Leibniz seminorms for "Matrix algebras converge to the sphere"

LEIBNIZ SEMINORMS F OR “MA TRIX ALGEBRAS CONVERG E TO THE SPHERE” MAR C A. RIEFFEL Abstract. In an ear lier pap er o f mine r elating vector bundles and Gromov–Hausdorﬀ distance for ordinary compac t metric spaces, it was crucial that the Lipschitz seminor ms from the metrics satisfy a str ong Leibniz pr op erty . In the present paper , for the now non- commutativ e situation of matrix algebras con verging to the sphere (or to other spaces) for quantum Gro mov–Hausdorﬀ distance, we show ho w to construct suitable seminorms that also satisfy the strong Leibniz prop erty . This is in preparation for ma k ing pr e cise certain statements in the literature of high-energy physics con- cerning “vector bundles” ov er ma tr ix algebra s that “co rresp ond” to monop ole bundles over the spher e . W e show that a fairly gen- eral sour ce of se mino rms that s a tisfy the stro ng Leibniz pr op erty consists of deriv ations in to nor med bimo dules . F o r matrix algebras our main technical to ols a r e coherent states and Berezin symbols . Intr oduction In a previous pap er [29] I sho w ed how to giv e a precise meaning to statemen ts in the literature of high- energy phys ics and string theory of the kind “Matrix algebras conv erge to the sp here”. (See [29] for n umerous references to the relev an t phy sics literature.) I did this by in tro ducing the concept of “compact quantum metric spaces”, in whic h the metric data is give n by a seminorm on the non- comm utative “alge- bra of functions”. This seminorm plays the role o f the usual Lipsc hitz seminorm on the algebra of con tinuous functions on an or dina r y com- pact metric space. Ho wev er, I was somewhat puzzled by t he fact that I needed virtually no a lgebraic conditions on the seminorm, only an imp ortant analytic condition. But when I later b egan trying to giv e precise meaning to further statemen ts in the ph ysics literat ur e of the 1991 Mathematics Subje ct Classiﬁc ation. Primar y 46L87 ; Secondary 53 C23, 58B34 , 81R15, 81R30 . Key wor ds and phr ases. quantum metric space, Gromov–Hausdor ﬀ distance, Lipschitz, Leibniz seminor m, coadjoint orbits, coherent states, Bere z in sym b ols . The res earch r e po rted here was supp orted in pa r t by National Science F o unda- tion grant DMS-050 0501. 1 2 MARC A. RIEFFEL kind “ here are the ve ctor bundles ov er the matrix algebras that cor - resp ond to the monop o le bundles ov er the sphere” (see [31] f or many references), I found that for o rdinary metric spaces a strong f o rm of t he Leibniz inequality for the seminorm pla y ed a crucial role [31]. (See, for example, the pro of of prop osition 2 . 3 of [31].) How ev er, on returning to the non- comm utative case o f matrix algebras conv erging to the sphere (or to other spaces), f or some time I did not see how to construct useful seminorms that brough t the matrix algebras and sphere close together while also having the strong Leibniz prop ert y . The main purpo se of t his pap er is to show ho w to construct suc h seminorms. As in the earlier pap er [2 9 ], the setting is that of coadjoin t orbits of compact semisimple Lie groups, of whic h the 2-sphere is the simplest example. The main tec hnical to ols con t inue to b e coherent states and Berezin sym b ols. In the ﬁrst f our sections of this pap er w e sho w that a fairly general setting for obtaining seminorms that p ossess the strong Leibniz prop- ert y that w e need consists of deriv ations in to normed bimo dules, and w e examine v arious a sp ects of this topic. The strong Leibniz prop erty for a seminorm L on a normed unital algebra A consists o f the usual Leibniz inequalit y together with the inequalit y L ( a − 1 ) ≤ k a − 1 k 2 L ( a ) whenev er a is in ve rtible in A . I ha v e not seen this latter inequalit y discusse d in the literature. In Section 4 we put together the v arious conditions that w e hav e found to b e imp orta n t, a nd there-b y give a ten tative deﬁnition for a “ compact C ∗ -metric space”. In Section 5 we examine the use of seminorms with the strong Leib- niz prop erty in connection with quan tum G romo v–Hausdorﬀ distance. (I exp ect that many of the ideas and tec hniques dev elop ed in this pap er will apply to man y o ther classes o f examples b ey ond “Matrix a lgebras con v erg e to the sphere”.) In Section 6 w e extend to the case of strongly Leibniz seminorms t he construction tech nique in t r o duced in [28 ] that w e called “bridges”. Sections 7 and 8 con tain those pieces o f our dev el- opmen t that can b e carried out for certain homogeneous spaces o f any compact gro up (including ﬁnite ones). Section 9 g iv es the statemen t of our main theorem fo r coadjoin t orbits, while Sections 10 through 13 con tain the detailed technic a l dev elopmen t needed to pro ve our main theorem. Finally , in Section 14 w e relate our results to other v ariants of quan tum Gromov –Hausdorﬀ distance that ha v e b een dev elop ed b y Da vid Kerr, Hanfeng Li, and W ei W u [13, 14, 17, 18, 38, 39, 40]. W e can describ e our ba sic setup a nd our main t heorem somewhat more sp eciﬁcally a s follows, where deﬁnitions for v arious terms are giv en in later sections. Let G b e a compact semis imple Lie group, LEIBNIZ SEMIN ORMS 3 let ( U, H ) b e an irreducible unitary represen tatio n of G , and let P b e the rank-one pro jection along a highest w eight v ector for ( U, H ). Let α b e the action of G o n L ( H ) b y conjuga tion b y U , and let H b e the α -stabilit y g roup of P . Let A = C ( G/H ). L et ω b e the highest w eigh t for U , and for eac h n ∈ Z > 0 let ( U n , H n ) b e the irreducible represen tation of G of highest w eigh t nω . Let α also denote the action of G on B n = L ( H n ) b y conjugation by U n . Cho ose o n G a con tinuous length-function ℓ . Then ℓ and the trans- lation action of G o n A , as w ell as the actio ns α of G on eac h B n , determine seminorms L A on A a nd L B n on B n that mak e ( A, L A ) and eac h ( B n , L B n ) into compact C ∗ -metric spaces. Main Theorem (sk etc h y statemen t of Theorem 9.1). F or any ε > 0 ther e exists an N such that for any n ≥ N we c an explicitly c onstruct a str ongly L eibniz se minorm, L n , on A ⊕ B n making A ⊕ B n into a c om p act C ∗ -metric sp ac e, such that the quotients of L n on A an d B n ar e L A and L B n , and for which the quantum Gr om ov-Hausdorﬀ distanc e b etwe en A and B n is no gr e ater than ε . I plan to a pply the results of this pap er in a future pap er to discuss v ector bundles ov er non-commu tativ e spaces (e.g., mono p ole bundles), along the lines used for ordinary spaces in [31]. I dev elop ed part of the mat erial presen ted here during a ten-w eek visit at the Isaac Newton Institute in Cam bridge, Engla nd, in the F all of 2006. I a m v ery appreciativ e of the stimu lating a nd enjo yable con- ditions prov ided b y the Isaac Newton Institute. I am grateful to Hanfeng Li for some imp ortan t commen ts on the ﬁrst v ersion of this pap er, whic h led to some substan tia l impro ve ments giv en in the presen t v ersion. 1. Stron gl y Leibniz seminorms F rom m y in v estigation of the relation b et w een v ector bundles on com- pact metric spaces that are close tog ether, b oth for ordinary spaces [31] and for non- comm utative spaces (a con tinuing in ves tig ation), I hav e found that the follo wing prop erties are v ery imp ortant when consid- ering the seminorms that play the role of the Lipsc hit z seminorms of ordinary metric spaces. Unless the con trary is stated, w e allow our seminorms to t a k e the v alue + ∞ , but w e require that they ta k e v alue 0 at 0. W e use the usual conv entions for calculating with + ∞ . The follo wing deﬁnition is close to deﬁnition 2.1 of [31]. Deﬁnition 1.1. Let A b e a normed unital algebra o ver R or C , and let L b e a seminorm on A . W e sa y that: 4 MARC A. RIEFFEL 1) L is L ei b niz if it satisﬁes the inequalit y L ( ab ) ≤ L ( a ) k b k + k a k L ( b ) for all a, b ∈ A . 2) L is s tr ongly L eib n iz if it is Leibniz, and L (1) = 0, and if for an y a ∈ A that has an in v erse in A , w e ha ve L ( a − 1 ) ≤ k a − 1 k 2 L ( a ) . 3) L is ﬁnite if L ( a ) < ∞ for all a ∈ A . 4) L is semiﬁnite if { a : L ( a ) < ∞} is norm-dense in A . 5) L is c ontinuous if it is norm-con tinuous. 6) L is low e r semic ontinuous if for one r ∈ R > 0 , hence for all r > 0, the set { a ∈ A : L ( a ) ≤ r } is norm-closed in A . If, furthermore, A is a ∗ - no rmed algebra (i.e., has an isometric in v olu- tion), then w e deﬁne L ∗ b y L ∗ ( a ) = L ( a ∗ ) for a ∈ A . W e then say that L is a ∗ -semin o rm if L = L ∗ . The pro of of the following prop osition is straigh tf orw ard. Prop osition 1.2. L e t A b e a unital norme d algeb r a. i. L et L b e a semino rm on A and le t r ∈ R + . If L satisﬁes one o f the pr o p erties 1–6 ab ove then r L sa tisﬁ e s that same pr op erty. ii. L et L 1 and L 2 b e two semino rm s on A . I f they ar e b oth L ei b n iz, or str on gly L eibniz, o r ﬁnite, or c ontinuous, or lower sem ic on - tinuous, then so is L 1 + L 2 . iii. L et { L α } b e a family of sem inorms on A , p ossibly i n ﬁnite, and let L b e the supr emum of this family. (I .e., L = W α L α , deﬁne d by L ( a ) = sup α { L α ( a ) } . F or two semin orms, L and L ′ , we wil l denote their maxim um by L ∨ L ′ .) Then L is a seminorm on A , and if e ach L α is L eibniz, or s tr ongly L eibniz, or low e r semic ontinuous, then so is L . iv. I f A is a ∗ - norme d algebr a and if L satisﬁes one of the pr op erties 1–6 ab ov e, then L ∗ satisﬁes that same pr op erty. I hav e seen no discussion of the strong Leibniz prop ert y in the liter- ature. I do not kno w of an example of a ﬁnite Leibniz seminorm whic h do es not satisfy the inequalit y fo r L ( a − 1 ) in the deﬁnition of “strongly Leibniz”. But if w e allow the v alue + ∞ then examples can b e con- structed in the following wa y . Let A b e a unita l normed algebra and let B b e a unital subalgebra of A . L et L 0 b e a ﬁnite Leibniz seminorm on B . D eﬁne a Leibniz seminorm, L , on A b y L ( a ) = L 0 ( a ) if a ∈ B LEIBNIZ SEMIN ORMS 5 and L ( a ) = + ∞ otherwise. If B contains an elemen t that is in v ertible in A but not in B then L is not strongly Leibniz. F or example, let A = C ( [0 , 1]) and let B b e its subalgebra of p olynomial f unctions, with L 0 ( f ) = k f ′ k . (This example is not lo we r semicon tin uo us.) Let A f = { a : L ( a ) < ∞} . It is clear that if L is Leibniz then A f is a subalgebra o f A . If L is in fact strongly Leibniz and a ∈ A f , then clearly a is in ve r t ible in A f if and o nly if it is inv ertible in A . It follow s that for an y a ∈ A f the sp ectrum of a in A f will b e the same a s its sp ectrum in A . In stupid examples we ma y ha ve 1 / ∈ A f , but with that understo o d, we see that: Prop osition 1.3. If L is str ongly L eibniz then A f is a sp e ctr al ly stable sub alge b r a of A . The imp ortance of this pro p osition will b e seen in Section 3. W e also remark tha t if A has an inv olution and if L is a Leibniz seminorm that is also a ∗ -seminorm, then A f is a ∗ -subalgebra of A . Simple arguments prov e the follo wing tw o prop ositions. Prop osition 1.4. L et A b e a no rm e d unital algebr a, an d let L b e a seminorm o n A . L et B b e a unital sub algebr a o f A , e quipp e d with the norm fr om A . If L is L eibniz, or str on gly L e i b n iz, or ﬁni te, or c ontinuous, or lower semic ontinuous, then so is the r estriction of L to B as a semino rm on B . (But if L is semiﬁnite, its r estriction to B ne e d not b e semiﬁnite.) Prop osition 1.5. L et A b e a ∗ -n o rme d unital algebr a and let L b e a seminorm on A . L et ˜ L = L ∨ ( L ∗ ) . T h en ˜ L is a ∗ -seminorm. If L is L eibn iz, or str ongly L eibn i z , or ﬁnite, or c ontinuous, or lower semic ontinuous, then s o is ˜ L . (B ut if L is semiﬁni te, ˜ L ne e d not b e semiﬁnite.) So in this w ay w e can usually arrange to w ork with ∗ -seminorms when dealing with ∗ -algebras. Here is ano ther wa y to combine seminorms: Prop osition 1.6. L et L 1 , . . . , L n b e seminorms on a norme d unital algebr a A , and let k · k 0 b e a norm on R n with the pr op erty that if ( r j ) , ( s j ) ∈ R n , and if 0 ≤ r j ≤ s j for al l j , 1 ≤ j ≤ n , then k ( r j ) k 0 ≤ k ( s j ) k 0 . Deﬁ ne a sem inorm, N , o n A by N ( a ) = k ( L j ( a )) k 0 , with the evid ent me aning if L j ( a ) = ∞ for some j . If e ach L j satisﬁes a p articular one of pr op erties 1 , 2 , 3 , 5 , 6 of D eﬁnition 1.1 then N satisﬁes that pr op erty to o. Pr o of. If eac h L j is Leibniz, then N ( ab ) = k ( L j ( ab )) k 0 ≤ k ( L j ( a ) k b k + k a k L j ( b )) k 0 6 MARC A. RIEFFEL ≤ k ( L j ( a )) k 0 k b k + k a kk ( L j ( b )) k 0 = N ( a ) k b k + k a k N ( b ) , and if eac h L j is strongly Leibniz, then also N ( a − 1 ) = k ( L j ( a − 1 )) k 0 ≤ k ( k a − 1 k 2 L j ( a )) k 0 = k a − 1 k 2 N ( a ) . It is clear that if each L j is ﬁnite, o r con tinuous, t hen so is N . Supp ose instead that eac h L j is lo we r semicon tin uous. Let ( a m ) b e a sequenc e in A whic h conv erges in norm to a ∈ A , and supp ose that there is a constant, K , suc h that N ( a m ) ≤ K for eac h m . F or eac h m let p m = ( L j ( a m )) ∈ R n , so that k p m k 0 ≤ K . Since the K - ba ll of R n for k · k 0 is compact, w e can pass to a conv ergent subsequ ence, so we can assume tha t t he sequence { p m } con v erges to a vec to r, p , in R n suc h that k p k 0 ≤ K and whose en tries are non-negative . Let ε > 0 b e giv en. Then there is an integer m ε suc h that if m ≥ m ε then L j ( a m ) ≤ p j + ε for eac h j . Since eac h L j is low er semicon t inuous, it follo ws that L j ( a ) ≤ p j + ε for eac h j . Then N ( a ) = k ( L j ( a )) k 0 ≤ k ( p j + ε ) k 0 ≤ k p k 0 + ε k (1 , . . . , 1 ) k 0 . Th us N ( a ) ≤ K since k p k 0 ≤ K and ε is ar bitrary .  2. General sources of strongl y Leibniz seminorms W e will no w examine general metho ds for cons tructing strongly Leib- niz sem inorms. W e recall ﬁrst [11] that a ﬁrst-or der diﬀer ential c a l c ulus o ve r a unital algebra A is a pair (Ω , d ) consisting of a bimo dule Ω ov er A a nd a deriv at io n d from A into Ω, that is, a linear map from A in t o Ω suc h that d ( ab ) = ( da ) b + a ( db ) for a ll a, b ∈ Ω. (W e will alw ays assume that our bimo dules are suc h that 1 A acts as the iden tity op erator on b oth left and right.) It is common to assume that Ω is generated as a bimo dule b y the range of d , but w e will not need to imp ose this requiremen t, thoug h it can alw ays b e arrang ed b y replacing Ω by its sub-bimo dule generated b y the range of d . Supp ose now that A is a normed unital algebra (with k 1 A k = 1), and that (Ω , d ) is a ﬁrst- order diﬀerential calculus for A . Assume fur- ther that Ω is equipp ed with a norm that mak es it into a normed A -bimo dule, that is, k aω b k Ω ≤ k a kk ω k Ω k b k for all a, b ∈ A and ω ∈ Ω. W e will then sa y that (Ω , d , k · k Ω ) is a norme d ﬁrst-or der diﬀer ential c alculus . W e do not require that d b e con tinuous f o r the norms on A a nd Ω. W e deﬁne L on A b y L ( a ) = k da k Ω . LEIBNIZ SEMIN ORMS 7 Notice that L is ﬁnite, and that L is con tinuous if d is. Prop osition 2.1. L et L on A b e deﬁne d as ab ove for a norme d ﬁrst- or der diﬀe r ential c alculus. Then L is str ongly L eibniz. Pr o of. That L is Leibniz follows immediately from the deﬁnitions of a deriv ation and of a normed bimo dule. T o see that L is strongly Leibniz, notice ﬁrst that from the deﬁnition of a deriv atio n w e obtain d (1 A ) = 0, so that L (1 A ) = 0. Supp o se no w that a is an in v ertible elemen t of A . Then 0 = d (1 A ) = d ( aa − 1 ) = ( da ) a − 1 + a ( d ( a − 1 )) . Th us d ( a − 1 ) = − a − 1 ( da ) a − 1 . On taking the nor m w e see that L ( a − 1 ) ≤ k a − 1 k 2 L ( a ).  W e remark that no eﬀectiv e c har acterization seems to b e known as to whic h Leibniz seminorms come from normed ﬁrst-order diﬀeren tial calculi (or of inner ones) as ab ov e. They fall within the scop e of the “ﬂat” diﬀeren tial seminorms deﬁned in deﬁnition 4.3 of [4 ], and for whic h equiv alen t conditions are giv en in theorem 4.4 of [4]. Necessary conditions for a diﬀeren tial seminorm to b e ﬂat a re given immediately after deﬁnition 4.3 and in prop osition 4.7 of [4 ]. F or L eibniz seminorms w e see ab ov e that a further necess a ry conditio n is that of b eing strongly Leibniz. Let us now give some examples. Example 2.2. Let ( X, ρ ) b e a compact metric space. F o r giv en x 0 , x 1 ∈ X with x 0 6 = x 1 let Ω x 0 ,x 1 b e R or C according to whether A = C ( X ) is o ver R or C , and deﬁne actions of A o n Ω x 0 ,x 1 b y f · ω = f ( x 0 ) ω , w · f = ω f ( x 1 ) . Deﬁne d b y d f = ( f ( x 1 ) − f ( x 0 )) /ρ ( x 1 , x 0 ) . It is easily ch ec k ed that (Ω x 0 ,x 1 , d ) is a ﬁrst-order diﬀeren tial calculus o ve r A . Giv e A = C ( X ) its suprem um norm, k · k ∞ , and giv e Ω x 0 ,x 1 the usual norm on R or C . Then Ω x 0 ,x 1 is a normed A -bimo dule. Cle arly d is con t inuous. W e set L x 0 ,x 1 ( f ) = k d f k = | f ( x 1 ) − f ( x 0 ) | /ρ ( x 1 , x 0 ) . Then from Propo sition 2.1 it follo ws that L x 0 ,x 1 is strongly Leibniz (and con tinuous ). No w let L b e the suprem um o f the L x 0 ,x 1 o ve r all pairs ( x 0 , x 1 ) with x 1 6 = x 0 . W e obtain in this w ay the usual L ipschitz seminorm, L ρ , on C ( X ). F rom Prop osition 1.2 it fo llows that L ρ is strongly Leibniz and 8 MARC A. RIEFFEL lo we r semicon tinuous. O f course L ρ is not con tinuous in general. But L ρ is semiﬁnite, since it is ﬁnite on the functions f x 0 ( x ) = ρ ( x, x 0 ), and these already generate a dense subalgebra, as seen b y means o f the Stone–W eierstrass theorem. This example can b e recast in a quite f a miliar form as follo ws. Let Z = ( X × X ) \ ∆ where ∆ is the diago nal of X × X . Thus Z is a lo cally compact space. Let Ω = C b ( Z ), the linear space of b ounded con tinuous functions on Z with the suprem um norm. Then Ω is a normed C ( X )- bimo dule for the actions ( f ω ) ( x 0 , x 1 ) = f ( x 0 ) ω ( x 0 , x 1 ) , ( ω f )( x 0 , x 1 ) = ω ( x 0 , x 1 ) f ( x 1 ) . Let A denote the subalgebra of C ( X ) consisting of the Lipsc hitz func- tions, and deﬁne a deriv ation d from A to C ( Z ) b y ( d f )( x 0 , x 1 ) = ( f ( x 1 ) − f ( x 0 )) /ρ ( x 1 , x 0 ) . Then the usual Lipsc hitz seminorm is giv en by L ρ ( f ) = k d f k ∞ . Al- ternativ ely , let Ω = C ( Z ), the space o f all con tinuous , p ossibly un- b ounded, functions o n Z , as a C ( X )-bimo dule in the ab ov e w ay . Then d can b e deﬁned on a ll of C ( X ) b y the a b ov e formula. W e can now consider the suprem um norm o n C ( Z ), taking v a lue + ∞ on un b o unded functions (so a bit b ey o nd our deﬁnitions a b o v e), and again s et L ρ ( f ) = k d f k ∞ . Example 2.3. Let us now consider some examples in whic h the normed unital algebra A may b e non- commutativ e. If Ω is an A -bimo dule, one alw ays has the corresp o nding inner deriv ations. That is, if ω ∈ Ω w e can set d ω ( a ) = ω a − aω . If Ω is a no rmed A - bimo dule then d ω is con tinuous , with k d ω k ≤ 2 k ω k . The corresp onding seminorm, L ω , deﬁned by L ω ( a ) = k d ω ( a ) k , is then a con tinuous strongly Leibniz seminorm. Supp ose no w that B is a unital normed algebra and that π is a unital homomorphism from A in to B . Then w e can view B a s a bimo dule o ve r A in the eviden t wa y , and obtain inner deriv ations and corresp onding strongly Leibniz seminorms, whic h are con tin uous if π is. Example 2.4. Supp ose no w tha t π is a non-degenerate represen tation of A as op erato rs on a normed space X , so that π can b e view ed as a unital homomorphism from A in to B ( X ), the algebra of b o unded op erators on X . Then B ( X ) can b e view ed in the eviden t wa y as a bimo dule ov er A , and any elemen t, D , of B ( X ) determines an inner deriv ation, and cor r esp o nding seminorm L ( a ) = k D π ( a ) − π ( a ) D k = k [ D , π ( a )] k . LEIBNIZ SEMIN ORMS 9 More g enerally , if one has t wo represen ta tions, π 1 and π 2 of A on X , then one can view B ( X ) a s an A -bimo dule via a · T · b = π 1 ( a ) T π 2 ( a ) , and again an y elemen t, D , of B ( X ) will determine an inner deriv atio n. (The twisted commutators in equation 2 . 4 and lemma 2 . 2 of [8] ﬁt in to this view, except t ha t there D is usually an unbounded op erator.) Alternately one can assem ble π 1 and π 2 in to one represen ta t ion on X ⊕ X , and use the o p erator  0 D D 0  on X ⊕ X . As an imp ortan t particular case, fo r X w e can take A itself and let π b e the left-r egula r represen tation of A on itself. As elemen t of B ( X ) w e can tak e an isometric algebra a utomorphism, α , of A . Then ( α ◦ π ( a ) − π ( a ) ◦ α )( b ) = α ( ab ) − aα ( b ) = ( α ( a ) − a ) α ( b ) . F rom this we see tha t k α ◦ π ( a ) − π ( a ) ◦ α k = k α ( a ) − a k , so that if w e set L ( a ) = k α ( a ) − a k then L will b e a contin uous strongly Leibniz seminorm. W e can view this in another w ay . View A as a bimo dule ov er A b y a · b · c = abα ( c ) , and set d ( a ) = α ( a ) − a . It is easily che c k ed that d is a (con tin uous) deriv ation, and so fro m Prop osition 2.1 w e see aga in that L is stro ng ly Leibniz. (This do es not require that α b e isometric.) Example 2.5. Now let G b e a group, and let α be an action of G on A , that is, a homomorphism from G in to Aut( A ). Let ℓ b e a length- function on G . F or each x ∈ G with x 6 = e G the map a 7→ k α x ( a ) − a k /ℓ ( x ) is a contin uous strong ly Leibniz seminorm. Let L b e the suprem um ov er G of all of these seminorms, so that L ( a ) = sup {k α x ( a ) − a k /ℓ ( x ) : x 6 = e G } . By Prop osition 1.2 we see that L is a low er-semicon tinuous strong ly- Leibniz seminorm. Of course L may not b e semiﬁnite. But if G is a lo cally compact group, if A is complete, so a Banach algebra, if α is a strongly contin uous action by isometric automor phisms of A , and if ℓ is a con tinuous length-function, then the discussion b efore theorem 2 . 2 of [27] sho ws that L is semiﬁnite. The discussion there is stated just for C ∗ -algebras, but it applies without c hange to Ba na c h algebras. 10 MARC A. RIEFFEL Example 2.6. Supp o se now that G is a connected Lie gro up, and that α is a strongly con tinuous action of G on A b y isometric auto- morphisms. Let g denote the Lie algebra of G , a nd let A ∞ denote the dense subalgebra of smo o th elemen ts of A for the action α . W e let α also denote the corresp onding inﬁnitesimal action of g o n A ∞ , deﬁned b y α X ( a ) = d dt     t =0 α exp( tX ) ( a ) for X ∈ g and a ∈ A ∞ . The arg ument in the pro o f of lemma 3 . 1 of [27] w orks here, and show s that k α X ( a ) k = sup {k α exp( tX ) ( a ) − a k / | t | : t 6 = 0 } . It fo llows from Prop osition 1.2 that the map a 7→ k α X ( a ) k is a ﬁ- nite low er-semicon tinuous strongly-Leibniz seminorm on A ∞ . Supp ose further that we are given a norm on g , and that w e set L ( a ) = sup {k α X ( a ) k : k X k ≤ 1 } . It f ollo ws again from Prop osition 1.2 that L is a low er-semicontin uous strongly-Leibniz seminorm on A ∞ , whic h is easily seen to b e ﬁnite, but whic h ma y well no t b e norm-contin uo us. Example 2.7. Supp o se no w that G is a Lie gro up and that ( U, H ) is a strongly con tin uous r epresen ta tion of G on a Hilb ert space H . As discusse d in section 3 of [27] we can deﬁne a n action, α , of G on B ( H ) b y α x ( T ) = U x T U ∗ x , and w e can let B b e the larg est subalgebra o f B ( H ) on whic h this action is strongly con tinuous. W e can then apply the discussion of the previous example to o btain a seminorm L on B ∞ . If A is a unital ∗ - subalgebra of B ∞ (whic h need not b e carried in to itself by α ), then according to Prop osition 1.4 the restriction of L to A is a low er-semicon tin uous strong ly-Leibniz ∗ - seminorm whic h is clearly ﬁnite. Example 2.8. Let us consider the ab ov e situation for the sp ecial case in whic h g = R . The n U is generated by a self-adjoint (often un- b ounded) op erato r , D , on H , that is, U t = e itD for all t ∈ R . Then it follo ws easily that for T ∈ B ∞ L ( T ) = k [ D , T ] k , and in particular that the comm utato r [ D , T ] is a b ounded op erator. All of this will then b e true for a n y T ∈ A ⊂ B ∞ . This applies in particular to the “Dir a c” op erators o n whic h Connes [6, 11] bases his approac h to metric non-commu tativ e diﬀerential geometry . LEIBNIZ SEMIN ORMS 11 3. Close d seminorms W e a da pt here some deﬁnitions from section 4 of [25]. Let A b e a normed unital a lg ebra, and let ¯ A denote its completion. Let L b e a seminorm on A ( v alue + ∞ allo we d) and let L 1 = { a ∈ A : L ( a ) ≤ 1 } . Let ¯ L 1 b e the closure of L 1 in ¯ A , and let ¯ L denote t he corresp onding “Mink ow ski functional” on ¯ A , deﬁned by setting, for c ∈ ¯ A , ¯ L ( c ) = inf { r ∈ R + : c ∈ r ¯ L 1 } . The v alue + ∞ m ust b e allo wed. Then ¯ L is a seminorm on ¯ A , a nd the pro of of prop osition 4 . 4 of [25] tells us that if L is lo w er semicon tin uous, then ¯ L is a n extension of L . W e call ¯ L the closur e of L . W e see that the set { c ∈ ¯ A : ¯ L ( c ) ≤ 1 } is closed in ¯ A . W e sa y that the o riginal seminorm L o n A is close d if L 1 is closed in ¯ A , or, equiv alently , is complete for the norm on A . Clearly if L is closed, then it is low er semicon tin uous. If L is closed and is not deﬁned on all of ¯ A , then ¯ L is obtained simply b y giving it v alue + ∞ on all the elemen ts of ¯ A that are not in A . It is clear that if L is semiﬁnite then so is ¯ L . W e r ecall that a unital subalgebra B of a unital algebra A is said to b e sp ectrally stable in A if for any b ∈ B the sp ectrum of b as an elemen t o f B is the same a s its sp ectrum as a n elemen t of A , or equiv alently , that any b that is inv ertible in A is inv ertible in B . Prop osition 3.1. L et L b e a L eibniz semino rm on a norme d unital algebr a A . Th e n ¯ L is L eibniz . Se t ¯ A f = { c ∈ ¯ A : ¯ L ( c ) < ∞} . If L (1) < ∞ , then ¯ A f is a unital s p e ctr al ly-stable sub a l g ebr a of the norm clos ur e of ¯ A f in ¯ A . If A is deﬁne d over C , then ¯ A f is stable under the holomorphic-func tion c alculus of its closur e. Pr o of. Let c, d ∈ ¯ A . If ¯ L ( c ) = ∞ or ¯ L ( d ) = ∞ there is nothing to show for the Leibniz condition. Otherwise , we can ﬁnd seque nces { a n } and { b n } in A suc h that { a n } con v erge to c while { L ( a n ) } conv erges to ¯ L ( c ) and L ( a n ) ≤ ¯ L ( c ) for all n , and similarly for { b n } and d . Then a n b n con v erg es to c d and L ( a n b n ) ≤ L ( a n ) k b n k + k a n k L ( b n ) ≤ ¯ L ( c ) k b n k + k a n k ¯ L ( d ) , and the rig h t-hand side conv erges to ¯ L ( c ) k d k + k c k ¯ L ( d ). Th us ¯ L is Leibniz. If L (1) < ∞ so that ¯ A f is a unital subalgebra of ¯ A , it follo ws from prop osition 3.12 of [4 ] (or prop osition 1.7 and theorem 1.17 of [34 ] or 12 MARC A. RIEFFEL lemma 1.6 .1 of [9]) that ¯ A f is sp ectrally stable in its closure in ¯ A , a nd in f a ct is stable under the holomorphic-function calculus there. W e sk etc h the pro of in our simpler setting. Deﬁne a new norm, M , on ¯ A f b y M ( c ) = k c k + ¯ L ( c ) . Then, as men tioned aft er deﬁnition 4 . 5 of [25], ¯ A f will b e complete for the norm M b ecause ¯ L is closed. (See the pro of of prop osition 1 . 6 . 2 of [37].) Because ¯ L is Leibniz, M is easily seen to b e an algebra norm, so that ¯ A f b ecomes a Banac h a lgebra. Let c ∈ ¯ A f . F rom the Leibniz rule w e ﬁnd that ¯ L ( c n ) ≤ n k c k n − 1 ¯ L ( c ), so that M ( c n ) ≤ k c k n + n k c k n − 1 ¯ L ( c ) . F rom this it follows that if k c k < 1 then the series P ∞ n =0 c n con v erg es for M to an elemen t of ¯ A f . Thus 1 − c is inv ertible in ¯ A f . It follows that if instead k 1 − c k < 1 then c is in vertible in ¯ A f . F rom this it is then easily seen (e.g. lemma 3.38 of [11 ]) that if c ∈ ¯ A f and if c is in v ertible in the norm-closure of ¯ A f in ¯ A , then c is in v ertible in ¯ A f . Consequen tly ¯ A f is sp ectrally stable in its closure in ¯ A . Assume no w that A is deﬁned ov er C . F or the deﬁnition and prop er- ties of the holomorphic-function (or “sym b olic”) calculus see [12 , 32]. It is w ell-kno wn tha t a dense subalgebra that is sp ectrally stable and is a Banach algebra fo r a norm stronger tha t the norm of the bigger algebra, is stable under the holomorphic-function calculus. (See the commen ts after deﬁnition 3.25 of [1 1].) W e brieﬂy recall the reason, for our contex t. F or notat ional simplicit y w e assume t hat ¯ A f is dense in ¯ A . Let c ∈ ¯ A f , and let f b e a C -v alued function deﬁned and holo- morphic on an op en neighborho o d O of the sp ectrum σ ¯ A ( c ). Let γ b e a union of a ﬁnite n um b er of curv es in O that surrounds σ ¯ A ( c ) in the usual wa y such that the Cauc h y in tegral formula using γ g iv es f on a neigh b or ho o d of σ ¯ A ( c ). Since ¯ A f is sp ectrally stable in ¯ A , the f unction z 7→ ( z − c ) − 1 , w ell-deﬁned on γ , has v alues in ¯ A f . Since ¯ A f is a Banach algebra for M , this function is con tin uous for M , and the in tegral f ( c ) = 1 2 π i Z γ f ( z )( z − c ) − 1 dz is well deﬁned in ¯ A f . Since the homomorphism from ¯ A f with no rm M to ¯ A with its o riginal norm is clearly con tinu ous, the image of f ( c ) in ¯ A will b e expressed by the same integral but no w interpreted in ¯ A . But f ( c ) ∈ ¯ A f . So the ab o v e in tegra l, but inte r preted in ¯ A , gives an elemen t o f ¯ A f as w as to b e show n.  LEIBNIZ SEMIN ORMS 13 F or the use of the holo morphic-function calculus when dealing with algebras ov er R see prop osition 2.4 of [31]. One r eason that the prop erty of b eing closed under the holomorphic- function calculus is imp or t an t is that it implies t ha t ¯ A f and its closure, sa y B , in ¯ A ha v e essen tially the same ﬁnitely-generated pro j ective mo d- ules (“ve cto r bundles”) in the sense that any suc h r igh t mo dule V for B is of the for m V = W ⊗ ¯ A f B f o r suc h a right mo dule W ov er ¯ A f , unique up to isomorphism. This is crucial to [31], and to our prop osed discussion of pro jectiv e mo dules and quantum Gromov–Hausdorﬀ dis- tance for non-comm utativ e C ∗ -algebras. The inclusion o f ¯ A f in to B also give s an isomorphism of their K - groups. (See app endix I I IC of [6] and theorem 3 . 4 4 of [11].) Prop osition 3.2. L e t L b e a str ongl y-L eibniz semi n orm on a norm e d algebr a A . Assume that A f is dense and sp e ctr a l ly stable in ¯ A . Then the cl o sur e, ¯ L , of L is str o n gly L eib niz. Pr o of. It is clear that ¯ L (1) = 0. F rom Prop osition 3.1 we kno w that ¯ L is Leibniz. Th us we only need to verify the condition on in vers es. Supp ose no w that c ∈ ¯ A and that c is in vertible in ¯ A . If ¯ L ( c ) = ∞ there is no t hing to sho w, so assume that c ∈ ¯ A f . Then there is a sequenc e { a n } in A that con verges to c while { L ( a n ) } conv erges to ¯ L ( c ) with L ( a n ) ≤ ¯ L ( c ) for all n (so a n ∈ A f ). Since c is inv ertible in ¯ A , and the set of in ve rtible elemen ts of a unital Banach algebra is op en, the elemen ts a n are ev en t ua lly inv ertible in ¯ A . Since A f is assumed to b e sp ectrally stable in ¯ A the elemen ts a n are ev en t ua lly inv ertible in A f . Th us w e can adjust the sequence { a n } so that eac h elemen t is inv ertible in A f . Then t he sequence { a − 1 n } con v erges to c − 1 , while f or eac h n L ( a − 1 n ) ≤ k a − 1 n k 2 L ( a n ) ≤ k a − 1 n k 2 ¯ L ( c ) . It follo ws easily that ¯ L ( c − 1 ) ≤ k c − 1 k 2 ¯ L ( c ). Th us ¯ L is strongly Leibniz.  4. C ∗ -metrics Up to this p oin t w e ha ve ignored the crucial analytic prop erty of the seminorms that deﬁne quan tum metric spaces, i.e., Lip-norms. W e recall this prop ert y here, fo r our sp ecial context of unital C ∗ -algebras. Let A b e a unital ∗ -algebra equipp ed with a C ∗ -norm ( but not assumed to b e complete). L et L b e a seminorm on A suc h that L (1) = 0. Deﬁne a metric, ρ L , on the state space, S ( A ), o f A b y ρ L ( µ, ν ) = sup {| µ ( a ) − ν ( a ) | : a = a ∗ and L ( a ) ≤ 1 } . 14 MARC A. RIEFFEL (Without f ur t her h yp otheses ρ L migh t ta k e t he v alue + ∞ .) W e will sa y that L is a Lip-norm if the top ology on S ( A ) from ρ L coincides with the we a k- ∗ top ology on S ( A ). In our deﬁnition of Lip-norms in deﬁnition 2 . 1 of [28] w e, in eﬀect, assumed tha t our seminorms L w ere deﬁned only on t he self-a djoin t part of A , but still deﬁned ρ L as ab ov e. The commen ts b efore deﬁnition 2 . 1 of [28] sho w that if L is a ∗ -seminorm then ρ L w ould not c hange if the condition “ a = a ∗ ” ab ov e w ere omitted. W e no w come to the deﬁnition that seems to b e dictated b y our in- v estigation of v ector bundles and G r omo v–Hausdorﬀ distance, b ot h for ordinary metric spaces [31 ] and for quan t um ones. It should b e view ed as ten tativ e, since future exp erience may require a dditio nal h yp o t heses. Deﬁnition 4.1. Let A b e a unital C ∗ -normed algebra and let L b e a seminorm on A (p o ssibly taking v alue + ∞ ) . W e will say that L is a C ∗ -metric on A if a) L is a lo w er-semicon tinuous strongly-Leibniz ∗ -seminorm, b) L (restricted to A sa ) is a Lip- norm, c) A f is sp ectrally stable in the completion, ¯ A , of A . By a c omp act C ∗ -metric sp ac e we mean a pair ( A, L ) consisting of a unital C ∗ -normed algebra A and a C ∗ -metric L on A . In using the w or d “ space” ab ov e, we should logically b e referring to ob jects in the dual to the category of unital C ∗ -algebras. But w e will not make this distinction explicit during our discussions in this pap er. W e need condition c) in Deﬁnition 4.1 so that w e can a pply Prop osi- tion 3.2 to conclude that the closure of a C ∗ -metric is strongly Leibniz and itself satisﬁes condition c). Hanfeng Li has p oin ted out to me that the subalgebra of p olynomials in the algebra of contin uous func- tions on the unit in t erv al with the usual Lipsc hitz seminorm sho ws that condition c) is indep endent of conditions a ) and b). A t this time it is not clear to me how b est to deﬁne C ∗ -metric spaces that are lo cally compact but not compact, though some substan tial indications can b e gleaned from t he results in [15]. Recall [4] that a ∗ -subalgebra B of a C ∗ -algebra A is said to be stable under the C 2 -function calculus for self-adjoint elemen ts if for any b ∈ B with b ∗ = b a nd a n y twice con tinuously diﬀeren tiable function f on R , the elemen t f ( b ) of A , deﬁned b y the con tinuous-function calculus on self-adjoin t elemen ts o f A , is in fact again in B . Prop osition 4.2. Every C ∗ -metric on a unital C ∗ -norme d algebr a is semiﬁnite. L et L b e a C ∗ -metric on a unital C ∗ -norme d alg e br a A , an d let ¯ L b e its closur e on the c ompletion ¯ A of A (so ¯ L is an ex tens ion of LEIBNIZ SEMIN ORMS 15 L ). Then ¯ L is a C ∗ -metric. L et ¯ A f b e deﬁne d as e arlier (so no w ¯ A f is dense in ¯ A ). Then ¯ A f is stable b oth under the h o lomorphic-function c alc ulus of ¯ A and the C 2 -c al c ulus on sel f - a djoint elements of ¯ A . Pr o of. Let L b e a C ∗ -metric on a unital C ∗ -normed algebra A , and let A f b e deﬁned as ab ov e. Supp ose that A f is not dense in A . Then it is easily seen that there is an a ∈ A with a ∗ = a that is not in the closure of A f . By the Hahn–Banach theorem there is a linear functional of norm 1 on the self-adjoint part of A that has v a lue 0 on all of the self- adjoin t part of A f . F rom lemma 2 . 1 of [2 5 ] it then follow s t ha t there are tw o distinct states o f A whic h agree on A f . Then t he distance b et wee n these tw o states f or the metric ρ L determined by L is 0, whic h con tradicts the requiremen t that the top o logy on S ( A ) determined b y ρ L coincides with the weak - ∗ top ology . The fact that ¯ L is a C ∗ -metric is seen as follows. By deﬁnition, ¯ L is closed so lo wer semi-con tin uo us. As remark ed ab o ve , ¯ L is strongly Leibniz b y condition c) and Prop osition 3.2. The closure o f a Lip-norm is again a Lip-norm, g iving the same metric on the state-space, as seen in prop osition 4.4 of [2 5]. That ¯ L satisﬁes condition c) f ollo ws from Prop osition 3.1. The fact that ¯ A f is stable under the holomor phic-f unction calculus of ¯ A f o llo ws immediately fro m the semiﬁniteness of ¯ L and Prop osition 3.1. The fact that ¯ A f is stable for the C 2 -function calculus on self- adjoin t elemen ts of ¯ A follows quic kly f rom prop osition 6.4 o f [4], whic h actually giv es a sligh tly stronger f act.  The condition that L b e a Lip-norm is o f ten a diﬃcult one to v erify for v ar ious sp eciﬁc examples. But most of the Lip-norms tha t ha v e b een constructed on C ∗ -algebras so far are in fact C ∗ -metrics. W e explain this now for sev eral of the classes of examples describ ed in sections 2 and 3 of [27]. Example 4.3. Let A be a unital C ∗ -algebra, let G b e a compact group, and let α b e an action of G o n A that is ergo dic in the sense that if an a ∈ A satisﬁes α x ( a ) = a for a ll x ∈ G then a ∈ C 1 A . Let ℓ b e a con tinuous length function on G , a nd deﬁne a seminorm L on A , a s in Example 2.5, by L ( a ) = sup {k α x ( a ) − a k /ℓ ( x ) : x / ∈ e G } . It is sho wn in [27] that L is a Lip-norm. But w e sa w in Example 2.5 that L is low er semicon tin uous and strongly Leibniz. Since L is deﬁned on all of A , the sp ectral stabilit y of ¯ A f in A fo llows from Prop osition 3.1. 16 MARC A. RIEFFEL The next sev eral examples in volv e “Dirac” o p erators in v arious set- tings. Example 4.4. This class of examples is the main class discussed in Connes’s ﬁrst pap er [5] on metric asp ects of non-commutativ e g eome- try . It is discussed brieﬂy a s example 3 . 6 of [27]. Let G b e a discrete group and let A = C ∗ r ( G ) b e its reduced group C ∗ -algebra acting on ℓ 2 ( G ). Let ℓ b e a length function on G . As Dira c op erator take the op erator D = M ℓ of p oin twise multiplication b y ℓ on ℓ 2 ( G ). The one- parameter unitary gro up generated b y D simply sends t to the op erator of p oint wise m ultiplication b y the function e itℓ . W e are then in the con- text of Examples 2.7 and 2.8. It is easily seen that the dense subalgebra C c ( G ) of functions of ﬁnite supp o rt is in the smo oth algebra B ∞ for the action of G on L ( ℓ 2 ( G )). As in Example 2.8 we thus obtain a low er- semicon tin uous strongly-Leibniz semiﬁnite ∗ -seminorm on A , whic h for an y f ∈ C c ( G ) is giv en b y L ( f ) = k [ D , f ] k . F rom Prop osition 3 .1 it follows that A f (for the closure of L ) is stable for the holomorphic-function calculus. But for stupid length functions L can fail to b e a Lip-norm, and it is not easy to see when it is a Lip- norm, and th us a C ∗ -metric. In [26], b y means of a long and in teresting argumen t, it is sho wn that L is a L ip- norm, and th us a C ∗ -metric, fo r G = Z d (and ev en for the t wisted group algebra C ∗ ( Z d , γ ) where γ is a bic haracter on Z d ) when ℓ is either a w ord- length function or the restriction to Z d of a no rm on R d . In [21] it is sho wn, by tec hniques en tirely diﬀeren t from those used for the case of Z d , that if G is a h yp erb o lic group and ℓ is a word-length function on G then L is a Lip-norm, and th us a C ∗ -metric. F or other classes of inﬁnite discrete groups, e.g., nilp oten t ones, it remains a m ystery as to whether L is a Lip-norm if ℓ is a w ord-length function. Some related examples can b e found in [1]. Example 4.5. Let α b e an a ctio n of the d -dimensional t orus T d , d ≥ 2, on a unital C ∗ -algebra A . In [23] it is shown that for an y sk ew- symmetric real d × d matrix θ one can deform the pro duct on A to get a new C ∗ -algebra, A θ . Connes and Landi [7] sho w that when M is a compact spin Riemannian manifold and α is a smo oth action of T d on M , so on A = C ( M ), lea ving the Riemannian metric inv aria nt, and lifting to the spin bundle, then there is a natural Dirac op erator for the (usually non-commu tativ e) defor med algebra A θ . As in Examples 2.8 and 4.3, this Dirac o p erator determines a ∗ - seminorm, L , on A θ whic h LEIBNIZ SEMIN ORMS 17 is low er semicon tin uous, strongly Leibniz, and semiﬁnite. Hanfeng Li [16] sho we d that L is a Lip ∗ -norm. Thus L is a C ∗ -metric. 5. Quotient seminorms and pro ximity W e now try to mo dify the deﬁnition of quantum Gromov – Hausdorﬀ distance so as to use the ab o ve deﬁnition of C ∗ -metrics. This in volv es quotien t seminorms, so we b egin by exploring them. There are at least three diﬃculties that confront us, na mely that the quotien t of a Leibniz seminorm may not b e Leibniz, that the quotien t of a strongly Leibniz seminorm, ev en if it is Leibniz, may not b e strongly Leibniz, and that reasonable ∗ - seminorms can agree on self-a djoin t elemen ts but still b e distinct. W e b egin by considering the ﬁrst diﬃculty . Let L b e a Leibniz seminorm o n a unital normed algebra C , and let π : C ։ A b e a unital homomorphism from C on to a unital normed algebra A . Let ˜ L A b e the quotient seminorm on A , deﬁned b y ˜ L A ( a ) = inf { L ( c ) : c ∈ C and π ( c ) = a } . It is kno wn [4] that ˜ L A need not b e Leibniz. (See also lemma 4 . 3 of [18] and the commen ts just b efore it.) But t he situation can b e partly rescued b y the follow ing deﬁnition. Deﬁnition 5.1. Let C , A , π and L b e as ab ov e, and assume tha t π is norm non-increasing. W e sa y that L is π -c omp atible if for ev ery a ∈ A and ev ery ε > 0 there is a c ∈ C suc h tha t π ( c ) = a and sim ultaneously L ( c ) ≤ ˜ L A ( a ) + ε and k c k ≤ k a k + ε. Prop osition 5.2. L et C , A , π and L b e a s ab o v e. If L is π -c omp atible then the norm on A c oinc i d es with the quotient no rm fr om C , and ˜ L A is L eibniz. Pr o of. The statemen t ab out the norms is easily veriﬁe d. Supp ose no w that a, b ∈ A and ε > 0 are giv en. Since L is π -compatible, we can ﬁnd c, d ∈ C suc h that π ( c ) = a a nd π ( d ) = b and the conditions of Deﬁnition 5.1 a r e satisﬁed. Then π ( cd ) = π ( ab ), and so ˜ L A ( ab ) ≤ L ( cd ) ≤ L ( c ) k d k + k c k L ( d ) ≤ ( ˜ L A ( a ) + ε )( k b k + ε ) + ( k a k + ε )( ˜ L A ( b ) + ε ) . Since ε is a r bit r a ry , w e see that ˜ L A is Leibniz.  Ho we ver, I do not kno w of a useful wa y to partly rescue the diﬃcult y that if L is strongly L eibniz and ˜ L A is Leibniz there seems to b e no reason that ˜ L A need b e strongly Leibniz (though I do not ha ve an example sho wing this diﬃculty ). 18 MARC A. RIEFFEL W e now consider the t hird diﬃcult y . It is quite instructiv e to ﬁrst consider ordinary metric spaces. F or this purp ose π -compatibility is useful. Prop osition 5.3. L e t ( Z , ρ ) b e a c omp act me tric sp ac e , and le t C = C ( Z ) b e its C ∗ -algebr a of c ontinuous c omplex-val ue d functions. L et X b e a close d subset of Z , let A = C ( X ) , and let π : C → A b e the usual r estriction homo m orphism. Then the L eibniz seminorm L ρ for ρ is π -c omp atible. Pr o of. Let f ∈ A . Let Q b e t he radial retraction of C on to its ba ll of radius k f k ∞ cen tered at 0. It is easily seen that the Lipsc hitz constant of Q is 1. Then for a ny h ∈ C with π ( h ) = f w e can set g = Q ◦ h and w e will hav e π ( g ) = f and L ( g ) ≤ L ( h ) while k g k = k f k . This quic kly giv es the desired result.  W e remark that the ab ov e arg ument do es not w ork for matrix-v alued functions, a s employ ed in [31], since the radial r etra ction no longer ha s Lipsc hitz constan t 1 [30]. While Prop o sition 5 .3 app ears fa vorable, the diﬃculty is that the quotien t of L ρ on A need not ag ree with t he Lipsc hitz seminorm from the metric ρ X on X coming from restricting ρ : Example 5.4. (See [3 7, 30 ].) Let ( X , ρ X ) b e the metric space con- taining exactly 3 p oints, at distance 2 from eac h other. W e can ask what the Gromo v–Hausdorﬀ distance is from ( X , ρ X ) to a metric space consisting of one p oint, say p . It is easily seen that the answ er is 1, with the metric ρ on Z = X ∪ { p } that extends ρ X giving p distance 1 to eac h p oint of X . No w let f be the function on X whic h sends the three p oin ts of X to the three diﬀerent cub e ro ots o f 1 in C . It is not diﬃcult to see that the extension of f to Z that has the smallest Lipsc hitz norm is the extension g that sends p to 0. But L ρ ( g ) is easily seen to b e substan tia lly larger than L ρ X ( f ). As remark ed in [31, 30], this is p ossible b ecause the metric on Z is somewhat h yp erb olic. On the other hand, for an y compact metric space ( Z , ρ ), any closed subset X of Z , a nd for a n y f ∈ C R ( X ), there is a g ∈ C R ( Z ) with g | X = f , k g k = k f k and L ρ ( g ) = L ρ X ( f ) [30]. This sho ws in pa r t icular that here L ρ X do es coincide with the quotien t seminorm from L ρ . It also means that for the situation of Example 5.4 w e ha ve tw o Leib- niz seminorms on C ( X ) whic h a gree on r eal-v alued functions but are nev ertheless distinct. (F o r a related phenomenon see [22].) F rom the commen ts at the end of the ﬁrst pa r a graph of Section 4 w e see that these t w o seminorms will giv e the same metrics on t he set of pro babilit y measures on X , and in particular the same metrics on X . LEIBNIZ SEMIN ORMS 19 W e no w turn our atten tion to Gro mo v–Hausdorﬀ distance. L et ( A, L A ) and ( B , L B ) b e C ∗ -metric spaces. The eviden t w ay to a dapt the deﬁnition of quan tum Gro mov–Hausdorﬀ distance giv en in deﬁni- tion 4 . 2 of [2 8] is to require t ha t the seminorms L considered on A ⊕ B b e C ∗ -metrics. Example 5.4 shows that we can not require the quotient of L on A to agree with L A , except on self-adjoint elemen ts (tho ug h for the main class of examples considered in later sections they will agree, so those examples are b etter-b eha v ed than Example 5 .4 ). Then w e do not know whether t he quotien t is Leibniz. W e could imp ose π - compatibilit y t o ensure this, but then w e still ma y not ha v e the strong Leibniz prop erty , so it is not clear that it is useful to imp ose this. P erhaps as our topic dev elops in the future it will b ecome clearer what a re the b est conditions to imp ose. An yw ay , guided b y the ab o v e observ ations, we set, parallel to notat io n 4 . 1 of [28]: Notation 5.5. Let ( A, L A ) and ( B , L B ) b e compact C ∗ -metric spaces. W e let M C ( L A , L B ) denote the collection of all C ∗ -metrics, L , on A ⊕ B suc h that the quotien t of L on A agrees with L A on self-adjoin t elemen t s of A , and similarly for the quotien t of L on B . W e w ant to mo dify the deﬁnition of quan t um Gromo v–Hausdorﬀ distance, dist q , giv en in deﬁnition 4 . 2 of [28] by requiring that the seminorms in volv ed there are in M C ( L A , L B ). But I am not able to sho w that the resulting notio n satisﬁes the triangle inequalit y . When one tries to imitate the pro of of the triangle inequalit y for dist q giv en in theorem 4 . 3 of [28], one of the main obstacles is in showing that the Lip-norm L AC of lemma 4 . 6, whic h is deﬁned as a quotient seminorm, is a C ∗ -metric. I would not b e surprised if the t r iangle inequalit y fails. So the term “distance” should not b e used. I will use instead the term “pro ximity”. Th us: Deﬁnition 5.6. Let ( A, L A ) and ( B , L B ) be compact C ∗ -metric spaces. W e deﬁne their proximit y b y pro x ( A, B ) = inf { dist ρ L H ( S ( A ) , S ( B )) : L ∈ M C ( L A , L B ) } . This deﬁnition mak es sense in the f ollo wing w ay . Both S ( A ) and S ( B ) ar e closed subsets of S ( A ⊕ B ). Muc h a s at the b eginning of Section 4, ρ L is a metric on S ( A ⊕ B ), and dist ρ L H is ordinary Hausdorﬀ distance with resp ect to ρ L . W e note that the h yp otheses in the deﬁ- nition of M C ( L A , L B ) a re suc h that prop osition 3 . 1 of [28] applies, so that for any L ∈ M C ( L A , L B ) the r estrictions of ρ L to S ( A ) and S ( B ) coincide with ρ L A and ρ L B . Put another wa y , when w e asso ciate to eac h L ∈ M C ( L A , L B ) its restriction to the self-adjoin t part of A ⊕ B w e obtain a map from M C ( L A , L B ) to M ( L s A , L s B ), where L s A denotes 20 MARC A. RIEFFEL the restriction of L A to the self-adjoint part of A , and similarly for L s B . This ma p need not b e either injectiv e or surjectiv e. It is clear t hat dist q ( A, B ) ≤ pro x ( A, B ) , since pro x( A, B ) is a n inﬁmum ov er a subset of the seminorms used to deﬁne dist q ( A, B ). Th us if w e hav e a sequence ( B n , L B n ) of C ∗ - metric spaces for whic h the sequenc e prox( A, B n ) con v erges to 0, then it follows that ( B n , L B n ) con verges to ( A, L A ) for quantum Gromov – Hausdorﬀ distance. F or this reason the absence o f the t r ia ngle in- equalit y will no t b e to o serious a problem. The a dv antage of pro x , as men tioned earlier, is tha t the use of seminorms L on A ⊕ B that are C ∗ -metrics p ermits one to try t o generalize to C ∗ -metric spaces the re- sults ab out vec tor bundles obtained in [31] fo r ordinary metric spaces. (W e plan to discuss this in a future pap er.) 6. Bimodule-bridges In the dev elopmen t of quan tum G romo v–Hausdorﬀ distance giv en in [28] and used in [29], a v ery con v enien t metho d for constructing suitable seminorms L on A ⊕ B inv olved suitable contin uous seminorms N on A ⊕ B that we called “ bridges”, with L then deﬁned as L ( a, b ) = L A ( a ) ∨ L B ( b ) ∨ N ( a, b ) . Within t he contex t of the presen t pap er it is na t ural to r equire t hat N satisfy a suitable Leibniz condition. There is an eviden t condition to consider, coming from viewing N as a seminorm on the algebra A ⊕ B . But it seems more appro pria te to require the stronger condition N (( a, b )( a ′ , b ′ )) ≤ N ( a, b ) k b ′ k + k a k N ( a ′ , b ′ ) . Examples sho w that this condition can b e in terpreted as indicating that N only pro vides metric dat a b etw een A and B , a nd not within A or within B . W e will ﬁnd it v ery useful to use bridges that come from normed bimo dules. Suc h bridges will satisfy the Leibniz condition stated abov e. Let A and B b e unital C ∗ -algebras, and let Ω b e an A - B -bimo dule. W e say that Ω is a normed bimo dule if it is equipp ed with a norm that satisﬁes, m uch as in Section 2, k aω b k ≤ k a kk ω kk b k for all a ∈ A , b ∈ B and ω ∈ Ω. W e assume tha t the identit y elemen ts of A and B b oth act as the identit y o p erator on Ω. LEIBNIZ SEMIN ORMS 21 Deﬁnition 6.1. Let ( A, L A ) and ( B , L B ) b e C ∗ -metric spaces. By a bimo dule bridge for ( A, L A ) and ( B , L B ) w e mean a normed A - B - bimo dule Ω together with a distinguished elemen t ω 0 6 = 0 suc h that when w e form the seminorm N on A ⊕ B deﬁned b y N ( a, b ) = k aω 0 − ω 0 b k , it has the prop ert y that for an y a ∈ A with a = a ∗ and any ε > 0 there is a b ∈ B with b ∗ = b such that L B ( b ) ∨ N ( a, b ) ≤ L A ( a ) + ε, and similarly for A and B in t erc hanged. Theorem 6.2. L et (Ω , ω 0 ) b e a bimo dule bridge fo r the C ∗ -metric sp ac es ( A, L A ) and ( B , L B ) , and let N b e deﬁne d as ab ove in terms of (Ω , ω 0 ) . Deﬁne L on A ⊕ B by L ( a, b ) = L A ( a ) ∨ L B ( b ) ∨ N ( a, b ) ∨ N ( a ∗ , b ∗ ) . Then L ∈ M C ( L A , L B ) . Pr o of. One can sho w directly that N is strongly Leibniz, or view Ω as an ( A ⊕ B ) - bimo dule in the eviden t w ay and apply Prop osition 2.1. Since N is also con tinuous, it follo ws from Prop osition 1.2 that L is lo we r semicon tin uous and strongly Leibniz. Clearly L is a ∗ -seminorm. Th us condition a) of Deﬁnition 4.1 is satisﬁed. W e now w a nt to apply theorem 5 . 2 o f [28] to show that L is a Lip ∗ - norm. W e m ust thus sho w that N ∨ N ∗ , restricted to t he self-adjoin t part of A ⊕ B is a bridge a s deﬁned in deﬁnition 5 . 1 of [28]. F rom its bimo dule source it is clear that N (1 A , 1 B ) = 0, while N (1 A , 0 ) 6 = 0 since ω 0 6 = 0. Since also N is con tinuous, it follows that the ﬁrst tw o conditions of deﬁnition 5 . 1 are satisﬁed. The main tec hnical condition of Deﬁnition 6.1 directly implies that condition 3 of deﬁnition 5 . 1 of [28] is satisﬁed, so that N ∨ N ∗ is indeed a bridge, and so L , restricted to self-adjoin t elemen ts, is a Lip-norm. Th us L is a Lip ∗ -norm, and so condition b) of Deﬁnition 4.1 is satisﬁed. Because N is clearly ﬁnite, ( A ⊕ B ) f , as deﬁned for L , coincides with A f ⊕ B f . F rom the fa ct tha t A f and B f are b y assumption sp ectrally stable in their completion it follo ws easily that ( A ⊕ B ) f is sp ectrally stable in its completion. Th us L satisﬁes condition c) of Deﬁnition 4.1 , so tha t L is a C ∗ -metric. Supp ose no w that w e are give n a ∈ A with a = a ∗ . F rom the formula for L it is clear that L ( a, b ) ≥ L A ( a ) for all b ∈ B . Let ε > 0 b e given . Then b y Deﬁnition 6.1 t here is a b ∈ B with b = b ∗ suc h that L B ( b ) ∨ N ( a, b ) ≤ L A ( a ) + ε. 22 MARC A. RIEFFEL Since N and N ∗ agree on self-adjoint elemen ts, it follows that L ( a, b ) ≤ L A ( a ) + ε . Since ε is arbitrary , it follo ws that the quotient of L on A applied to a giv es L A ( a ). In the same w ay the quotient o f L on B , restricted to self-adjoint elemen ts, giv es L B on self-adjoin t elemen t s. Th us L ∈ M C ( L A , L B ).  In the next sections we will see ho w to construct useful bimo dule bridges for “matrix algebras con verging to the sphere”. Hanfeng Li has p ointed out to me that prox is dominated by the “n uclear Gromov - Hausdorﬀ distance” dist nu that he deﬁned in remark 5.5 of [18] and studied f ur t her in section 5 of [14]. He giv es a pro o f of this in the app endix of [19]. (He uses the term “n uclear” b ecause this distance has fav ora ble prop erties for n uclear C ∗ -algebras.) W e sk etch here how this w orks, so that it can b e easily compared with what w e ha ve done ab ov e. The crux o f Li’s a pproac h is tha t he restricts atten tion to bimo dules of a quite special kind. Sp eciﬁcally , for unital C ∗ -algebras A and B let H ( A, B ) denote the collection of all triples ( D , ι A , ι B ) consisting of a unital C ∗ -algebra D and injectiv e (so isometric) unital homomorphisms ι A and ι B from A and B in to D . W e can then view D as a n A - B - bimo dule in the eviden t w ay . F o r a C ∗ -metric L A on A Li sets E ( L A ) = { a ∈ A sa : L A ( a ) ≤ 1 } , the L A -unit-ball in A sa . Then for a n y ( D , ι A , ι B ) ∈ H ( A, B ) he consid- ers dist H ( ι A ( E ( L A )) , ι B ( E ( L B ))) , the or dina r y Hausdorﬀ distance in D for the norm of D . Ev en tho ug h E ( L A ) and E ( L B ) are unbounded, this distance is ﬁnite, for the follow- ing reason. Let r A b e the ra dius of ( A, L A ), as deﬁned in section 2 of [25], so that k ˜ a k ˜ ≤ r A L A ( a ) for any a ∈ A sa , where ˜ denotes image in the quotient A sa / R 1 A , with the quotien t norm. Then if a ∈ E ( L A ) so that L A ( a ) ≤ 1, it follow s that a = a ′ + t 1 A for some t ∈ R a nd a ′ ∈ A sa with k a ′ k ≤ r A . Let b = t 1 B , so that b ∈ E ( L B ). Then k ι A ( a ) − ι B ( b ) k = k a ′ k ≤ r A . Th us ι A ( E ( L A )) is in the r A -neigh b orho o d of ι B ( E ( L B )). By also inter- c hanging the roles of a and b w e see that dist H ( ι A ( E ( L A )) , ι B ( E ( L B )) ≤ max( r A , r B ) . Then Li deﬁnes dist nu ( A, B ) (or, more precisely , dist nu ( L A , L B )) to b e inf { dist H ( ι A ( E ( L A )) , ι B ( E ( L B ))) : ( D , ι A , ι B ) ∈ H ( A, B ) } . Li sho ws as follow s that dist nu satisﬁes the triangle inequality . Sup- p ose that a third compact C ∗ -metric space ( C, L C ) is giv en. Let d AB = LEIBNIZ SEMIN ORMS 23 dist nu ( A, B ), and similarly for d B C and d AC . Giv en ε > 0 we can ﬁnd ( D , ι A , ι B ) ∈ H ( A, B ) and ( E , ρ B , ρ C ) ∈ H ( B , C ) suc h that dist H ( ι A ( E ( L A )) , ι B ( E ( L B )) ≤ d AB + ε, and similarly for d B C . Let F = D ∗ B E b e an amalgamated pro duct of D and E ov er B (using the inclusions ι B and ρ B ). This means that there are unital injectiv e homomorphisms σ D and σ E of D and E in to F suc h that σ D ◦ ι B = σ E ◦ ρ B . (It is natural to cut do wn to the subalgebra generated by t he images of D and E in F .) Before con tin uing, w e remark that it is easy to construct a univ ersal amalgamated free pro duct, A ∗ C B , if one do es not insist that the homomorphisms in to it from A and B are injectiv e. One take s the quotien t of the univ ersal ( i.e. f ull) f r ee pro duct A ∗ B by the ideal generated b y the desired relatio ns from C . See [20]. What is not as simple is to sho w that the eviden t homomorphisms of A and B into the unive r sal A ∗ C B are injectiv e. This was ﬁrst shown by Blac k adar in [3]. In a commen t added in pro of in that pap er, Blac k adar sa ys t ha t John Phillips has sho wn him a preferable pro o f . Black a da r ha s show n me this pro of of John Phillips, and since it seems no t to hav e app eared in print up to no w, w e sk etch it here. Hanfeng Li has p oin ted out to me that a ve rsion of the argumen t in a substan tially more complicated situation app ears in t he pro of of pro p osition 2.2 of [2]. T o simplify notation we simply view C as a unital subalgebra of eac h of A and B . The crux of the matt er is to sho w that there are faithful (non-degenerate) represen tatio ns of A and B on the same Hilb ert space whose restrictions to C ar e equal. W e construct suc h represen tations as f o llo ws. (1) L et ( π 1 , H 1 ) b e a faithful represen ta tion o f A . F orm the re- stricted represen ta tion ( π 1 | C , H 1 ) of C , and extend it to a rep- resen tation ( ρ 1 , H 1 ⊕ K 1 ) of B . (This can b e done b y decom- p osing in to cyclic represen t a tions and extending their states – see lemma 2.1 of [2] .) (2) No tice that ρ 1 | C carries H 1 in to itself a nd so carries K 1 in to itself. Extend ( ρ 1 | C , K 1 ) to a represen tatio n ( π 2 , K 1 ⊕ H 2 ) of A . (3) Extend ( π 2 | C , H 2 ) to a represen tat io n ( ρ 2 , H 2 ⊕ K 2 ) of B . (4) Contin ue this pro cess through all the positive integers, and form H = L ∞ 1 ( H j ⊕ K j ). The π j ’s a nd ρ j ’s combine to giv e repre- sen tations π and ρ of A and B on H whic h can b e chec k ed to agree on C . Since π 1 w as c hosen t o b e a f a ithful represen tation of A , so is π . Th us the homomorphism from A into A ∗ C B m ust b e injectiv e. The situation is symmetric fo r A and B , so the homomo r phism from B in t o A ∗ C B m ust also b e injectiv e. 24 MARC A. RIEFFEL W e return to demonstrating the triangle inequalit y for dist nu . Let τ A = σ D ◦ ι A and τ C = σ E ◦ ρ C . Then ( F , τ A , τ C ) ∈ H ( A, C ). F urther- more, if a ∈ E ( L A ) then there is a b ∈ E ( L B ) su c h that k ι A ( a ) − ι B ( b ) k ≤ d AB + ε , so that k τ A ( a ) − σ D ( ι B ( b )) k ≤ d AB + ε . In the same wa y there exists c ∈ E ( L C ) suc h that k σ E ( ρ B ( b )) − τ C ( c )) k ≤ d B C + ε . But σ D ( ι B ( b )) = σ E ( ρ B ( b )), a nd so k τ A ( a ) − τ C ( c ) k ≤ d AB + d B C + 2 ε. In this w ay w e ﬁnd that dist nu ( L A , L C ) ≤ dist nu ( L A , L B ) + dist nu ( L B , L C ) . F urther fav ora ble prop erties of dist nu are presen ted in [18 , 1 4] that we will not discuss here. Giv en ( D , ι A , ι B ) ∈ H ( A, B ), w e can view D as a normed A - B - bimo dule in the eviden t w a y , and as sp ecial elemen t we can choose ω 0 = 1 D . The cor r esp o nding b ounded seminorm N D on A ⊕ B is then simply deﬁned by N D ( a, b ) = k ι A ( a ) − ι B ( b ) k . Giv en C ∗ -metrics L A and L B on A and B , w e can se ek constan ts γ su c h that γ − 1 N D is a bimo dule bridge for L A and L B . Let δ = dist H ( ι A ( E ( L A )) , ι B ( E ( L B ))). Given an y ε > 0 w e sho w that δ + ε is suc h a constan t. Let a ∈ A sa with L A ( a ) = 1. Then there is a b ∈ B sa suc h that L B ( b ) ≤ 1 and k ι A ( a ) − ι B ( b ) k ≤ δ + ε , so that L B ( b ) ∨ ( δ + ε ) − 1 N D ( a, b ) ≤ 1 = L A ( a ) . W e can interc hange t he roles of A and B . Thus w e see that ( δ + ε ) − 1 N D is indeed a bimo dule bridge. Notice tha t f or any a ∈ A and b ∈ B w e ha ve N ( a ∗ , b ∗ ) = N ( a, b ). Th us when w e deﬁne L on A ⊕ B by L ( a, b ) = L A ( a ) ∨ L B ( b ) ∨ ( δ + ε ) − 1 N D ( a, b ) it follo ws from Theorem 6.2 that L ∈ M C ( L A , L B ). But eve n more is true. As suggested b y Li, w e will f ollo w the argu- men t in the last para graph of the pro of of pro p osition 4.7 of [17 ]. Let µ ∈ S ( A ). View A and B as subalgebras of D via ι A and ι B . By the Hahn-Banac h theorem, extend µ to a state ˜ ν of D , a nd then restrict ˜ ν to B to get ν ∈ S ( B ). Then for a ∈ A sa and b ∈ B sa w e ha ve | µ ( a ) − ν ( b ) | = | ˜ ν ( a ) − ˜ ν ( b ) | ≤ k ι A ( a ) − ι B ( b ) k ≤ ( δ + ε ) L ( a, b ) , where L is deﬁned a s ab ov e. Consequen tly if L ( a, b ) ≤ 1 t hen w e hav e | µ ( a ) − ν ( b ) | ≤ δ + ε . Th us µ is in the δ + ε - neigh b or ho o d of S ( B ) for LEIBNIZ SEMIN ORMS 25 the metric ρ L on S ( A ⊕ B ). The same argumen t w o r ks with the roles of A and B revers ed. Since ε is a r bit r a ry , we see fro m this that pro x( A, B ) ≤ dist nu ( A, B ) , as a sserted. In [18 , 14] Li indicates that dist nu w orks ve ry w ell with man y o f the classes of sp eciﬁc examples whose metric asp ects hav e b een studied. In particular, he p oin ted out t o me that dist nu can b e used to give a n alternate pro of of our Main Theorem (in a qualitativ e w ay). This alter- nate pro of is attractiv e b ecause of its quite general appro ac h. How ev er, a pro of via dist nu app ears to me to b e less concrete and quan titative than that whic h we giv e in the next sections, b oth b ecause the pro of via dist nu uses a somewhat deep theorem o f Blanch ard on the subtriv- ialization o f con tinuous ﬁelds of nuclear C ∗ -algebras (as discusse d in remark 5.5 of [18 ]), and b ecause of its use of the Hahn- Banac h theo- rem seen just ab o ve . The pro of w e will give provides sp eciﬁc estimates for the appro ximation, and pro vides a constructiv e w ay of ﬁnding a state fo r o ne of the alg ebras that is close to a giv en state of the other algebra. 7. M a trix algebras and homogeneous sp aces In this section w e b egin the study of our main example. Our dis- cussion will b e f airly parallel to that in [29] but with some imp ortan t diﬀerences. F or the reader’s con ve nience w e will include here some fragmen ts of [29] in order to make precise our setting. W e will usually use the notation used in [29]. Let G b e a compact group (p erhaps ev en ﬁnite at ﬁrst). Let U b e an irreducible unitary r epresen ta tion of G on a Hilb ert space H . Let B = L ( H ) denote the C ∗ -algebra of linear op erators on H ( a “full matrix algebra”). There is a natural action, α , of G o n B b y conjugation by U . That is, α x ( T ) = U x T U ∗ x for x ∈ G and T ∈ B . W e in tro duce metric data in to the picture b y choosing a con tin uous length- function, ℓ , on G . W e require that ℓ satisfy the additional condition that ℓ ( xy x − 1 ) = ℓ ( y ) for x, y ∈ G . This ensures that the metric on G deﬁned b y ℓ is in v a rian t under b o th left and right translations. As in Example 2.5 w e deﬁne a seminorm, L B , o n B b y L B ( T ) = sup {k α x ( T ) − T k /ℓ ( x ) : x 6 = e G } . Then L B is a C ∗ -metric on B fo r t he reasons giv en in Example 4.3. Let P b e a rank-one pro jection in B . Let H = { x ∈ G : α x ( P ) = P } , the stabilit y subgroup for P . Let A = C ( G/H ), the C ∗ -algebra of con tinuous complex-v alued functions on G/ H . W e let λ denote the 26 MARC A. RIEFFEL usual a ction of G on G/H , and so on A , b y translation. W e deﬁne a seminorm, L A , on A a s in Example 2.5 by L A ( f ) = sup {k λ x ( f ) − f k / ℓ ( x ) : x 6 = e G } . Again, L A is a C ∗ -metric for the r easons giv en in Example 4 .3. W e can then ask for estimates of pro x( A, B ). T o obtain suc h an estimate w e need to construct a suitable C ∗ -metric on A ⊕ B . W e do this a s f ollo ws. F or any T ∈ B its Berezin cov ar ia n t sym b ol, σ T , is deﬁned b y σ T ( x ) = tr( T α x ( P )) , for x ∈ G . Here tr is the usual unnor ma lized tr ace on B . Because of the deﬁnition of H we see that σ T ∈ C ( G/H ) = A . When the α x ( P )’s are view ed a s giving states of B via tr as ab ov e, they form a “coheren t state”, assigning a pure state of B to each pure state of A . Once w e note t hat tr is α - in v ariant, it is easy to see that σ is a unital, p ositiv e, norm-non-increasing α - λ - equiv ariant op erator from B to A . How ev er ev en tually one really w ants also the prop erty that if σ T = 0 then T = 0. This is equiv alen t to the linear span o f the α x ( P )’s in B b eing all of B . It is an in teresting question a s to whic h represen tations U admit suc h a P , and ho w many suc h P ’s, ev en for ﬁnite groups. W e let Ω = L ( B , A ), the Banach space of linear op erators from B to A , equipp ed with the op erator norm corr esp o nding to the C ∗ -norms on A a nd B . (P erhaps w e should b e using the space of completely b o unded op erators here.) W e let M and Λ denote the left regular represe n tations of A and B . Then Ω is a n A - B -bimo dule for the op erations f ω = M f ◦ ω and ω T = ω ◦ Λ T . It is easily che ck ed that Ω is a normed A - B -bimo dule. Of course σ ∈ Ω. W e will tak e our bimo dule bridge for ( A, L A ) and ( B , L B ) to b e of the form ( Ω , γ − 1 σ ) where γ is a p ositiv e real n um b er that is y et to b e determined. Set N σ ( f , T ) = k M f ◦ σ − σ ◦ Λ T k . Then the seminorm N from (Ω , γ − 1 σ ) is deﬁned by N ( f , T ) = γ − 1 N σ ( f , T ) . W e need to determine the v a lues of γ for whic h ( Ω , γ − 1 σ ) is a bimodule bridge so that, in particular, the corresp onding seminorm L has L A and L B as quotien ts for self-adjoin t elemen t s. But, as a ﬁrst step in sho wing what t he implication f o r pro ximity will b e, w e hav e: Prop osition 7.1. Supp os e that γ is such that (Ω , γ − 1 σ ) is a bimo dule bridge for L A and L B , an d let N b e the sem inorm it determines. L et LEIBNIZ SEMIN ORMS 27 L = L A ∨ L B ∨ N ∨ N ∗ and let ρ L b e the metric on S ( A ⊕ B ) that L determines. Then S ( A ) is in the γ -neighb orho o d of S ( B ) for ρ L . Pr o of. Let µ ∈ S ( A ). W e m ust ﬁnd a ν ∈ S ( B ) suc h that ρ L ( µ, ν ) ≤ γ . W e c ho o se ν = µ ◦ σ . Let ( f , T ) ∈ A ⊕ B b e suc h tha t L ( f , T ) ≤ 1, so that N ( f , T ) ≤ 1 and t h us N σ ( f , T ) ≤ γ . Then | µ ( f , T ) − ν ( f , T ) | = | µ ( f ) − µ ( σ T ) | ≤ k f − σ T k = k ( M f ◦ σ − σ ◦ Λ T )( I ) k ≤ N σ ( f , T ) ≤ γ , where I is the iden tit y elemen t in B . F rom the deﬁnition of ρ L it follo ws that ρ L ( µ, ν ) ≤ γ .  W e remark tha t in o ur earlier pa p er on “matr ix a lgebras con ve r ge to the sphere” [2 9] the bridge N that we had used w as N ( f , T ) = γ − 1 k f − σ T k . The ab ov e calculation rev eals that this old N is r elat ed to our new one just b y a pplying our M f ◦ σ − σ ◦ Λ T to the iden tit y op erator. The old N is not Leibniz. T o pro ceed further we now obtain another express io n for N σ whic h will b e more con ve nient for some purp oses. W e no t e that for S, T ∈ B and f ∈ A we hav e ( M f ◦ σ − σ ◦ Λ T )( S ) = f σ S − σ T S , and that when this is ev alua t ed at x ∈ G/H w e obta in f ( x ) σ S ( x ) − σ T S ( x ) = f ( x ) tr( S α x ( P )) − tr( T S α x ( P )) = tr( α x ( P )( f ( x ) I − T ) S ) . The op erator norm of M f ◦ σ − σ ◦ Λ T is then the suprem um of the absolute v a lue of the ab o ve expression take n o ver all x ∈ G/H and S ∈ B with k S k ≤ 1. But tr gives a pairing that expresses the dual of B with its op erator norm as B with the trace-class norm, whic h w e denote b y k · k 1 . F rom this fact w e see that k M f ◦ σ − σ · Λ T k = sup {k α x ( P )( f ( x ) I − T ) k 1 : x ∈ G/H } . But if R is a rank-one op erator then R ∗ R = r 2 Q for some rank-o ne pro jection Q a nd some r ∈ R + , so that k R k 1 = tr(( R ∗ R ) 1 / 2 ) = r = k R ∗ R k 1 / 2 = k R k , where the norm o n the right is the op erator norm. In this wa y we obtain: Prop osition 7.2. F or f ∈ A and T ∈ B we have N σ ( f , T ) = sup { N x ( f , T ) : x ∈ G/H } wher e N x ( f , T ) = k α x ( P )( f ( x ) I − T ) k . 28 MARC A. RIEFFEL W e remark that N x ( f , T ) can easily b e che c k ed to b e stro ngly Leib- niz. 8. The choice of the const ant γ Let us ﬁrst see what c ho ices of γ ensure that L has L A as a quotien t. It suﬃces to choose γ suc h that for any f ∈ A w e can ﬁnd T ∈ B suc h that L B ( T ) ∨ N ( f , T ) ≤ L A ( f ). On G/H let us momen tarily use the G - in v arian t measure of mass 1 to giv e A the norm from L 2 ( G/H ). Similarly , on B w e put the Hilb ert–Sc hmidt norm from the n ormalize d trace. Then σ has an adjoin t op erator, whic h w e denote b y ˘ σ . It is easily computed [29] to b e deﬁned by ˘ σ f = d Z G/H f ( x ) α x ( P ) dx, where d is the dimension of H . One can easily v erify t ha t ˘ σ is a p ositiv e and λ - α -equiv ariant map from A to B . F urthermore, ˘ σ 1 = d R α x ( P ) dx , whic h is clearly α - in v ariant, and so is a scalar m ultiple of I since U is irreducible. But clearly the usual trace o f d R α x ( P ) dx is d . Th us ˘ σ 1 = I , that is, ˘ σ is unital. (This is wh y w e used the normalized traces in deﬁning ˘ σ .) It follo ws t ha t ˘ σ is a lso norm non- increasing. Then, g iv en f ∈ A , w e will c ho ose T to b e T = ˘ σ f . It is easily seen (as in the pro of of prop osition 1 . 1 of [29]) t ha t L B ( ˘ σ f ) ≤ L A ( f ). F or an y x ∈ G/ H w e hav e by equiv ariance of ˘ σ N x ( f , ˘ σ f ) = k α x ( P )( f ( x ) I − ˘ σ f ) k = k P (( λ − 1 x f )( e ) I − ˘ σ λ − 1 x f ) k . Since f is arbit r a ry a nd L A is λ -inv ariant, it suﬃces for us to consider k P ( f ( e ) I − ˘ σ f ) k . But k P ( f ( e ) I − ˘ σ f ) k =     P  f ( e ) d Z α y ( P ) dy − d Z f ( y ) α y ( P ) dy      = d     Z ( f ( e ) − f ( y )) P α y ( P ) dy     ≤ L A ( f ) d Z ρ G/H ( e, y ) k P α y ( P ) k dy , where ρ G/H is the ordinary metric on G/H from L A . F rom all of this w e obtain: Prop osition 8.1. Set γ A = d R ρ G/H ( e, y ) k P α y ( P ) k dy . Then for any γ ≥ γ A the seminorm L = L A ∨ L B ∨ γ − 1 ( N σ ∨ N ∗ σ ) on A ⊕ B has L A as its quotient on A . LEIBNIZ SEMIN ORMS 29 W e remark that in the ab o ve pro p osition we do not hav e to restrict atten tion to self-adjoin t elemen ts, in con t rast to the r equiremen t in Deﬁnition 6.1. Note that ˘ σ ¯ f = ( ˘ σ f ) ∗ . I do not kno w whether the ab ov e condition on γ is the b est that can b e obtained in the absence of further h yp otheses on G , U , P and ℓ . W e no w consider the quotien t of L on B . G iv en T ∈ B w e seek f ∈ A suc h that L A ( f ) ∨ N ( f , T ) ≤ L B ( T ). W e c ho ose f = σ T , and seek what requiremen t this puts on γ . As ab ov e, it is easy to c hec k that L A ( σ T ) ≤ L B ( T ). Again b y equiv ariance w e hav e N x ( σ T , T ) = k α x ( P )(tr( T α x ( P )) I − T ) k = k P (tr( P α − 1 x ( T )) I − α − 1 x ( T )) k . Since T is arbitrary and L B is α -in v ariant, it suﬃces t o c ho ose γ large enough that k P tr( P T ) − P T k ≤ γ L B ( T ) for all T ∈ B . Notice that the left-hand side giv es a seminorm (with v alue 0 for T = P or I ) on the quotien t space ˜ B = B / C I , while L B giv es a norm on ˜ B . Since B is ﬁnite-dimensional, there do es exist a ﬁnite γ suc h that t he ab ov e inequalit y is satisﬁed. Notice also that σ T ∗ = ( σ T ) − . Thus we obtain: Prop osition 8.2. Deﬁne γ B by γ B = sup {k P tr( P T ) − P T k : T ∈ B a nd L B ( T ) ≤ 1 } . Then γ B is ﬁnite, a n d for any γ ≥ γ B the seminorm L = L A ∨ L B ∨ γ − 1 ( N σ ∨ N ∗ σ ) o n A ⊕ B h as L B as its quotient on B . F or later use w e no w express k P (tr( P T ) I − T ) k in a diﬀeren t form. Since taking adjoin ts is an isometry , and b y the C ∗ -relation, and b y the fact that if R is a p o sitiv e op erator then k P RP k = tr( P RP ) b ecause P is of rank 1, w e hav e k P (tr( P T ) I − T ) k 2 = k P (tr( P T ) I − T )(tr( P T ) I − T ) ∗ P k = tr  P (tr( P T ) I − T )(tr( P T ) I − T ) ∗ P )  = | tr ( P T ) | 2 − tr( P T P ) tr( P T ) − tr( P T ) tr( P T ∗ P ) + tr( P T T ∗ P ) = tr( P T T ∗ P ) − | tr( P T ) | 2 . Th us: Prop osition 8.3. F or any T ∈ B we have k P (tr( P T ) I − T ) k = (tr( P T T ∗ P ) − | tr( P T ) | 2 ) 1 / 2 . W e remark that if ξ is a unit v ector in the rang e of P then tr( P T T ∗ P ) − | tr( P T ) | 2 = h T T ∗ ξ , ξ i − |h T ∗ ξ , ξ i| 2 . 30 MARC A. RIEFFEL When T is self-adjoin t this is the “mean-square deviation” of T in the state determined by ξ [35]. W e no w need to consider ho w small a neighbor ho o d of S ( A ) contains S ( B ). Let ν ∈ S ( B ) b e giv en. W e choose µ = ν ◦ ˘ σ , and observ e that µ ∈ S ( A ). Let ( f , T ) ∈ A ⊕ B b e suc h that L ( f , T ) ≤ 1 , so tha t N σ ( f , T ) ≤ γ . Then | µ ( f , T ) − ν ( f , T ) | = | ν ( ˘ σ f ) − ν ( T ) | ≤ k ˘ σ f − T k =     d Z f ( x ) α x ( P ) dx − d Z α x ( P ) T dx     = d     Z α x ( P )( f ( x ) I − T ) dx     ≤ d Z N x ( f , T ) d x ≤ dN σ ( f , T ) ≤ dγ . But t he presence of d here causes us diﬃculties later, so w e take another path, namely that used near the end of section 2 of [2 9 ]. W e hav e k ˘ σ f − T k ≤ k ˘ σ f − ˘ σ ( σ T ) k + k ˘ σ ( σ T ) − T k ≤ k f − σ T k + k ˘ σ ( σ T ) − T k ≤ γ A + k ˘ σ ( σ T ) − T k , where w e hav e used that k f − σ T k ≤ N σ ( f , T ), a s seen in the pro of of Prop osition 7.1. Notice that T 7→ k ˘ σ ( σ T ) − T k is a seminorm on B whic h takes v alue 0 for T = I , and so drops t o a seminorm on ˜ B = B / C I , where L B b ecomes a norm. Notation 8.4. W e set δ B = sup {k T − ˘ σ ( σ T ) k : L B ( T ) ≤ 1 } . With this notatio n the ab o ve discussion g iv es: Prop osition 8.5. Supp o s e that γ ≥ γ A ∨ γ B , so that L has L A and L B as quotients (wher e L = L A ∨ L B ∨ γ − 1 ( N σ ∨ N ∗ σ ) ). T hen S ( B ) is in the ( γ A + δ B ) -neighb orho o d of S ( A ) . 9. The set-up fo r comp act Lie groups W e now sp ecialize to the case in whic h G is a compact connected semisimple Lie group. W e use man y of the tec hniques used in sections 6 and 7 of [2 9], and we usually use the notation established in sections 5 and 6 of [29]. W e now review that not a tion. W e let g 0 denote the Lie algebra of G , while g denotes t he complexiﬁcation of g 0 . W e ch o ose a maximal torus in G , with corresponding Cartan subalgebra of g , its set of r o ots, and a choice of p ositive ro ots. W e let ( U, H ) b e an irreducible unitary represen tation of G , and w e let U also denote the corresp onding represen tation of g . W e c ho ose a highest w eight v ector, LEIBNIZ SEMIN ORMS 31 ξ , for ( U, H ) with k ξ k = 1 . F or a ny n ∈ Z ≥ 1 w e set ξ n = ξ ⊗ n in H ⊗ n , and w e let ( U n , H n ) b e the restriction o f U ⊗ n to the U ⊗ n -in v ariant subspace, H n , of H ⊗ n whic h is generated b y ξ n . Then ( U n , H n ) is an irreducible represen ta tion of G with highest weigh t v ector ξ n , and its highest we ig h t is just n times the highest w eigh t of ( U, H ). W e denote the dimension of H n b y d n . W e let B n = L ( H n ). The action of G on B n b y conjugation b y U n will b e denoted simply b y α . W e assume that a con tinuous length function, ℓ , has b een c hosen for G , and we denote the cor r esp o nding C ∗ -metric on B n b y L B n . W e let P n denote the rank-one pro jection along ξ n . Then the α -stability subgroup H for P = P 1 will also b e the stability subgroup f o r each P n . L et γ A n and γ B n b e the constan ts deﬁned in Prop ositions 8.1 and 8 .2 but for P n . As done earlier, w e let A = C ( G/H ) , and w e let L A b e the seminorm on A for ℓ a nd the action of G . W e can now state the main theorem of this pap er. Theorem 9.1. L et notation b e as ab ov e . Set γ n = max { γ A n , γ B n } for e ach n , and let L n b e d e ﬁ ne d on A ⊕ B n as in Pr op os ition 7.1 b ut using γ n . Th en L n ∈ M C ( L A , L B ) , and the se quenc e { L n } shows that the se quenc e { prox( A, B n ) } c onver ges to 0 as n go es to ∞ . The next three sections will b e dev oted to the pro o f of t his theorem. 10. The proof tha t γ A n → 0 Consisten t with the notation o f Prop osition 8 .1 , w e ha ve set γ A n = d n Z ρ G/H ( e, x ) k P n α x ( P n ) k dx. Prop osition 10.1. The se quenc e { γ A n } c onver ges to 0 . Pr o of. F or an y t wo v ectors η , ζ w e let h η , ζ i 0 denote the rank-one o p- erator that they determine. Then for an y n w e ha ve k P n α x ( P n ) k = kh ξ n , ξ n i 0 h U n x ξ n , U n x ξ n i 0 k = |h U n x ξ n , ξ n i| = |h U x ξ , ξ i| n = k P α x ( P ) k n . W e use the ana logous treatmen t given in lemma 3 . 3 and theorem 3 . 4 of [29], where w e see that d n |h U x ξ , ξ i| 2 n dx (= d n k P n α x ( P n ) k 2 dx ) is a probabilit y measure on G/H , and that the sequence of these pro babilit y measures con v erges in the w eak- ∗ t op ology to the δ -measure on G/H supp orted at eH . Since ρ G/H ( e, e ) = 0, it follow s that the sequence 32 MARC A. RIEFFEL d n R ρ G/H ( e, x ) k P n α x ( P n ) k 2 dx conv erg es to 0 . Now γ A 2 n = d 2 n Z ρ G/H ( e, x ) k P α x ( P ) k 2 n dx = ( d 2 n /d n ) d n Z ρ G/H ( e, x ) k P n α x ( P n ) k 2 dx, and so if we can sho w that ( d 2 n /d n ) is b ounded, then w e ﬁnd tha t the sequence { γ 2 n } con ve rges to 0. W e use the W eyl dimension form ula, as presen ted f o r example in theorem 4 . 14 . 6 of [36], to show that { d 2 n /d n } is b ounded. W e let ω b e the highest we ig h t o f U for our choice P of p ositiv e ro o ts. If one examines the dimension for m ula, it is eviden t that one only needs to use those p ositiv e ro ots α suc h that h ω , α i > 0. W e denote this set b y P ω , and w e denote its cardinality by p . It is clear that for an y n ∈ Z > 0 w e ha ve P nω = P ω . The W eyl dimension formula then tells us that d n =  Y h nω + δ , α i .  Y h δ, α i  where b oth pro ducts are taken o v er P ω , and δ is half the sum of the p ositiv e ro ot s. Th us d 2 n /d n =  Y h 2 nω + δ , α i .  Y h nω + δ , α i  = Y (1 + h nω , α i / h nω + δ , α i ) ≤ 2 p , so that the sequence d 2 n /d n is b ounded as needed, and consequen tly the sequence { γ A 2 n } con verges to 0. In the same w ay , we ﬁnd that d n +1 /d n ≤ (1 + n − 1 ) p . Since 0 ≤ k P α x ( P ) k ≤ 1, w e hav e k P α x ( P ) k n ≥ k P α x ( P ) k n +1 . Thu s the in tegrals deﬁning γ A n are non-increasing. It fol- lo ws that γ A 2 n +1 ≤ (1 + (2 n ) − 1 ) p γ A 2 n . Since the seque nce { γ A 2 n } con v erges to 0, it follo ws that the seq uence { γ A 2 n +1 } does also, so that the sequence { γ A n } conv erges to 0.  11. Pr ope r t ies of Berezin symbols W e no w need results related to those giv en in sections 4 a nd 5 of [29], leading to the pro of of theorem 6.1 of [29], and we will shortly also need theorem 6.1 of [2 9 ] itself. But Jerem y Sain has fo und a substan tial simpliﬁcation of the pro o f of theorem 6.1 of [29]. He give s his arg ument in section 4 .4 of [33] in the more complicated con text of quan tum groups. W e will use his a rgumen ts here in our presen t con text. This will in particular pr ovide Sain’s pro of of theorem 6.1 of [29]. LEIBNIZ SEMIN ORMS 33 As in [29], we denote the Berezin sym b ol map f rom B n to A = C ( G/H ) by σ n . F rom theorem 3 . 1 o f [29] w e ﬁnd that σ n is injectiv e b ecause ξ n is a highest w eigh t v ector. C o nsisten t with the notation deﬁned near the b eginning of Section 8, w e denote the adj o in t of σ n b y ˘ σ n . W e let (11.1) δ A n = Z G/H ρ G/H ( e, x ) d n tr( P n α x ( P n )) dx. In section 3 o f [29 ] δ A n w as denoted b y γ n , and theorem 3.4 of [29] sho ws b oth that the sequence { δ A n } conv erges to 0, and that (11.2) k f − σ n ( ˘ σ n ( f )) k ∞ ≤ δ A n L A ( f ) for all f ∈ A and all n . W e remark that σ n ◦ ˘ σ n is often called the “Berezin transform” (f or a giv en n ). As in section 4 of [2 9] w e let ˆ G denote the set of equiv alence classes of irreducible unitary represen ta t io ns o f G . F or an y ﬁnite subset S of ˆ G w e let A S and B n S denote the direct sum o f the isotypic comp onents of A and B n for the represen tat io ns in S and for the actions of G on A and B n (and similarly f o r actions on ot her Bana c h spaces). Since σ n is equiv arian t, it carries B n S in to A S . Since σ n is injectiv e, it follo ws that the dimension of B n S is no larger than that of A S , whic h is ﬁnite. Since { δ A n } con ve r g es to 0 , it follows from equation 11.2 that σ n ◦ ˘ σ n con v erg es strongly to the identit y op erator o n the space of functions f for whic h L A ( f ) < ∞ . But A S is contained in this space and is ﬁnite- dimensional, and σ n ◦ ˘ σ n carries A S in to itself for eac h n . Consequen t ly σ n ◦ ˘ σ n restricted to A S con v erg es in norm to the iden tity op erator on A S . It fo llows that there is an in t eger, N S , suc h that σ n ◦ ˘ σ n on A S is inv ertible and k ( σ n ◦ ˘ σ n ) − 1 k < 2 for ev ery n > N S . In particular, σ n from B n S to A S will b e surjectiv e for n > N S . Since, as mentioned ab ov e, σ n is alw ays injectiv e, and k σ n k = 1 = k ˘ σ n k for a ll n , w e can quic kly see t ha t: Lemma 11.3. (Se e c or ol lary 4.1 7 of [33] .) F or n > N S b oth σ n and ˘ σ n going b etwe en A S and B n S ar e invertible an d their inverses have op er ator-norm no bigge r than 2 . Fix n > N S , and let T ∈ B n S b e given. Set f = ( ˘ σ n ) − 1 ( T ). Note tha t f is we ll- deﬁned, a nd that k f k ∞ ≤ 2 k T k b y Lemma 11.3 . Then k T − ˘ σ n ( σ n T ) k = k ˘ σ n ( f ) − ˘ σ n ( σ n ( ˘ σ n f )) k ≤ k f − σ n ( ˘ σ n f ) k ≤ δ A n L A ( f ) , where we ha ve used inequality 11.2 for the last inequality ab ov e. Be- cause ( ˘ σ n ) − 1 is α - λ - equiv arian t and k ( ˘ σ n ) − 1 k ≤ 2 , w e ha ve L A ( f ) ≤ 2 L B n ( T ). W e hav e th us obtained: 34 MARC A. RIEFFEL Lemma 11.4. (Se e pr op osition 4.19 o f [33] .) F or any n > N S and any T ∈ B n S we have k T − ˘ σ n ( σ n T ) k ≤ 2 δ A n L B n ( T ) . Cho ose a faithful ﬁnite-dimensional unitary represen tation, π 0 , o f G that con tains the trivial represen tation, and let π = π 0 ⊗ ¯ π 0 , where ¯ π 0 is the con tragra dien t represen tation for π 0 . Let χ b e t he c haracter of π . Then χ is a non-negativ e real-v a lued function on G . Since π is fait hful, w e hav e the strict inequalit y χ ( x ) < χ ( e ) for any x ∈ G with x 6 = e . Let χ m denote the c haracter of π ⊗ m , so tha t equally w ell it is the m th p oin t wise p o w er of χ . Set ϕ m = χ m / k χ m k 1 . Then the sequence { ϕ m } is a norm-1 approximate identit y for the con volution algebra L 1 ( G ), as seen in the pro of of theorem 8 . 2 of [28]. F urt hermore, eac h ϕ m is cen tral in L 1 ( G ). L et β b e an isometric strongly contin uous action o f G on a Banac h space D , and let L D b e the corresp onding seminorm for ℓ . Let β ϕ n denote the correspo nding “integrated form” op erator. As in the pro of o f lemma 8 . 3 o f [28], for eac h d ∈ D we ha ve k d − β ϕ m ( d ) k = k d Z ϕ m ( x ) dx − Z ϕ m ( x ) β x ( d ) dx k ≤ Z ϕ m k d − β x ( d ) k dx ≤  Z ϕ m ( x ) ℓ ( x ) dx  L D ( d ) , and the sequence R ϕ m ( x ) ℓ ( x ) dx  con v erg es to 0. W e can now ar g ue exactly as in the rest of the pro of of theorem 6.1 of [29] to obtain: Theorem 11.5. (The or em 6.1 of [29] ) F or e ach n ≥ 1 let δ B n b e as deﬁne d in Notation 8.4 but for B n , so that i t is the sm a l lest c onstant such that k T − ˘ σ n ( σ n T ) k ≤ δ B n L B n ( T ) for al l T ∈ B n . Then the se quenc e { δ B n } c onver ges to 0 . Pr o of of The or em 11.5. Let ε > 0 b e giv en. W e can choose ϕ = ϕ m as just ab ov e suc h that f or a ny ergo dic action β of G on an y unital C ∗ -algebra C w e ha ve k c − β ϕ ( c ) k ≤ ( ε/ 3) L ( c ) f o r all c ∈ C . Now ϕ is a p ositiv e function, a nd is a linear combination of the c hara cters of a ﬁnite subset S of ˆ G , and so the integrated o p erator β ϕ is a completely p ositiv e unital equiv ariant map of C onto its S -isot ypic comp o nen t. Then for ev ery n a nd ev ery T ∈ B n w e ha ve α ϕ ( T ) ∈ B n S and k T − ˘ σ n ( σ n T ) k ≤ ( ε/ 3) L B n ( T ) + k α ϕ ( T ) − ˘ σ n ( σ n α ϕ ( T ) ) k + ( ε/ 3) L B n ( T ) . LEIBNIZ SEMIN ORMS 35 F rom Lemma 11.4 there is an in teger N ε suc h that for an y n > N ε and an y T ′ ∈ B n S w e ha ve k T ′ − ˘ σ n ( σ n ( T ′ )) k ≤ ( ε/ 3) L B n ( T ′ ) . Since α ϕ ( T ) ∈ B n S , w e can apply this to T ′ = α ϕ ( T ). When w e use the fact that L B n ( α ϕ ( T )) ≤ L B n ( T ), we see that for an y n > N ε and any T ∈ B n w e hav e k T − ˘ σ n ( σ n T ) k ≤ εL B n ( T ) . This immediately implies the statemen t ab out the sequenc e { δ B n } .  12. The proof tha t γ B n → 0 Consisten t with the notation o f Prop osition 8 .2 , w e ha ve set γ B n = sup {k P n tr( P n T ) − P n T k : T ∈ B n and L B n ( T ) ≤ 1 } . Prop osition 12.1. The se quenc e { γ B n } c onver ges to 0 . Pr o of. Let ε > 0 be giv en. With t he notation that w e used just b efore Theorem 11.5, c ho ose m 0 suc h that for ϕ = ϕ m 0 w e hav e R ϕ ( x ) ℓ ( x ) dx ≤ ε/ 4. Then by the calculation do ne there w e ha ve k T − α ϕ ( T ) k ≤ ( ε/ 4) L B n ( T ) for all n and for all T ∈ B n . Then f o r an y n and any T ∈ B n k ( P n tr( P n T ) − P n T ) − ( P n tr( P n α ϕ ( T )) − P n α ϕ ( T )) k ≤ | tr( P n ( T − α ϕ ( T ))) | + k T − α ϕ ( T ) k ≤ 2 k T − α ϕ ( T ) k ≤ ( ε/ 2) L B n ( T ) , where for the next-to-last inequalit y w e ha ve used the fa ct that P n ( T − α ϕ ( T )) is o f rank 1. No w a s discussed in the pro of of Theorem 1 1 .5, ϕ is a linear com bi- nation of the characters of a ﬁnite subset S of ˆ G . Th us α ϕ ( T ) ∈ B n S and L B n ( α ϕ ( T )) ≤ L B n ( T ), and so we now see that it suﬃces to pro ve: Main Lemma 12.2. L et S b e given. F o r any ε > 0 ther e is an in te ger N ε such that for any n ≥ N ε and any T ∈ B n S we have k P n tr( P n T ) − P n T k ≤ ( ε/ 2 ) L B n ( T ) . Pr o of. Let f ∈ A , and let n b e given. Because A is comm uta tiv e and ˘ σ n is p ositive , it follows from K a dison’s generalized Sc hw arz inequalit y (e.g. 10.5.9 of [12]) that w e hav e ˘ σ n f ( ˘ σ n f ) ∗ ≤ ˘ σ n f ¯ f 36 MARC A. RIEFFEL for the usual order on p ositiv e op erators. When w e combine this with Prop osition 8.3 w e obtain k P n (tr( P n ˘ σ n f ) I − ˘ σ n f ) k 2 = tr( P n ˘ σ n f ( ˘ σ n f ) ∗ P n ) − | tr( P n ˘ σ n f ) | 2 ≤ tr( P n ˘ σ n f ¯ f ) − | tr( P n ˘ σ n f ) | 2 = ( σ n ( ˘ σ n f ¯ f ))( e ) − | σ n ( ˘ σ n f )( e ) | 2 , whic h b y equation 11.2 ab ov e and theorem 3.4 of [29] conv erges to ( f ¯ f )( e ) − | f ( e ) | 2 = 0 as n increases. F or each n deﬁne an op erator, J n , o n B n b y J n ( T ) = P n (tr( P n T ) I − T ) . The calculation ab ov e show s that t he sequence J n ( ˘ σ n f ) con verges to 0 for an y f ∈ A with L A ( f ) < ∞ . F or S as ab o ve it follo ws that the sequence of restrictions of J n ◦ ˘ σ n to A S con v erg es to 0 in op erator norm. Let N S b e as in Lemma 11.3 , so tha t k ( ˘ σ n ) − 1 k ≤ 2 fo r n > N S . It follows that for n > N S w e ha ve k J n k ≤ 2 k J n ◦ ˘ σ n k , so that the sequence of restrictions of J n to B n S con v erg es to 0 in norm. Thus for an y ε ′ > 0 we can ﬁnd a n n ε ′ suc h that for n > n ε ′ and all T ∈ B n S w e ha ve k J n ( T ) k ≤ ε ′ k T k . No w J n ( I ) = 0, and so it follow s that k J n ( T ) k ≤ ε ′ k ˜ T k ∼ , where m uc h as b efore k · k ∼ denotes the quotien t norm o n ˜ B n = B n / C I . But b y lemma 2 . 4 of [24] the radius of eac h of the a lg ebras B n is no larger than r = R ℓ ( x ) dx , in the sense tha t k ˜ T k ∼ ≤ r L B n ( T ) for a ll T ∈ B n . W e include a slightly simpler pro of here. F or T ∈ B n let η ( T ) = R α x ( T ) dx , so that η ( T ) ∈ C I since U n is irreducible. Then k ˜ T k ∼ ≤ k T − η ( T ) k = k Z ( T − α x ( T )) dx k ≤ L B n ( T ) Z ℓ ( x ) dx. It f ollo ws that J n ( T ) ≤ r ε ′ L B n ( T ) . Consequen tly , if we c ho ose ε ′ = ε/ (2 r ), and set N ε = n ε ′ ∨ N S , w e ﬁnd that for n ≥ N ε w e ha ve k P n tr( P n T ) − P n T k ≤ ( ε/ 2) L B n ( T ) for all T ∈ B n S , a s needed.   LEIBNIZ SEMIN ORMS 37 13. The proof of the main theorem W e no w use the results of the previous sections to pro v e Theorem 9.1. F or an y n set γ n = max( γ A n , γ B n ), and deﬁne L n on A ⊕ B n b y L n ( f , T ) = L A ( f ) ∨ L B n ( T ) ∨ γ − 1 n ( N σ n ( f , T ) ∨ N σ n ( ¯ f , T ∗ )) . Then for eac h n w e hav e γ n ≥ γ A n so that the quotien t of L n on A is L A b y Prop osition 8.1 , and w e hav e γ n ≥ γ B n so that the quotien t of L n on B n is L B n b y Prop osition 8.2. Th us L n is in M C ( L A , L B n ) as deﬁned in Notation 5.5. Then according to Prop osition 7.1 (with notation a s in Prop osi- tion 8 .1 and in the sente nce b efor e Prop osition 10.1), S ( A ) is in the γ n -neigh b orho o d of S ( B n ) for ρ L n . F urthermore, according to Prop osi- tion 8.5 (with notation as in Theorem 11 .5 ) S ( B n ) is in the ( γ A n + δ B n )- neigh b or ho o d of S ( A ). It follo ws that dist ρ L n H ( S ( A ) , S ( B n )) ≤ max { γ A n + δ B n , γ n } ≤ max { γ A n + δ B n , γ B n } , and so pro x( A, B n ) ≤ max { γ A n + δ B n , γ B n } . But γ A n , δ B n and γ B n all con verge to 0 as n go es to ∞ , according to Prop o- sition 10.1, Theorem 11 .5 (theorem 6 . 1 of [29]), and Prop o sition 12.1 resp ectiv ely . Consequen tly prox( A, B n ) conv erges to 0 as n go es to ∞ , as desired. 14. Ma tricial s eminorms In t his section w e will brieﬂy describe the relations b et wee n the pre- vious sections o f this pap er and sev eral v ariants of quan tum G romo v– Hausdorﬀ distance. The ﬁrst v ariant is the matricial quan tum Gromov–Hausdorﬀ dis- tance introduced b y Kerr [1 3]. It has the adv antage that if t wo C ∗ - algebras with L ip- norms are at distance 0 fo r his distance then the C ∗ -algebras are isomorphic. W e will not rep eat here Kerr’s deﬁnitions and results for general op erator systems; ra ther w e will only indicate, somewhat sk etc hily , what Kerr’s v ariant sa ys in the con text o f the presen t pap er. F or an y unital C ∗ -algebra A and eac h q ∈ Z > 0 the ∗ -algebra M q ( A ) of q × q matrices with entries in A has a unique C ∗ - norm. The collection of these C ∗ -norms forms a “ma t ricial norm” for A . Giv en unital C ∗ -algebras A and B , a linear map ϕ : A → B deter- mines for eac h q a linear map, ϕ q , from M q ( A ) to M q ( B ), b y en try- wise application. One sa ys that ϕ is “completely p ositive ” if each ϕ q is p os- itiv e as a map b et w een C ∗ -algebras. F or eac h q let U C P q ( A ) denote the collection of unital completely p ositive maps fro m A into M q ( C ). 38 MARC A. RIEFFEL The U C P q ( A )’s are called the “matricial state-spaces” o f A . All these considerations apply equally well to unital C ∗ -normed algebras, where “p ositiv e” is with resp ect to the completions. Let a Lip ∗ -norm, L , on A b e sp eciﬁed. Then Kerr deﬁnes a metric, ρ q L , on U C P q ( A ) b y ρ q L ( ϕ, ψ ) = sup {k ϕ ( a ) − ψ ( a ) k : L ( a ) ≤ 1 } , and he sho ws that the top ology on U C P q ( A ) determined b y ρ q L agrees with the p oin t- norm to p ology (and so is compact). Now let ( A, L A ) and ( B , L B ) b e unital C ∗ -algebras with Lip ∗ -norms. Essen tially as in deﬁnition 4 . 2 of [2 8 ] let M ( L A , L B ) denote the set of Lip ∗ -norms on A ⊕ B whose quotients on the self-adjoint part a gree with L A and L B . No t e that U C P n ( A ) a nd U C P n ( B ) can b e view ed as subsets o f U C P n ( A ⊕ B ) in an eviden t w ay . Then for eac h q Kerr deﬁnes the q -distance, dist q s , b et w een A and B by dist q s ( A, B ) = inf { dist ρ q L H ( U C P q ( A ) , U C P q ( B )) : L ∈ M ( A, B ) } , and he deﬁnes t he complete distance, dist s , b y dist s ( A, B ) = sup q { dist q s ( A, B ) } . Finally (for our purp oses), he show s t ha t for our setting of coadjoint orbits with A = C ( G/H ) and B n = L ( H n ) with their Lip ∗ -norms fro m a length function ℓ , one has lim n →∞ dist s ( A, B n ) = 0 . W e can quic kly adapt Kerr’s argumen ts to our Leibniz setting. F o r C ∗ -algebras A a nd B equipp ed with C ∗ -metrics, w e deﬁne M C ( L A , L B ) exactly as in Notatio n 5.5. An y L in M C ( L A , L B ) is, in particular, a Lip ∗ -norm, and so deﬁnes for eac h q the metric ρ q L on U C P q ( A ⊕ B ). W e can then deﬁne, for each q , pro x q ( A, B ) = inf { dist ρ q L H ( U C P q ( A ) , U C P q ( B )) : L ∈ M C ( A ⊕ B ) } . Then w e can deﬁne “complete proximit y” by pro x s ( A, B ) = sup q { pro x q ( A, B ) } . Of course, we ha v e dist s ( A, B ) ≤ pro x s ( A, B ) . LEIBNIZ SEMIN ORMS 39 Theorem 14.1. F or A = C ( G/H ) a n d B n = L ( H n ) with their C ∗ - metrics L A and L n B as deﬁne d e arlier in terms of a le ngth function on G , we have lim n →∞ pro x s ( A, B n ) = 0 . Pr o of. W e follow the outline of Kerr’s example 3 . 13 of [13], but for a giv en n w e set, as earlier, L n = L A ∨ L B n ∨ N n ∨ N ∗ n with N n = γ − 1 n N σ n and with γ n c hosen exactly as in the pro o f of Theorem 9.1 that is completed in Section 13 . Th us L n ∈ M C ( L A , L B n ). The k ey o bserv ation, f or Kerr and for us, is tha t σ n and ˘ σ n are (unital) completely p ositive maps, so that if ϕ ∈ U C P q ( A ) then ϕ ◦ σ n is in U C P q ( B n ), and similarly for ˘ σ n . Give n ϕ ∈ U C P q ( A ), set ψ = ϕ ◦ σ n . Then exactly as in the pro of of Propo sition 7 .1 we see that if L n ( f , T ) ≤ 1 then k ϕ ( f ) − ψ ( T ) k ≤ k f − σ T k ≤ γ n , so that U C P q ( A ) is in the γ n neigh b or ho o d of U C P q ( B n ). On the other hand, for any ψ ∈ U C P q ( B n ) set ϕ = ψ ◦ ˘ σ . Then in the somewhat more complicated wa y g iv en in Section 13 w e ﬁnd that U C P q ( B n ) is in as small a neigh b orho o d of U C P q ( A ) a s desired if n is suﬃcien tly large.  W e remark that in section 5 of [13] Kerr considers a weak form of the Leibniz prop erty which he calls “ f -Leibniz” (f o r which he commen ts that the corresp onding distance may not satisfy the triangle inequalit y). In [17 ] Hanfeng Li intro duced a quite ﬂexible v ariant of quan tum Gromov – Ha usdorﬀ distance that in a suitable w ay uses the Ha usdorﬀ distance b etw een the unit L -balls of tw o quan tum metric spaces. Li called this “ order-unit quan t um Gro mo v–Hausdorﬀ distance”. In [14] Kerr and Li dev elop ed a matricial v ersion of Li’s v ar ia n t, whic h they called “op erator G r o mo v–Hausdorﬀ distance”. They then sho w (the- orem 3 . 7) that this vers ion coincides with Kerr’s matricial quan tum Gromov – Ha usdorﬀ distance. It w ould b e interes ting to hav e a v ersion of our complete proximit y ab o v e that is deﬁned in terms of the unit L -balls, since it migh t w ell ha ve certain tec hnical adv antages similar to those p ossessed by Li’s o r der- unit Gromov–Hausdorﬀ distance. F or the sp eciﬁc case o f C ∗ -algebras, Li in t ro duced [18 ] y et a nother v ariant of quan tum Gromov – Hausdorﬀ distance that explicitly uses the algebra m ultiplication. He calls this “ C ∗ -algebraic quan tum Gromo v– Hausdorﬀ distance”. It would b e in teresting to kno w ho w this v ersion relates to Leibniz seminorms and pro ximit y . W e should men t io n that 40 MARC A. RIEFFEL in sev eral places t he later pap ers of Kerr and of Li discusse d in t his section again consider the f - Leibniz prop ert y that Kerr introduced in [13]. Hanfeng Li has p o in ted out to me that muc h the same argumen ts as giv en in the last part of Section 6 showing that prox is dominated b y his dist nu , also sho w that our “complete pro ximit y” prox s is dominated b y dist nu ; and since, as mentioned in Section 6, the examples that ha ve b een studied so far for con v ergence for quan tum G romo v-Ha usdorﬀ dis- tance all in volv e nuc lear C ∗ -algebras, and conv ergence for them holds for dist nu , this gives for them a pro of of conv ergence for pro x s . The pap ers discussed ab o v e all b egin just with a Lip -norm. In a diﬀeren t direction W ei W u has deﬁned and studied matricial Lipschitz seminorms [38, 39, 40]. Again, we will not rep eat here his general deﬁnitions and results; ra ther w e will only indicate somewhat sk etc hily ho w they can b e adapted to the con text o f the presen t pap er, I thank W ei W u for answ ering sev eral questions that I had ab out his pap ers. Let G b e a compact group equipp ed with a length function ℓ , and let α b e an action of G on a unital C ∗ -algebra A . Then G has an eviden t en try-wise action on M q ( A ) fo r each q ∈ Z > 0 , and w e can then use ℓ to deﬁne a seminorm, L q , on eac h M q ( A ) as in Example 2.5. This family of seminorms satisﬁes Ruan-ty p e axioms [1 0], in particular, L ( T ij ) ≤ L q ( T ) fo r T = { T ij } ∈ M q ( A ). W u presen ts this family as one example of what he calls a “ matrix Lipschitz seminorm” on A . It is a v ery natural example, and it indicates how natura l it is to consider matrix Lipsc hitz seminorms quite generally . Ho we ver W u do es no t mak e use o f the fact tha t eac h o f the seminorms L q ab ov e is Leibniz (in fact, strongly Leibniz), and he uses the bridge from [29], whic h is not Leibniz. F or A = C ( G/H ) and B n as earlier w e denote the seminorms by L q A and L n,q B . As W u no t es, the Berezin sym b ol map σ n giv es, b y entry-wis e application, a completely po sitiv e map from M q ( B n ) to M q ( A ) for eac h q ∈ Z > 0 . W e denote these maps still b y σ n . Muc h as in Section 7 w e can then deﬁne a seminorm on M q ( A ⊕ B n ) b y k M f ◦ σ n − σ n ◦ Λ T k . But the analogue of the alternativ e description in terms of seminorms N x giv en in Prop osition 7 .2 is no w more complicated, and so I ha v e found it b est just to w or k directly with the a nalogs of the N x ’s. Sp ecif- ically , we write diag ( α x ( P n )) for the matrix in M q ( B n ) eac h of whose diagonal en tries is α x ( P n ), with a ll o t her entries b eing 0. F o r eac h LEIBNIZ SEMIN ORMS 41 x ∈ G (o r G/H ) we set N n,q x ( f , T ) = k dia g ( α x ( P ))( f ( x ) ⊗ I n − T ) k for an y ( f , T ) ∈ M q ( A ⊕ B n ). It is easily seen that N n,q x is strongly Leibniz. W e then set N n,q σ ( f , T ) = sup { N n,q x ( f , T ) : x ∈ G } . Then w e set N n,q ( f , T ) = γ − 1 N n,q σ ( f , T ) , where γ remains to b e ch osen for each n . Finally we set L n,q ( f , T ) = L q A ( f ) ∨ L n,q B ( T ) ∨ N n,q ( f , T ) ∨ N ∗ n,q ( f , T ) . It is easily v eriﬁed that the family { L n,q } is a “matrix L ipschitz semi- norm” as deﬁned in deﬁnition 3 . 1 of [40]. W e w ould lik e to c ho ose γ in suc h a w ay that the quotients of L n,q on M q ( A ) and M q ( B n ) are L q A and L n,q B . W e consider the quotien t on M q ( A ) ﬁrst. W e note, as do es W u, that ˘ σ n giv es, by en try-wise application, a unital completely p ositiv e map from M q ( A ) to M q ( B n ). Giv en f ∈ M q ( A ), w e set T = ˘ σ n f . The n, m uc h as in Section 8, N n,q x ( f , T ) =      α x ( P n )( f ij ( x ) I n − d Z f ij ( y ) α y ( P n ) dy      , where {·} denotes a matrix. As in Section 8, the translation-inv a riance of L q A and the arbit r a riness of f p ermit us to consider just the case in whic h x = e . Then, with manipulations a s in Section 8, we see that N n,q e ( f , T ) ≤ d Z k{ f ij ( e ) − f ij ( y ) }kk diag( P n α y ( P n )) k dy ≤ L q A ( f ) Z ρ ( e, y ) d k P n α y ( P n ) k dy = L q A ( f ) γ A n , where γ A n is deﬁned at the b eginning of Section 9 . Th us if γ ≥ γ A n then the quotien t of L n,q on M q ( A ) will b e L q A , whic h is exactly the same condition as fo r the case of q = 1 treated in Section 8. W e now consider the quotien t on M q ( B n ). Giv en T ∈ M q ( B n ), w e set f = σ n T . Then N n,q x ( f , T ) = k{ α x ( P )(tr( α x ( P ) T ij ) I n − T ij ) }k . I don’t see a go o d wa y t o estimate this except b y the en try-wise esti- mate ≤ q sup i,j k α x ( P )(tr( α x ( P ) T ij ) I n − T ij ) k 42 MARC A. RIEFFEL ≤ q γ B n sup i,j L B n ( T ij ) ≤ q γ B n L n,q B ( T ) , where γ B n is deﬁned at t he b eginning of Section 12, and where we hav e used the α - in v ariance of L B n , and the f a ct that for an y R ∈ M q ( B n ) w e ha ve k R k ≤ q sup i,j {k R ij k} . (T o see t his la t t er, express R as the sum of the q matrices whose only no n- zero entries a re the en tries R ij of R for whic h i − j is constan t mo dulo q .) Th us if γ ≥ q γ B n then the quotien t of L n,q on M q ( B n ) will b e L n,q B . The factor of q in this estimate has the quite undesirable eﬀect t ha t we seem not to b e able to say that for a suﬃcien t ly large γ it is true that for all q sim ultaneously t he quotien t of L n,q on M q ( B n ) is L n,q B . Th us the family { L n,q } can not b e used to estimate the “ quantized Gromov–Hausdorﬀ distance” deﬁned b y W u in deﬁnition 4 . 5 of [4 0 ]. But for ﬁxed q we will still hav e that q γ B n con v erg es to 0 a s n → ∞ , and this may still b e useful, for instance in dealing with ve cto r bundles along the lines discuss ed in [31]. According to W u’s deﬁnition of “quan tized Gromov –Hausdorﬀ dis- tance” w e mus t now sho w that U C P q ( A ) and U C P q ( B n ) are within suitable neighborho o ds of eac h other in U C P q ( A ⊕ B ) (once w e ha v e c hosen γ ≥ γ A n ∨ q γ B n ). G iven f ∈ M q ( A ) and ϕ ∈ U C P q ( A ) (whic h W u denotes b y C S q ( A )), let hh ϕ, f ii denote the elemen t of M q 2 ( C ) whose en tries are the ϕ ij ( f k l )’s. (See 1 . 1 . 27 of [10].) Equiv alen tly , view f as in M q ⊗ A , and let ˜ ϕ = I q ⊗ ϕ so that ˜ ϕ : M q ⊗ A → M q ⊗ M q . Then hh ϕ , f i i = ˜ ϕ ( f ). W e can thus use L q A to deﬁne a metric, D L q A , on U C P q ( A ), deﬁned by D L q A ( ϕ 1 , ϕ 2 ) = sup {khh ϕ 1 , f i i − hh ϕ 2 , f i ik : f ∈ M q ( A ) , L q A ( f ) ≤ 1 } . (See prop osition 3 . 1 of [39].) W u show s tha t the top ology on U C P q ( A ) from the metric D L q A coincides with the p oin t-no r m top ology . In the same w ay L n,q B deﬁnes a metric on U C P q ( B n ), and L n,q deﬁnes a met- ric on U C P q ( A ⊕ B n ). F urthermore, when we view U C P q ( A ) and U C P q ( B n ) as subsets of U C P q ( A ⊕ B n ), the r estriction of D L n,q to them will a gree with D L q A and D L n,q B if the quotients of L n,q on M q ( A ) and M q ( B n ) agree with L q A and L n,q B . (See prop osition 3 . 6 of [40 ].) W e no w sho w that U C P q ( A ) is in a suitably small neighborho o d of U C P q ( B n ) for D L q n . Lemma 14.2. F or a ny ( f , T ) ∈ M q ( A ⊕ B n ) we have k f − σ n T k ≤ q N n,q σ ( f , T ) . Pr o of. k f − σ n T k = sup x k{ f ij ( x ) − tr( α x ( P ) T ij ) }k LEIBNIZ SEMIN ORMS 43 ≤ q sup x,i,j | tr( α x ( P )( f ij ( x ) I n − T ij )) | ≤ q sup x,i,j k α x ( P )( f ij ( x ) I n − T ij ) k ≤ q sup x k{ α x ( P )( f ij ( x ) I n − T ij ) }k = q N n,q σ ( f , T ) .  W e can now pro ceed muc h as in the ﬁrst ha lf of W u’s pro of of theo- rem 8 . 6 of [40]. Let q b e ﬁxed, and no w set γ n = γ A n ∨ q γ B n in t he deﬁ- nition of L n,q , so that L n,q has the righ t quotien ts. Let ϕ ∈ U C P q ( A ) b e giv en. Set ψ = ϕ ◦ σ n , so that ψ ∈ U C P q ( B n ). Supp ose that ( f , T ) ∈ M q ( A ⊕ B n ) and that L q n ( f , T ) ≤ 1, so that N n,q σ ( f , T ) ≤ γ n . Then b y Lemma 14.2 khh ϕ, f ii − hh ψ , T iik = khh ϕ, f − σ n T iik ≤ k f − σ n T k ≤ q N n,q σ ( f , T ) ≤ q γ n . Th us U C P q ( A ) is in the q γ n -neigh b orho o d of U C P q ( B n ). Since γ A n ∨ q γ B n con v erg es to 0 as n → ∞ we can mak e q γ n as small as desired by c ho o sing n large enough. W e no w show that U C P q ( B n ) is in a suitably small neigh b o rho o d of U C P q ( A ). W e can pro ceed as in the second half of W u’s pro of of his theorem 8 . 6 o f [40]. Let ψ ∈ U C P q ( B n ) b e giv en. Set ϕ = ψ ◦ ˘ σ n , so that ϕ ∈ U C P q ( A ). F or L ( f , T ) ≤ 1 as ab ov e w e hav e, muc h as in the pro of of Prop osition 8.5, khh ϕ, f ii − hh ψ , T iik = khh ψ , ˘ σ n f − T iik ≤ k ˘ σ n f − T k ≤ k ˘ σ n f − ˘ σ n ( σ n T ) k + k ˘ σ n ( σ n T ) − T k ≤ k f − σ n T k + k ˘ σ n ( σ n T ) − T k ≤ q γ n + k ˘ σ n ( σ n T ) − T k . W e can deal with the second of these terms m uc h as w e do in Section 13, just as W u do es. One then sees that for a giv en ε > 0 one can (for a ﬁxed q ) c ho ose N large enough that U C P q ( A ) and U C P q ( B n ) are in eac h ot her’s ε -neigh b orho o d fo r n ≥ N . 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