′ h ( a ) − σ ′ h ′ ( a ) k < ε . Therefore the restriction of σ ′ on A is strongly continuous with respect to p . (cid:3) By Lemma 2.4 the actions α and β on C ∗ ( ˆ G/ Γ , ρ ) commute witheach other and are strongly continuous. Denote by C ( ˆ G/ Γ , ρ ) thespace of once differentiable elements of C ∗ ( ˆ G/ Γ , ρ ) with respect to theaction α . Recall the B s defined in (4). Proposition 3.3.
Let X , · · · , X n be a basis of g . For Y ∈ k say Ad x ( Y ) = X j F j,Y ( x ) X j , where Ad denotes the adjoint action of G on g . Then F j,Y ∈ B . Any f ∈ C ( ˆ G/ Γ , ρ ) is once differentiable with respect to the action β and β Y ( f ) = − X j F j,Y ∗ α X j ( f ) . (5) Proof.
Clearly F j,Y is a smooth function on G . Since the subgroups Γ, K and ρ ( ˆ K ) commute with K , if y is in any of these subgroups, thenAd y ( Y ) = Y , and hence X j F j,Y ( x ) X j = Ad x ( Y ) = Ad x (Ad y ( Y )) = Ad xy ( Y ) = X j F j,Y ( xy ) X j , which means that F j,Y is invariant under the right translation of y .Thus F j,Y ∈ C ( G/K
Γ) = C ( G/K
Γ) = B . For each X ∈ g denote ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 13 by X ( X resp.) the corresponding right (left resp.) translationinvariant vector field on G . Then Y = P j F j,Y X j .Let f ∈ C ( ˆ G/ Γ , ρ ) ∩ B s for some s ∈ ˆ K . By Lemma 2.4 the normon B s ⊆ C ∗ ( ˆ G/ Γ , ρ ) is exactly the supremum norm. Thus f belongsto the space C ( G ) of continuously differentiable functions on G . Forany continuous vector field Z on G denote by ∂ Z the correspondingderivation map C ( G ) → C ( G ). Then ∂ Y ( f ) = X j F j,Y ∂ X j ( f ) = − X j F j,Y α X j ( f ) . Since F j,Y is invariant under the right translation of Γ and ρ ( K ), wehave F j,Y ( x ) g t ( x ) = F j,Y ∗ g t ( x ) for any g t ∈ B t and x ∈ G . ByLemma 2.4 the actions α and β on C ∗ ( ˆ G/ Γ , ρ ) commute with eachother. Thus α preserves B s , and hence α X ( f ) ∈ B s for every X ∈ g .Therefore ∂ Y ( f ) = − P j F j,Y ∗ α X j ( f ).Let ̺ : [0 , → K be a continuously differentiable curve in K . Thenlim z → f ( x̺ ν + z ) − f ( x̺ ν ) z = ( ∂ ( ̺ ′ ν ) ( f ))( x̺ ν ) = ( − X j F j,̺ ′ ν ∗ α X j ( f ))( x̺ ν )for all ν ∈ [0 ,
1] and x ∈ G , and hence we have the integral form f ( x̺ ν ) − f ( x̺ ) = Z ν ( − X j F j,̺ ′ z ∗ α X j ( f ))( x̺ z ) dz (6)for all ν ∈ [0 ,
1] and x ∈ G . The left hand side of (6) is the valueof β ̺ ν ( f ) − β ̺ ( f ) at x , while the right hand side of (6) is the valueof R ν β ̺ z ( − P j F j,̺ ′ z ∗ α X j ( f )) dz at x , where the integral is taken in B s ⊆ C ∗ ( ˆ G/ Γ , ρ ). Therefore β ̺ ν ( f ) − β ̺ ( f ) = Z ν β ̺ z ( − X j F j,̺ ′ z ∗ α X j ( f )) dz (7)for all ν ∈ [0 , f ∈ ⊕ s ∈ ˆ K ( C ( ˆ G/ Γ , ρ ) ∩ B s ). By Lemma 3.2 C ( ˆ G/ Γ , ρ ) is a Banach space with norm p ( · ) = L ( · ) + k · k , β preserves C ( ˆ G/ Γ , ρ ) and p , and the restriction of β on C ( ˆ G/ Γ , ρ ) is stronglycontinuous on C ( ˆ G/ Γ , ρ ) with respect to p . By Lemma 2.4 the spec-tral subspace of C ∗ ( ˆ G/ Γ , ρ ) corresponding to s ∈ ˆ K for the action β is equal to B s . It follows that the spectral subspace of C ( ˆ G/ Γ , ρ )corresponding to s ∈ ˆ K for the restriction of β on C ( ˆ G/ Γ , ρ ) is exactly C ( ˆ G/ Γ , ρ ) ∩ B s . Then standard techniques tell us that ⊕ s ∈ ˆ K ( C ( ˆ G/ Γ , ρ ) ∩ B s ) is dense in C ( ˆ G/ Γ , ρ ) with respect to p . Notice that both sides of (7) define continuous maps from C ( ˆ G/ Γ , ρ ) to C ∗ ( ˆ G/ Γ , ρ ). Therefore(7) holds for all f ∈ C ( ˆ G/ Γ , ρ ). Consequently,lim z → β ̺ z ( f ) − β ̺ ( f ) z = β ̺ ( − X j F j,̺ ′ ∗ α X j ( f ))for all f ∈ C ( ˆ G/ Γ , ρ ). It follows easily that f is once differen-tiable with respect to β and β Y ( f ) = − P j F j,Y ∗ α X j ( f ) for all f ∈ C ( ˆ G/ Γ , ρ ) and Y ∈ k . (cid:3) We shall need the following lemma (compare [34, Proposition 2.5]).
Lemma 3.4.
Let σ be a strongly continuous action of G on a Banachspace A as isometric automorphisms. Let a ∈ A . Then for any ε > ,there is some b ∈ A such that b is smooth with respect to σ , k b k ≤ k a k , k b − a k ≤ ε , and sup = X ∈ g k σ X ( b ) kk X k ≤ sup = X ∈ g k σ eX ( a ) − a kk X k . If A hasan isometric involution being invariant under σ , then when a is self-adjoint, we can choose b also to be self-adjoint.Proof. Endow G with a left-invariant Haar measure. Let U be a smallopen neighborhood of the identity element in G with compact clo-sure, which we shall determine later. Let ϕ be a non-negative smoothfunction on G with support contained in U such that R G ϕ ( x ) dx = 1.Set b = R G ϕ ( x ) σ x ( a ) dx . Then b is smooth with respect to σ , and k b k ≤ k a k . When U is small enough, we have k a − b k ≤ ε/
2. For any X ∈ g , setting ψ ( x ) = Ad x − ( X ), we have k σ e X ( b ) − b k = k Z G ϕ ( x )( σ e X x ( a ) − σ x ( a )) dx k = k Z G ϕ ( x ) σ x ( σ e ψ ( x ) ( a ) − a ) dx k≤ Z G ϕ ( x ) k σ x ( σ e ψ ( x ) ( a ) − a ) k dx ≤ sup x ∈ U k σ e ψ ( x ) ( a ) − a k≤ sup = Y ∈ g k σ e Y ( a ) − a kk Y k · sup x ∈ U k ψ ( x ) k . Set δ = ε/ (2 + 2 k a k ). When U is small enough, we have k Ad x − ( X ) k ≤ (1 + δ ) k X k for all X ∈ g and x ∈ U . Then k σ e X ( b ) − b k ≤ (1 + δ ) k X k sup = Y ∈ g k σ eY ( a ) − a kk Y k for all X ∈ g . By Lemma 3.1 we getsup = X ∈ g k σ X ( b ) kk X k = sup = X ∈ g k σ e X ( b ) − b kk X k ≤ (1 + δ ) sup = X ∈ g k σ e X ( a ) − a kk X k . ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 15
Now it is clear that b ′ = b/ (1 + δ ) satisfies the requirement. Note that b ′ is self-adjoint if a is so. (cid:3) Nondeformed case
In this section we consider the nondeformed case, i.e., the case ρ is the trivial homomorphism ρ sending the whole ˆ K to the identityelement of G . In Proposition 4.2 we identify L ρ on C ( ˆ G/ Γ , ρ ) withthe Lipschitz seminorm for certain metric on G/ Γ.Note that C , ( G/ Γ , ρ ) is sub- ∗ -algebra of C ( G/ Γ) = C ( G/ Γ). Bythe universality of C ∗ ( ˆ G/ Γ , ρ ) we have a natural ∗ -homomorphism ψ of C ∗ ( ˆ G/ Γ , ρ ) into C ( G/ Γ), extending the inclusion C , ( G/ Γ , ρ ) ֒ → C ( G/ Γ). The right translation of K on G induces a strongly continuousaction β ′′ of K on C ( G/ Γ), and clearly ψ intertwines β and β ′′ . Anapplication of Lemmas 2.6 and 2.4 tells us that ψ is injective. Bydefinition B s is the spectral subspace of C ( G/ Γ) corresponding to s ∈ ˆ K . Thus ⊕ s ∈ ˆ K B s is dense in C ( G/ Γ). As ⊕ s ∈ ˆ K B s is in the image of ψ , we see that ψ is surjective and hence is an isomorphism. We shallidentify C ∗ ( ˆ G/ Γ , ρ ) and C ( G/ Γ) via ψ .The seminorm L ρ describes the size of derivatives of f ∈ C ( ˆ G/ Γ , ρ ).If it corresponds to some metric on G/ Γ, this metric should be kindof geodesic distance. In order for the geodesic distance to be defined,throughout the rest of this paper we assume:(S5) G/ Γ is connected.The examples in Section 2 all satisfy this condition.Fix an inner product on g . Then we obtain a right translation in-variant Riemannian metric on G in the usual way. Denote by d G thegeodesic distance on connected components of G . We extend d G to asemi-distance on G via setting d G ( x, y ) = ∞ if x and y lie in differentconnected components of G . Lemma 4.1.
The function d on G/ Γ × G/ Γ defined by d ( x Γ , y Γ) :=inf x ′ ∈ x Γ ,y ′ ∈ y Γ d G ( x ′ , y ′ ) is equal to inf y ′ ∈ y Γ d G ( x, y ′ ) . It is a metric on G/ Γ and induces the quotient topology on G/ Γ .Proof. Let V be a connected component of G . Then V Γ is clopen in G , and hence V Γ / Γ is clopen in G/ Γ for the quotient topology. As G/ Γ is connected, we conclude that V Γ / Γ = G/ Γ. Therefore d is finitevalued.Since d G is right translation invariant, we have inf x ′ ∈ x Γ ,y ′ ∈ y Γ d G ( x ′ , y ′ ) =inf y ′ ∈ y Γ d G ( x, y ′ ). It follows easily that d is a metric on G/ Γ.Let x ∈ G . Let W be a neighborhood of x Γ in G/ Γ for the quotienttopology. Then there exists ε > d G ( x, y ) < ε , then y Γ ∈ W . It follows that if d ( x Γ , y Γ) < ε , then y Γ ∈ W . Thereforethe topology induced by d on G/ Γ is finer than the quotient topology.For any ε ′ >
0, set U = { y ∈ G | d G ( x, y ) < ε ′ } . Then U is an openneighborhood of x . Thus U Γ / Γ is an open neighborhood of x Γ forthe quotient topology. For any z Γ ∈ U Γ / Γ, we can find z ′ ∈ z Γ ∩ U and hence d ( x Γ , z Γ) ≤ d G ( x, z ′ ) < ε ′ . Therefore the quotient topologyon G/ Γ is finer than the topology induced by d . We conclude that d induces the quotient topology. (cid:3) Proposition 4.2.
For any f ∈ C ( ˆ G/ Γ , ρ ) ⊆ C ∗ ( ˆ G/ Γ , ρ ) = C ( G/ Γ) ,we have L ρ ( f ) = sup x Γ = y Γ | f ( x Γ) − f ( y Γ) | d ( x Γ , y Γ) . Proof.
The right hand side of the above equation is equal to sup x = y | f ( x ) − f ( y ) | d G ( x,y ) .So it suffices to show L ρ ( f ) = sup x = y | f ( x ) − f ( y ) | d G ( x, y ) . (8)The proof is similar to that of [31, Proposition 8.6]. Let ̺ : [0 , → G be a continuously differentiable curve. Denote by ℓ ( ̺ ) the length of ̺ .Then ( f ◦ ̺ ) ′ ( ν ) = ( α − Ad ̺ν ( ̺ ′ ν ) f )( ̺ ν ) for all ν ∈ [0 , | f ( ̺ ) − f ( ̺ ) | = | Z ( f ◦ ̺ ) ′ ( ν ) dν | ≤ Z | ( f ◦ ̺ ) ′ ( ν ) | dν = Z | ( α − Ad ̺ν ( ̺ ′ ν ) f )( ̺ ν ) | dν ≤ Z k α Ad ̺ν ( ̺ ′ ν ) f k dν ≤ L ρ ( f ) Z k Ad ̺ ν ( ̺ ′ ν ) k dν = L ρ ( f ) ℓ ( ̺ ) , where in the last equality we use the fact that the Riemannian metric on G is right translation invariant. It follows easily that | f ( ̺ ) − f ( ̺ ) | ≤ L ρ ( f ) ℓ ( ̺ ) holds if ̺ is only piecewise continuously differentiable. Con-sidering all piecewise continuously differentiable curves connecting x and y we obtain | f ( x ) − f ( y ) | ≤ L ρ ( f ) d G ( x, y ) for all x, y ∈ G . ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 17
Denote by e G the identity element of G . For any 0 = X ∈ g , we havesup x = y | f ( x ) − f ( y ) | d G ( x, y ) ≥ sup x sup ν =0 | f ( x ) − f ( e νX x ) | d G ( x, e νX x )= sup ν =0 sup x | f ( x ) − f ( e νX x ) | d G ( e G , e νX ) ≥ sup ν =0 sup x | f ( x ) − f ( e νX x ) || ν |k X k = sup ν =0 k f − α e − νX f k| ν |k X k = sup ν =0 k α e νX f − f k| ν |k X k ≥ k α X ( f ) kk X k . Therefore sup x = y | f ( x ) − f ( y ) | d G ( x,y ) ≥ L ρ ( f ). This proves (8). (cid:3) Lip-norms and compact group actions
In this section we recall the definition of compact quantum metricspaces and prove Theorem 5.2, which enables one to show that certainseminorm defines a quantum metric, via the help of a compact groupaction.Rieffel has set up the theory of compact quantum metric spaces inthe general framework of order-unit spaces [31, Defintion 2.1]. Weshall need it only for C ∗ -algebras. By a C ∗ -algebraic compact quantummetric space we mean a pair ( A, L ) consisting of a unital C ∗ -algebra A and a (possibly + ∞ -valued) seminorm L on A satisfying the realitycondition L ( a ) = L ( a ∗ )(9)for all a ∈ A , such that L vanishes exactly on C and the metric d L onthe state space S ( A ) defined by d L ( ψ, φ ) = sup L ( a ) ≤ | ψ ( a ) − φ ( a ) | (10)induces the weak ∗ -topology. The radius of ( A, L ), denote by r A , isdefined to be the radius of ( S ( A ) , d L ). We say that L is a Lip-norm .Let A be a unital C ∗ -algebra and let L be a (possibly + ∞ -valued)seminorm on A vanishing on C . Then L and k · k induce (semi)norms˜ L and k · k ∼ respectively on the quotient space ˜ A = A/ C .Recall that a character of a compact group is the trace function of afinite-dimensional complex representation of the group [7, Section II.4]. Lemma 5.1.
Let H be a compact group and H be a closed normalsubgroup of H of finite index. Then for any linear combination offinitely many characters of H , its multiplication with the characteristicfunction of H is also a linear combination of finitely many charactersof H .Proof. The products and sums of characters of H are still characters[7, Proposition II.4.10]. Thus it suffices to show that the characteristicfunction of H on H is a linear combination of finitely many charactersof H .Since H/H is finite, every C -valued class function on H/H , i.e.,functions being constant on conjugate classes, is a linear combinationof characters of H/H [16, Proposition 2.30]. Thus the characteristicfunction of { e H/H } on H/H , where e H/H denotes the identity elementof H/H , is a linear combination of characters of H/H . Then thecharacteristic function H on H is a linear combination of charactersof H . (cid:3) Recall that a length function on a topological group H is a continuous R ≥ -valued function, ℓ , on H such that ℓ ( h ) = 0 if and only if h is equalto the identity element e H of H , that ℓ ( h h ) ≤ ℓ ( h ) + ℓ ( h ) for all h , h ∈ H , and that ℓ ( h − ) = ℓ ( h ) for all h ∈ H .Suppose that a compact group H has a strongly continuous action σ on a Banach space A as isometric automorphisms. Endow H withits normalized Haar measure. For any continuous C -valued function ϕ on H , define a linear map σ ϕ : A → A by σ ϕ ( a ) = Z H ϕ ( h ) σ h ( a ) dh for a ∈ A . Denote by ˆ H the set of isomorphism classes of irreduciblerepresentations of H . For each s ∈ ˆ H , denote by A s the spectralsubspace of A corresponding to s . For a finite subset J of ˆ H , set A J = P s ∈ J A s .The main tool we use for the proof of Theorem 1.1 will be the fol-lowing slight generalization of [23, Theorem 4.1]. Theorem 5.2.
Let A be a unital C ∗ -algebra, let L be a (possibly + ∞ -valued) seminorm on A satisfying the reality condition (9), and let σ be a strongly continuous action of a compact group H on A by auto-morphisms. Assume that L takes finite values on a dense subspace of A , and that L vanishes on C . Suppose that the following conditions aresatisfied: ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 19 (1) there are some length function ℓ on a closed normal subgroup H of H of finite index and some constant C > such that L ℓ ≤ C · L on A , where L ℓ is the (possibly + ∞ -valued) seminormon A defined by L ℓ ( a ) = sup { k σ h ( a ) − a k ℓ ( h ) | h ∈ H , h = e H } . (11)(2) for any linear combination ϕ of finitely many characters on H we have L ◦ σ ϕ ≤ k ϕ k · L on A , where k ϕ k denotes the L norm of ϕ ; (3) for each s ∈ ˆ H not being the trivial representation s of H , theset { a ∈ A s | L ( a ) ≤ , k a k ≤ r } is totally bounded for some r > , and the only element in A s vanishing under L is ; (4) there is a unital C ∗ -algebra A containing the fixed-point subal-gebra A σ , with a Lip-norm L A , such that L A extends the re-striction of L to A σ ; (5) for each s ∈ \ H/H ⊆ ˆ H not equal to s , there exists someconstant C s > such that k · k ≤ C s L on A s .Then ( A, L ) is a C ∗ -algebraic compact quantum metric space with r A ≤ C R H ℓ ( h ) dh + P s = s ∈ \ H/H C s (dim( s )) + r A , where H is endowed withits normalized Haar measure. We need some preparation for the proof of Theorem 5.2. The follow-ing lemma generalizes [23, Lemma 3.4].
Lemma 5.3.
Let H be a compact group, and let H be a closed normalsubgroup of H of finite index. Let f be a continuous C -valued functionon H with f ( e H ) = 0 . Then for any ε > there is a nonnegativefunction ϕ on H with support contained in H such that ϕ is a linearcombination of finitely many characters of H , k ϕ k = 1 , and k ϕ · f k <ǫ .Proof. Denote by χ the characteristic function of H on H . Set g = f χ + ε (1 − χ ). Then g ∈ C ( H ) and g ( e H ) = 0. By [23, Lemma 3.4]we can find a nonnegative function φ on H such that φ is a linearcombination of finitely many characters, k φ k = 1, and k φ · g k < ε/ ε R H \ H φ ( h ) dh ≤ k φ · g k < ε/
2, and hence k χφ k = k φ k − Z H \ H φ ( h ) dh > − / / . Set ϕ = χφ/ k χφ k . By Lemma 5.1 ϕ is a linear combination of finitelymany characters of H . One has k ϕ · f k = k χφf k / k χφ k = k χφg k / k χφ k < ( ε/ / (1 /
2) = ε. (cid:3) For a compact group H and a finite subset J of ˆ H , set ¯ J = { ¯ s | s ∈ J } , where ¯ s denotes the contragradient representation. Replacing [23,Lemma 3.4] by Lemma 5.3 in the proof of [23, Lemma 4.4], we get: Lemma 5.4.
Let H be a compact group. For any ε > there isa finite subset J = ¯ J in ˆ H , containing the trivial representation s ,depending only on ℓ and ε/C , such that for any strongly continuousisometric action σ of H on a complex Banach space A with a (possibly + ∞ -valued) seminorm L on A satisfying conditions (1) and (2) inTheorem 5.2, and any a ∈ A , there is some a ′ ∈ A J with k a ′ k ≤ k a k , L ( a ′ ) ≤ L ( a ) , and k a − a ′ k ≤ εL ( a ) . If A has an isometric involution being invariant under σ , then when a is self-adjoint we can choose a ′ also to be self-adjoint. We are ready to prove Theorem 5.2.
Proof of Theorem 5.2.
Most part of the proof of [23, Theorem 4.1] car-ries over here. In fact, conditions (2)-(4) here are the same as theconditions (2)-(4) in [23, Theorem 4.1]. Since the proof of Lemma 4.5in [23] does not involve condition (1) there, this lemma still holds inour current situation. Replacing [23, Lemma 4.4] by Lemma 5.4 in theproof of Lemma 4.6 of [23], we see that the latter also holds in ourcurrent situation. To finish the proof of Theorem 5.2, we only need toprove the following analogue of Lemma 4.7 of [23]:
Lemma 5.5.
We have k · k ∼ ≤ ( C Z H ℓ ( h ) dh + X s = s ∈ \ H/H C s (dim( s )) + r A ) L ∼ on ( ˜ A ) sa , where H is endowed with its normalized Haar measure.Proof. By Lemma 5.1 the characteristic function ϕ of H on H is alinear combination of characters of H . Set n = | H/H | . Let a ∈ A sa .Then σ nϕ ( a ) belongs to A sa and is fixed by σ | H . We have k a − σ nϕ ( a ) k = k Z H a dh − Z H σ h ( a ) dh k ≤ Z H k a − σ h ( a ) k dh ≤ L ℓ ( a ) Z H ℓ ( h ) dh ≤ C · L ( a ) Z H ℓ ( h ) dh, where the last inequality comes from the condition (1). By the condi-tion (2) we have L ( σ nϕ ( a )) ≤ k nϕ k · L ( a ) = L ( a ) . ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 21
Note that A σ | H = ⊕ s ∈ \ H/H A s . Say, σ nϕ ( a ) = P s ∈ \ H/H a s with a s ∈ A s .For each s ∈ \ H/H , denote by χ s the corresponding character of H/H ,thought of as a character of H . Then a s = σ dim( s ) χ s ( σ nϕ ( a )) [23, Lemma3.2]. Thus L ( a s ) = L ( σ dim( s ) χ s ( σ nϕ ( a )) ≤ k dim( s ) χ s k L ( σ nϕ ( a )) ≤ (dim( s )) L ( a ) , where the first inequality comes from the condition (2). Note that a s ∈ A sa . By the condition (5) we have k a s k ≤ C s L ( a s ) ≤ C s (dim( s )) L ( a )for each s ∈ \ H/H not equal to s . By the condition (4), we have k b k ∼ ≤ r A L ∼ ( b )for all b ∈ ( A s ) sa = ( A σ ) sa [30, Proposition 1.6, Theorem 1.9] [23,Proposition 2.11]. Thus k a s k ∼ ≤ r A L ∼ ( a s ) = r A L ( a s ) ≤ r A L ( a ) . Therefore we have k a k ∼ ≤ k a − σ nϕ ( a ) k + k a s k ∼ + X s = s ∈ \ H/H k a s k≤ C · L ( a ) Z H ℓ ( h ) dh + r A L ( a ) + X s = s ∈ \ H/H C s (dim( s )) L ( a )as desired. (cid:3) This finishes the proof of Theorem 5.2. (cid:3) Proof of Theorem 1.1
In this section we prove Theorem 1.1.Denote by K the connected component of K containing the identityelement e K . Take an inner product on k and use it to get a translationinvariant Riemannian metric on K in the usual way. For each x ∈ K set ℓ ( x ) to be the geodesic distance form e K to x . Then ℓ is a lengthfunction on K .In order to prove Theorem 1.1, we just need to verify the conditions inTheorem 5.2 for ( A, L, H, H , σ ) = ( C ∗ ( ˆ G/ Γ , ρ ) , L ρ , K, K , β ). Recallthat we are given a norm on g , and(12) L ρ ( a ) = ( sup = X ∈ g k α X ( a ) kk X k , if a ∈ C ( ˆ G/ Γ , ρ ); ∞ , otherwise,for a ∈ C ∗ ( ˆ G/ Γ , ρ ). By Lemma 2.4 the actions α and β on C ∗ ( ˆ G/ Γ , ρ ) commute witheach other. Thus β preserves C ( ˆ G/ Γ , ρ ) and L ρ .Choose the basis X , . . . , X dim( G ) of g in Proposition 3.3 to be ofnorm 1. Denote by C the supremum of || F j,Y || for all 1 ≤ j ≤ dim( G )and Y in the unit sphere of k (with respect to the inner product on k above) in Proposition 3.3. Lemma 6.1.
We have L ℓ ≤ ( dim ( G ) C ) · L ρ on C ∗ ( ˆ G/ Γ , ρ ) .Proof. It suffices to show L ℓ ≤ (dim( G ) C ) · L ρ on C ( ˆ G/ Γ , ρ ). ByProposition 3.3 every a ∈ C ( ˆ G/ Γ , ρ ) is once differentiable with re-spect to the action β . By [31, Proposition 8.6] we have L ℓ ( a ) =sup Y ∈ k , || Y || =1 || β Y ( a ) || . Then from (5) in Proposition 3.3 we get L ℓ ( a ) ≤ (dim( G ) C ) L ρ ( a ). (cid:3) Lemma 6.2.
For any linear combination ϕ of finitely many charactersof K we have L ρ ◦ β ϕ ≤ || ϕ || · L ρ on C ∗ ( ˆ G/ Γ , ρ ) .Proof. We have remarked above that β preserves L ρ . By Lemma 3.1one has L ρ ( a ) = sup = X ∈ g k α e X ( a ) − a kk X k (13)for every a ∈ C ( ˆ G/ Γ , ρ ). It follows that L ρ is lower semi-continuouson C ( ˆ G/ Γ , ρ ) equipped with the relative topology from C ( ˆ G/ Γ , ρ ) ⊆ C ∗ ( ˆ G/ Γ , ρ ). By Lemma 3.2 the action β is also strongly continuous on C ( ˆ G/ Γ , ρ ) with respect to the norm defined in Lemma 3.2. Then β ψ isalso well-defined on C ( ˆ G/ Γ , ρ ) for any continuous C -valued function ψ on K . By [23, Remark 4.2.(3)] we get Lemma 6.2. (cid:3) The conditions (1) and (2) in Theorem 5.2 for (
A, L, H, H , σ ) =( C ∗ ( ˆ G/ Γ , ρ ) , L ρ , K, K , β ) follow from Lemmas 6.1 and 6.2 respectively.Fix an inner product on g , and denote by L ′ ρ the seminorm on C ∗ ( ˆ G/ Γ , ρ ) defined by (12) but using this inner product norm instead.Since g is finite dimensional, any two norms on g are equivalent. There-fore there exists some constant C > ρ such that L ′ ρ ≤ C L ρ .By Lemma 4.1 and Proposition 4.2 the restriction of L ′ ρ on C ( ˆ G/ Γ , ρ ) ⊆ C ( G/ Γ) is the Lipschitz seminorm associated to some metric d on G/ Γ.The Arzela-Ascoli theorem [12, Theorem VI.3.8] tells us that the set { a ∈ C ∗ ( ˆ G/ Γ , ρ ) | L ρ ( a ) ≤ r , k a k ≤ r } is totally bounded for any r , r >
0. Since for each s ∈ ˆ K neither the seminorm L ρ nor the ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 23 C ∗ -norm on B s ⊆ C ∗ ( ˆ G/ Γ , ρ ) depends on ρ , the condition (3) in The-orem 5.2 for ( A, L, H, H , σ ) = ( C ∗ ( ˆ G/ Γ , ρ ) , L ρ , K, K , β ) follows.From the criterion of Lip-norms in [30, Proposition 1.6, Theorem 1.9](see also [23, Proposition 2.11]) one sees that the Lipschitz seminormassociated to the metric on any compact metric space is a Lip-normon the C ∗ -algebra of continuous functions on this space. Since L ′ ρ on C ∗ ( ˆ G/ Γ , ρ ) = C ( G/ Γ) is no less than the Lipschitz seminorm associ-ated to the metric d on G/ Γ, from [30, Proposition 1.6, Theorem 1.9]one concludes that L ρ is also a Lip-norm on C ( G/ Γ). Therefore wemay take ( A , L A ) in condition (4) of Theorem 5.2 to be ( C ( G/ Γ) , L ρ )for ( A, L, H, H , σ ) = ( C ∗ ( ˆ G/ Γ , ρ ) , L ρ , K, K , β ).Let s ∈ ˆ K not being the trivial representation of K , and let a ∈ B s .Then L ′ ρ ( a ) ≤ C L ρ ( a ) = C L ρ ( a ). Thus for any λ in the range of a on G/ Γ one has k a − λ C ( G/ Γ) k C ( G/ Γ) ≤ C C L ρ ( a ), where C denotesthe diameter of G/ Γ under the metric d . We have k a k C ∗ ( ˆ G/ Γ ,ρ ) = k a k C ( G/ Γ) = k Z K h k, s i β k ( a − λ C ( G/ Γ) ) dk k C ( G/ Γ) ≤ k a − λ C ( G/ Γ) k C ( G/ Γ) ≤ C C L ρ ( a ) . This establishes the condition (5) of Theorem 5.2 for (
A, L, H, H , σ ) =( C ∗ ( ˆ G/ Γ , ρ ) , L ρ , K, K , β ).We have shown that the conditions in Theorem 5.2 hold for ( A, L, H, H , σ ) =( C ∗ ( ˆ G/ Γ , ρ ) , L ρ , K, K , β ). Thus Theorem 1.1 follows from Theorem 5.2.7. Quantum Gromov-Hausdorff distance
In this section we prove Theorem 1.2.We recall first the definition of the distance dist nu from [19, Sec-tion 5]. To simplify the notation, for fixed unital C ∗ -algebras A and A , when we take infimum over unital C ∗ -algebras B containingboth A and A , we mean to take infimum over all unital isometric ∗ -homomorphisms of A and A into some unital C ∗ -algebra B . De-note by dist B H the Hausdorff distance between subsets of B . For a C ∗ -algebraic compact quantum metric spaces ( A, L A ), set E ( A ) = { a ∈ A sa | L A ( a ) ≤ } . For any C ∗ -algebraic compact quantum metric spaces ( A , L A ) and( A , L A ), the distance dist nu ( A , A ) is defined asdist nu ( A , A ) = inf dist B H ( E ( A ) , E ( A )) , where the infimum is taken over all unital C ∗ -algebras B containing A and A . Throughout the rest of this section, we fix G , Γ, K such that thereexits ρ satisfying the conditions (S1)-(S5). We also fix a norm on g .Denote by Ω the set of all ρ satisfying the conditions (S1) and (S2),equipped with the weakest topology making the maps Ω → G sending ρ to ρ ( s ) to be continuous for each s ∈ ˆ K .Every closed subgroup of a Lie group is also a Lie group [37, Theorem3.42]. Thus K is a compact abelian Lie group. Then K is the productof a torus and a finite abelian group [7, Corollary 3.7]. Therefore ˆ K is finitely generated. Let s , . . . , s n be a finite subset of ˆ K generatingˆ K . Then the map ϕ : Ω → Q nj =1 G sending ρ to ( ρ ( s ) , . . . , ρ ( s n )) isinjective, and its image is closed. Furthermore, it is easily checked thatthe topology on Ω is exactly the pullback of the relative topology of ϕ (Ω) in Q nj =1 G . Since G is a Lie group, it is locally compact metrizable.Thus Q nj =1 G and Ω are also locally compact metrizable.For clarity and convenience, we shall denote the actions α and β on C ∗ ( ˆ G/ Γ , ρ ) by α ρ and β ρ respectively, and denote the C ∗ -norm on C ∗ ( ˆ G/ Γ , ρ ) by k · k ρ . Consider the (possibly + ∞ -valued) auxiliaryseminorm L ′′ ρ on C ∗ ( ˆ G/ Γ , ρ ) defined by L ′′ ρ ( a ) = sup = X ∈ g k α ρ,e X ( a ) − a k ρ k X k . Lemma 7.1.
Let W be a locally compact Hausdorff space with a con-tinuous map W → Ω sending w to ρ w . Let f be a continuous sectionof the continuous field of C ∗ -algebras over W in Proposition 2.9. Thenthe function w L ′′ ρ w ( f w ) is lower semi-continuous on W .Proof. Let w ′ ∈ W . To show that the above function is lower semi-continuous at w ′ , we consider the case L ′′ ρ w ′ ( f w ′ ) < ∞ . The case L ′′ ρ w ′ ( f w ′ ) = ∞ can be dealt with similarly. Let ε >
0. Take 0 = X ∈ g such that L ′′ ρ w ′ ( f w ′ ) k X k < k α ρ w ′ ,e X ( f w ′ ) − f w ′ k ρ w ′ + ε k X k . It is easily checked that w α ρ w ,e X ( f w ) is also a continuous section ofthe continuous field. Then when w is close enough to w ′ , we have k α ρ w ′ ,e X ( f w ′ ) − f w ′ k ρ w ′ < k α ρ w ,e X ( f w ) − f w k ρ w + ε k X k and hence L ′′ ρ w ′ ( f w ′ ) k X k < k α ρ w ,e X ( f w ) − f w k ρ w + 2 ε k X k ≤ ( L ′′ ρ w ( f w ) + 2 ε ) k X k . Therefore L ′′ ρ w ′ ( f w ′ ) ≤ L ′′ ρ w ( f w ) + 2 ε . (cid:3) ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 25
Note that although the ∗ -algebra structure of C , ( G/ Γ , ρ ) ( C b , ( G, ρ )resp.) depends on ρ , the Banach space structure, the left transla-tion action of G and the right translation action of K on C , ( G/ Γ , ρ )( C b , ( G, ρ ) resp.) do not depend on ρ . Thus we may denote by C , ( G/ Γ), α and β this Banach space and these actions respectively.Also denote by C , ( G/ Γ) the set of once differentiable elements of C , ( G/ Γ) with respect to α . Lemma 7.2.
For any a in ⊕ s ∈ ˆ K ( B s ∩ C , ( G/ Γ)) , the function ρ L ρ ( a ) is continuous on Ω .Proof. Say, a = P s ∈ F a s for some finite subset F of ˆ K and a s ∈ B s ∩ C , ( G/ Γ) for each s ∈ F . Then L ρ ( a ) = sup X ∈ g , k X k =1 k P s ∈ F α X ( a s ) k ρ for each ρ ∈ Ω. Since α commutes with β , we have α X ( a s ) ∈ B s .By Proposition 2.9 the function ρ
7→ k P s ∈ F α X ( a s ) k ρ is continu-ous on Ω for each X ∈ g . Since g is a finite-dimensional vectorspace and α X ( a s ) depends on X linearly, it follows easily that thefunction ( X, ρ )
7→ k P s ∈ F α X ( a s ) k ρ is continuous on g × Ω. As theunit sphere of g is compact, one concludes that the function ρ sup X ∈ g , k X k =1 k P s ∈ F α X ( a s ) k ρ is continuous on Ω. (cid:3) Fix ρ ′ ∈ Ω. Let Z be a compact neighborhood of ρ ′ in Ω.Note that the linear span of ρ f ( ρ ) a ∈ C ∗ ( ˆ G/ Γ , ρ ) for a in some B s and f ∈ C ( Z ) is dense in the C ∗ -algebra of continuous sections ofthe continuous field over Z in Proposition 2.9. Since Z is a compactmetrizable space, C ( Z ) is separable. As G is a Lie group, it is separable.Then G/ Γ is separable, and hence is a compact metrizable space. Thus C ( G/ Γ) is separable, and hence B s is separable for each s ∈ ˆ K . On theother hand, since ˆ K is finitely generated, ˆ K is countable. Thereforethe C ∗ -algebra of continuous sections of the continuous field over Z inProposition 2.9 is separable.By Proposition 2.5 each C ∗ ( ˆ G/ Γ , ρ ) is nuclear. Every separable con-tinuous field of unital nuclear C ∗ -algebras over a compact metric spacecan be subtrivialized [6, Theorem 3.2]. Thus we can find a unital C ∗ -algebra B and unital embeddings C ∗ ( ˆ G/ Γ , ρ ) → B for all ρ ∈ Z suchthat, via identifying each C ∗ ( ˆ G/ Γ , ρ ) with its image in B , the con-tinuous sections of the continuous field over Z in Proposition 2.9 areexactly the continuous maps Z → B whose images at each ρ are in C ∗ ( ˆ G/ Γ , ρ ).For any C ∗ -algebraic compact quantum metric space ( A, L A ) and anyconstant R no less than the radius of ( A, L A ), the set D R ( A ) := { a ∈ A sa | L A ( a ) ≤ , k a k ≤ R } is totally bounded and every a ∈ E ( A ) canbe written as x + λ for some x ∈ D R ( A ) and λ ∈ R [30, Proposition A, L, H, H , σ ) = ( C ∗ ( ˆ G/ Γ , ρ ) , L ρ , K, K , β ) withsome C, C s and ( A , L A ) not depending on ρ . Thus, by Theorem 5.2there is some constant R such that the radius of ( C ∗ ( ˆ G/ Γ , ρ ρ ) , L ρ ) isno bigger than R for all ρ ∈ Ω. For any ε >
0, by Lemmas 5.4 and 2.4there is a finite subset F ⊆ ˆ K satisfying that for any ρ ∈ Ω and any x ∈ E ( C ∗ ( ˆ G/ Γ , ρ )) there is some y ∈ E ( C ∗ ( ˆ G/ Γ , ρ )) ∩ P s ∈ F B s with k y k ρ ≤ k x k ρ and k x − y k ρ < ε . Lemma 7.3.
Let ε > . Then there is a neighborhood U of ρ ′ in Z such that for any ρ ∈ U and any a ∈ E ( C ∗ ( ˆ G/ Γ , ρ ′ )) there is some b ∈ E ( C ∗ ( ˆ G/ Γ , ρ )) with k a − b k B < ε .Proof. According to the discussion above we can find a finite subset Y of E ( C ∗ ( ˆ G/ Γ , ρ ′ )) ∩ P s ∈ F B s such that for every a ∈ E ( C ∗ ( ˆ G/ Γ , ρ ′ ))there are some z ∈ Y and λ ∈ R with k a − ( z + λ ) k ρ ′ < ε . For each y ∈ Y , write y as P s ∈ F y s with y s ∈ B s . Since L ρ ′ ( y ) < ∞ , y is oncedifferentiable with respect to α ρ ′ . It is easy to see that each y s is oncedifferentiable with respect to α ρ ′ . Thus, by Lemma 7.2 the function ρ L ρ ( y ) is continuous on Ω. Then we can find a constant δ > U of ρ ′ in Z such that δ k y ρ k ρ < ε , k y ρ ′ − y ρ k B < ε ,and L ρ ( y ρ ) < δ for all y ∈ Y and ρ ∈ U , where y ρ denotes y as an element in C ∗ ( ˆ G/ Γ , ρ ). Fix ρ ∈ U . Set b = z ρ / (1 + δ ). Then L ρ ( b + λ ) = L ρ ( b ) <
1, and k a − ( b + λ ) k B ≤ k a − ( z ρ ′ + λ ) k ρ ′ + k z ρ ′ − z ρ k B + k z ρ − b k ρ < ε + ε + ε = 3 ε. (cid:3) Lemma 7.4.
Let ε > . Then there is a neighborhood U of ρ ′ in Z such that for any ρ ∈ U and any a ∈ E ( C ∗ ( ˆ G/ Γ , ρ )) there is some b ∈ E ( C ∗ ( ˆ G/ Γ , ρ ′ )) with k a − b k B < ε .Proof. According to the discussion before Lemma 7.3, it suffices toshow that there is a neighborhood U of ρ ′ in Z such that for any ρ ∈ U and any a ∈ E ( C ∗ ( ˆ G/ Γ , ρ )) ∩ ⊕ s ∈ F B s satisfying k a k ρ ≤ R there issome b ∈ E ( C ∗ ( ˆ G/ Γ , ρ ′ )) with k a − b k B < ε . Suppose that this fails.Then we can find a sequence { ρ n } n ∈ N in Z converging to ρ ′ and an a n ∈ E ( C ∗ ( ˆ G/ Γ , ρ n )) ∩ ⊕ s ∈ F B s satisfying k a n k ρ n ≤ R for each n ∈ N such that k a n − b k B ≥ ε for all n ∈ N and b ∈ E ( C ∗ ( ˆ G/ Γ , ρ ′ )). Write a n as P s ∈ F a n,s with a n,s ∈ B s . Then a n,s = R K h k, s i β ρ n ,k ( a n ) dk . Thus k a n,s k ρ n ≤ k a n k ρ n ≤ R and L ρ n ( a n,s ) ≤ L ρ n ( a n ) ≤ L ρ on B s does not depend on ρ , and the set { a ∈ ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 27 B s | L ρ ( a ) ≤ , k a k ≤ R } is totally bounded, passing to a subsequenceif necessary, we may assume that a n,s converges to some a s in B s when n → ∞ for each s ∈ F . Set a = P s ∈ F a s . Then ( a n ) ρ n converges to a ρ ′ in B as n → ∞ , where ( a n ) ρ n and a ρ ′ denote a n and a as elementsin C ∗ ( ˆ G/ Γ , ρ n ) and C ∗ ( ˆ G/ Γ , ρ ′ ) respectively. In particular, a is self-adjoint and k a k ρ ′ ≤ lim n →∞ k a n k ρ n ≤ R .By Lemma 3.1 we have L ′′ ρ n ( a n ) = L ρ n ( a n ) ≤ n ∈ N . On theone-point compactification W = N ∪{∞} of N , consider the continuousmap W → Ω sending n ∈ N to ρ n and ∞ to ρ ′ . Then the section f defined as f n = a n ∈ C ∗ ( ˆ G/ Γ , ρ n ) for n ∈ N and f ∞ = a ∈ C ∗ ( ˆ G/ Γ , ρ ′ )is a continuous section of the continuous field on W in Proposition 2.9.Thus, by Lemma 7.1 we have L ′′ ρ ′ ( a ) ≤ lim inf n →∞ L ′′ ρ n ( a n ) ≤
1. ByLemma 3.4 we can find some self-adjoint b ∈ C ( ˆ G/ Γ , ρ ′ ) with k b k ρ ′ ≤k a k ρ ′ ≤ R , k b − a k ρ ′ ≤ ε/
2, and L ρ ′ ( b ) ≤ L ′′ ρ ′ ( a ) ≤
1. Then b ∈E ( C ∗ ( ˆ G/ Γ , ρ ′ )), and k b − a n k B → k b − a k ρ ′ ≤ ε/ n → ∞ . Therefore, when n is large enough, we have k b − a n k B < ε ,contradicting our assumption. This finishes the proof of the lemma. (cid:3) From Lemmas 7.3 and 7.4 we conclude that Theorem 1.2 holds.
Appendix A. Comparison of dist nu and proxIn this appendix we compare the distance dist nu and the proximityRieffel introduced in [35].A (possibly + ∞ -valued) seminorm L on a unital (possibly incom-plete) C ∗ -norm algebra A is called a C ∗ -metric [35, Definition 4.1] if(1) L is lower semi-continuous, satisfies the reality condition (9),and is strongly-Leibniz in the sense that L ( ab ) ≤ L ( a ) k b k + k a k L ( b ) for all a, b ∈ A , L (1 A ) = 0, and L ( a − ) ≤ k a − k L ( a )for all a being invertible in A ,(2) L extended to the completion ¯ A of A by L ( a ) = ∞ for a ∈ ¯ A \ A is a Lip-norm on ¯ A ,(3) the algebra { a ∈ A | L ( a ) < ∞} is spectrally stable in ¯ A .In such case, the pair ( A, L ) is called a compact C ∗ -metric space .The seminorm L ρ in Theorem 1.1 may fail to be a C ∗ -metric sinceit may fail to be lower semi-continuous. However, it is lower semi-continuous on C ( ˆ G/ Γ , ρ ) by Lemma 3.1. Thus its restriction on thealgebra of smooth elements in C ∗ ( ˆ G/ Γ , ρ ) with respect to α is a C ∗ -metric. By [35, Proposition 3.2] its closure ¯ L ρ is a C ∗ -metric on C ∗ ( ˆ G/ Γ , ρ ). Lemma 3.4 tells us that¯ L ρ ( a ) = sup = X ∈ g k α e X ( a ) − a kk X k for all a ∈ C ∗ ( ˆ G/ Γ , ρ ).In [35, Definition 5.6, Section 14] Rieffel introduced the notions of proximity prox( A, B ) and complete proximity prox s ( A, B ) between twocompact C ∗ -metric spaces ( A, L A ) and ( B, L B ). In general, one hasprox s ( A, B ) ≥ prox( A, B ). For each q ∈ N , denote by UCP q ( A ) the setof unital completely positive linear maps from the completion ¯ A of A to M q ( C ). Define prox q ( A, B ) as the infimum of the Hausdorff distanceof UCP q ( A ) and UCP q ( B ) in UCP q ( A ⊕ B ) under the metric d qL , for L running through C ∗ -metrics L on A ⊕ B whose quotients on A and B agree with L A and L B on A sa and B sa respectively. Here the metric d qL is defined as d qL ( ϕ, ψ ) = sup L ( a,b ) ≤ k ϕ ( a, b ) − ψ ( a, b ) k . Then prox s ( A, B ) is defined as sup q prox q ( A, B ).Note that the definition of dist nu extends to compact C ∗ -metricspaces ( A, L A ) and ( B, L B ) directly. Theorem A.1.
For any compact C ∗ -metric spaces ( A, L A ) and ( B, L B ) ,one has dist nu ( A, B ) ≥ prox s ( A, B ) . Proof.
The proof is similar to those of [24, Proposition 4.7] and [19,Theorem 3.7]. Let A be a unital C ∗ -algebra containing ¯ A and ¯ B . Set c = dist A H ( E ( A ) , E ( B )). Let ε >
0. Define a seminorm L on A ⊕ B by L ( a, b ) = max( L A ( a ) , L B ( b ) , k a − b k c + ε ) . It was pointed in the proof of [24, Proposition 4.7] that L extendedto A ⊕ B = ¯ A ⊕ ¯ B as in the condition (2) of the definition of C ∗ -metrics above is a Lip-norm, and that the quotients of L on A and B agree with L A and L B on A sa and B sa respectively. It is readilychecked that L satisfies the conditions (1) and (3) in the definitionof C ∗ -metrics. Thus L is a C ∗ -metric on A ⊕ B . For any q ∈ N and ϕ ∈ UCP q ( A ), by Arveson’s extension theorem [8, Theorem 1.6.1]extend ϕ to a φ in UCP q ( A ). Set ψ to be the restriction of φ on ¯ B .For any ( a, b ) ∈ E ( A ⊕ B ) one has k ϕ ( a, b ) − ψ ( a, b ) k = k ϕ ( a ) − ψ ( b ) k = k φ ( a − b ) k ≤ k a − b k ≤ c + ε. ETRIC ASPECTS OF NONCOMMUTATIVE HOMOGENEOUS SPACES 29
Thus d qL ( ϕ, ψ ) ≤ c + ε . Similarly, for any ψ ′ ∈ UCP q ( B ), we can findsome ϕ ′ ∈ UCP q ( A ) with d qL ( ϕ ′ , ψ ′ ) ≤ c + ε . Therefore prox q ( A, B ) ≤ c + ε . It follows that prox q ( A, B ) ≤ dist nu ( A, B ), and hence prox s ( A, B ) ≤ dist nu ( A, B ) as desired. (cid:3)
It was pointed out in Section 5 of [19] that one has continuity ofquantum tori and θ -deformation, convergence of matrix algebras tointegral coadjoint orbits of compact connected semisimple Lie groups,and approximation of quantum tori by finite quantum tori with respectto dist nu . It follows from Theorem A.1 that we also have such conti-nuity, convergence and approximation with respect to prox s and prox.In particular, this yields a new proof for [35, Theorem 14.1]. References [1] B. Abadie. Generalized fixed-point algebras of certain actions on crossed prod-ucts.
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