Connes metric for states in group algebras

Connes’ metric for states in group algebras ∗ Esteban Andruc ho w and Gabriel Larotonda Abstract In this article we follow the main idea of A. Connes for the co nstruction of a metric in the state space of a C ∗ -algebra . W e fo cus in the r educed a lgebra of a discr ete g roup Γ , and prov e s o me equiv alences and re la tions betw een t wo cen tral ob jects of this ca tegory: the word-length growth (connected with the degree of the extension of Γ when the group is an e x tension of Z Z b y a finite group), and the top olog ical equiv alence b etw een the ω ∗ top ology and the one introduced with this metric in the state space of C ∗ r (Γ). Keyw ords: group C ∗ -algebra , state spa ce, non commutativ e metric spa ce. 1 In tro duc tion In [Connes1] and [Co nnes 2], A. Connes int ro duce d what he called no n commutativ e metric s pa ces, which co nsist of a triples ( A , D , H ) where A is a C ∗ -algebra , a cting on the Hilb ert space H , and D is an unbounded oper a tor in H, called the Dira c op erator, satisfying • ( D 2 + 1) − 1 is compact • the set { a ∈ A : [ D , a ] is bou nded } is nor m- dense in A W e a re interested in the c a se when Γ is a discrete gr oup with ident ity element e a nd the algebra A is the reduced C ∗ -algebra C ∗ r (Γ). The Hilbe r t space is ℓ 2 (Γ), with C ∗ r (Γ) acting as left convoluters (i.e. the left regula r representation). The Dir a c op era tor is defined in terms of a leng th function on Γ. A length function is a map L : Γ → I R + satisfying 1. L ( g h ) ≤ L ( g ) + L ( h ) for a ll g , h ∈ Γ. 2. L ( g − 1 ) = L ( g ) for all g ∈ Γ . 3. L ( e ) = 0 . If Γ is given b y genera tors a nd relatio ns, the pro to t ypica l leng th function is the map which assigns to ea ch w or d its (m inimal) length. W e shall fix this data L , a nd we will make the further assumption that the sets { g ∈ Γ : L ( g ) ≤ c } are finite for an y c > 0. The Dirac op erator [Connes2] is then defined as follows: D ( δ g ) = L ( g ) δ g ∗ 2000 Mathematics S ub ject Classification: 46L30, 46L05, 46L85. 1 where { δ g : g ∈ Γ } is the canonical orthonor mal basis of ℓ 2 (Γ). As is custom, we s ha ll denote by λ g the element δ g regar ded a s an o pe rator in ℓ 2 (Γ). The metric (of the no n commutativ e metric space) is defined in the state space S ( C ∗ r (Γ)) of C ∗ r (Γ) by means of the fo r mula d ( ψ , ϕ ) = sup { | ψ ( a ) − ϕ ( a ) | : a ∈ C ∗ r (Γ) with k [ a, D ] k ≤ 1 } . Here [ , ] denotes the usual conmutator of oper ators. This d is not necessar ily finite. In this note we study situations in which it is finite, and consider a problem p ose d by M. Rieffel, as k ing under which assumptions the metric th us defined induces on the state s pace a topolog y which is equiv alen t to the w ∗ top ology . The basic ex ample of this situation, which even justifies the na me ” non commutative metric space”, o ccurs when A is C ( M ), the a lg ebra of cont inuous functions on a spin manifold M [Connes2], [GL]. M Rieffel found [Rieffel] a natura l triple ass o ciated to the noncommutativ e tor i. Also he po int ed out that one can find a p os itive answer for matrix a lgebras. In this note we consider this problem for gr oup algebr as arising fr o m discrete gro ups and triples arising from length functions. Instead of dea ling with the d metric directly , we refer it to tw o metrics, d ∞ and d 2 , r elated with the a symptotic b ehaviour of the family { 1 L ( g ) : e 6 = g ∈ Γ } : d ∞ ( ϕ, ψ ) = sup g 6 = e | ϕ ( λ g ) − ψ ( λ g ) | L ( g ) , (1.1) and d 2 ( ϕ, ψ ) =  X g 6 = e | ϕ ( λ g ) − ψ ( λ g ) | 2 L ( g ) 2  1 / 2 . (1.2) First note that d ∞ is a well defined metric and that d ∞ ( ϕ, ψ ) ≤ d ( ϕ, ψ ). T he first fact is appar ent. T o pr ove the second, no te that k [ D , λ g ] k = L ( g ), and therefor e d ∞ ( ϕ, ψ ) = sup a = 1 L ( g ) ,g 6 = e | ϕ ( a ) − ψ ( a ) | ≤ d ( ϕ, ψ ) . Also note that d 2 may f ail to b e finite. Indeed, consider Γ = Z Z × Z Z. Then the family { 1 L ( g ) : g 6 = e } do es not b elong to ℓ 2 (Z Z × Z Z). Consider the p ositive definite functions f ( g ) = 1 for a ll g and h = δ e . These functions induce s tates ϕ f and ϕ h on C ∗ r (Z Z × Z Z ) satisfying ϕ f ( λ g ) = f ( g ) and ϕ h ( λ g ) = h ( g ). It follows tha t d 2 ( ϕ f , ϕ h ) = X e 6 = g ∈ Z Z × Z Z 1 L ( g ) 2 = ∞ . Denote by K (Γ) the g roup algebra of Γ, i.e. the set of elements o f the form P g ∈ F α g λ g , wher e α g ∈ C and F ⊂ Γ is a finite set. Lemma 1 .1 d ∞ is a met ric in S ( C ∗ r (Γ)) which induc es a top olo gy e quivale nt t o the w ∗ -top olo gy. Pro of. If d ∞ ( ϕ n , ϕ ) → 0, then cle a rly ϕ n ( λ g ) → ϕ ( λ g ) for all g 6 = e . Since ϕ n , ϕ ar e s ta tes, ϕ n ( λ e ) = ϕ n (1) = 1 = ϕ ( λ e ). It follows that ϕ n ( a ) → ϕ ( a ) fo r all a ∈ K (Γ). Since ϕ n , ϕ hav e their nor ms b ounded (by 1), and since K (Γ) is dens e in C ∗ r (Γ), it follows that ϕ n → ϕ in the w ∗ top ology . Conv ersely , supp ose tha t ϕ n ( a ) → ϕ ( a ) for a ll a ∈ C ∗ r (Γ) a nd fix ǫ > 0. Let F = { g ∈ Γ : L ( g ) < 4 /ǫ } , which is a finite set, say F = { g 1 , ..., g k } . If g ∈ F , o ne ha s | ϕ n ( λ g ) − ϕ ( λ g ) | L ( g ) ≤ | ϕ n ( λ g ) | + | ϕ ( λ g ) | 4 /ǫ ≤ ǫ/ 2 . 2 On the o ther hand, there ex ists n 0 such that for a ll n ≥ n 0 , | ϕ n ( λ g i ) − ϕ ( λ g i ) | < ǫ 2 min { L ( g 1 ) , ..., L ( g k ) } , for i = 1 , ..., k . It follows that | ϕ n ( λ g i ) − ϕ ( λ g i ) | L ( g i ) < ǫ/ 2 . Therefore, if n ≥ n 0 , then sup g 6 = e | ϕ n ( λ g ) − ϕ ( λ g ) | L ( g ) = d ∞ ( ϕ n , ϕ ) → 0 . ✷ 2 Comparison b et w een d , d ∞ and d 2 Here we esta blish the basic inequa lit y for thes e metrics, namely d ∞ ≤ d ≤ d 2 . Lemma 2 .1 L et a = P g ∈ F α g λ g ∈ K (Γ) , then k [ D , a ] k ≥  X g ∈ F | α g | 2 L ( g ) 2  1 / 2 . Pro of. Note that [ D , a ] δ e = X g ∈ F α g [ D , λ g ] δ e = − X g ∈ F α g L ( g ) δ g , bec ause D λ g δ e = D δ g = L ( g ) δ g , and in pa rticular D δ e = 0. Therefore k [ D , a ] δ e k 2 2 = X g ∈ F | α g | 2 L ( g ) 2 . It follows that k [ D , a ] k ≥ k [ D , a ] δ e k 2 =  P g ∈ F | α g | 2 L ( g ) 2  1 / 2 . ✷ Prop ositio n 2.2 d ∞ ( ϕ, ψ ) ≤ d ( ϕ, ψ ) ≤ d 2 ( ϕ, ψ ) . Pro of. Pick a = P g ∈ F α g λ g ∈ K (Γ), with k [ D , a ] k ≤ 1 (note that for a ny a ∈ K (Γ), [ D, a ] is a bo unded op era to r). Then | ϕ ( a ) − ψ ( a ) | = | X g ∈ F α g ( ϕ ( λ g ) − ψ ( λ g )) | = | X e 6 = g ∈ F α g L ( g ) ( ϕ ( λ g ) − ψ ( λ g )) L ( g ) | , which by the Ca uch y-Sch wartz inequa lity is less than o r equal to  X e 6 = g ∈ F | α g | 2 L ( g ) 2  1 / 2  X e 6 = g ∈ F | ϕ ( λ g ) − ψ ( λ g ) | 2 L ( g ) 2  1 / 2 ≤ k [ D , a ] k d 2 ( ϕ, ψ ) ≤ d 2 ( ϕ, ψ ) . The pr o of finishes by o bserving that the set of element s a ∈ K (Γ) with k [ D , a ] k ≤ 1 is dense among element s b ∈ C ∗ r (Γ) with k [ D, b ] k ≤ 1. Indeed, let b = P g ∈ Γ β g λ g ∈ C ∗ r (Γ) with k [ D, b ] k ≤ 1. F or finite sets F ⊂ Γ, the truncated e lement s b F = P g ∈ F β g λ g ∈ K (Γ) conv erg e in nor m to b . Clearly als o the (b ounded) commutan ts [ D , b F ] c onv erge in norm to [ D , b ]. Denote by N F = k [ D , b ] kk [ D , b F ] k − 1 (after droping the elements b F with [ D , b F ] = 0 ). Then N F b F lies in K (Γ), the c o mmu tants [ D , N F b F ] hav e norm less than or equal to 1, and con verge to b . ✷ 3 W e emphasize that d 2 might b e infinite. It would b e finite if for example the family { 1 L ( g ) : e 6 = g ∈ Γ } w ould lie in ℓ 2 (Γ). This imp os es a stro ng condition o n Γ, namely , that the gro up Γ has linear growth (p olinomial gr owth with degree 1), see [Gromov] a nd [Co nnes2]. This means, that there exists co nstants k , l such tha t # { g ∈ Γ : L ( g ) ≤ c } ∼ k c + l . Example 2.3 L et us c onsider the fol lowing examples, of gr oups Γ wich satisfy that the family { 1 L ( g ) : e 6 = g ∈ Γ } lies in ℓ 2 . 1. L et Γ = Z Z . Her e the length function is L ( m ) = | m | , m ∈ Z Z . The gr oup C ∗ -algebr a e quals in this c ase C ( S 1 ) . 2. L et Γ b e a finite ex tension of Z Z , i.e. a gr oup Γ which has a c opy of Z Z inside, as a normal sub gr oup, and the quotient F = Γ / Z Z is finite. Then, as a set, Γ is Z Z × F . L et F = { f 1 , ..., f n } . Then the classes of (1 , f 1 ) , ..., (1 , f n ) (i.e. these elements r e gar de d as element s of Γ ) ar e gener ators for Γ . L et u s c onsider the length funct ion L given by wor d length with r esp e ct to this set of gener ators. Note that for this L , ther e ar e at most 2 n elements of Γ with any given length. It fol lows t hat { 1 L ( g ) : e 6 = g ∈ Γ } lies in ℓ 2 . The (r e duc e d) C ∗ -algebr a of su ch Γ c an b e c ompute d. They c onsist of algebr as of n × n matric es with entries in C ( S 1 ) , se e chapter VIII of [ Davidso n ] for a c omplete descrition of this c omputation. L et us p oi nt out two sp e cial c ases of t his typ e (a) Γ = Z Z × F with the u sual pr o duct for p airs. In this c ase the C ∗ -algebr a is C ∗ r (Z Z × F ) ≃ C ( S 1 ) ⊗ C ∗ r ( F ) . The algebr a C ∗ r ( F ) is finite dimensional, ther efor e in this c ase C ∗ r (Γ) c onsists of a dir e ct sum of ful l matrix algebr as with entries in C ( S 1 ) . In p articular, if F = S k the gr oup of p ermutations of or der k , then C ∗ r (Γ) = M k ( C ( S 1 )) . (b) Consider the (unique) n ontrivial automorphism of Z Z , θ ( m ) = − m . Then one has a Z Z 2 extension of Z Z , Γ = Z Z × θ Z Z 2 , and the c orr esp onding C ∗ -algebr a C ∗ r (Γ) is the cr oss pr o du ct C ( S 1 ) × θ Z Z 2 , which identifies with the algebr a of 2 × 2 matric es with entries in C ( S 1 ) of the form  f ( z ) g ( z ) f ( z ) g ( z )  , wher e f and g ar e c ontinuous functions in S 1 . Prop ositio n 2.4 If Γ has a length function L which satisfi es that { 1 L ( g ) : e 6 = g ∈ Γ } is squ ar e summable, then the met ric d is wel l define d (is finite) and induc es on the state sp ac e of C ∗ r (Γ) the w ∗ top olo gy. Pro of. Since { 1 L ( g ) } ∈ ℓ 2 , d ≤ d 2 < ∞ . By the a bove res ults it suffices to prov e that if a sequence ϕ n conv erges to ϕ in the w ∗ top ology , then it conv erges in the d metric. W e claim that it co nv erges in the d 2 metric. Fix ǫ > 0. There exis ts a finite s et F = { g 1 , ..., g k } such that  X g ∈ Γ − F | ϕ n ( λ g ) − ϕ ( λ g ) | 2 L ( g ) 2  1 / 2 ≤ 2  X g ∈ Γ − F 1 L ( g ) 2  1 / 2 < ǫ/ 2 . Put c = ( P k i =1 1 L ( g i ) 2 ) 1 / 2 . Ther e exists n 0 such that for all n ≥ n 0 , one has | ϕ n ( λ g i ) − ϕ ( λ g i ) | < ǫ/ 2 c . Therefore  k X i =1 | ϕ n ( λ g i ) − ϕ ( λ g i ) | 2 L ( g i ) 2  1 / 2 < ǫ/ 2 . Then d 2 ( ϕ n , ϕ ) → 0. ✷ 4 Corollary 2 .5 If Γ is a finite extension of Z Z , then in S ( C ∗ r (Γ)) the d metric is wel l define d and induc es the ω ∗ -top olo gy. 3 Normal states whic h are b ou nded with resp ect to the trace W e s hall prov e tha t the metric d is finite on the set of no rmal s ta tes of C ∗ r (Γ) whic h a re b ounded with resp ect to the trace o f C ∗ r (Γ), i.e. the states ϕ which extend to no rmal s ta tes of the V on Neumann algebr a L Γ of Γ, and v erify that there exists a constant κ > 0 suc h that ϕ ( a ∗ a ) ≤ κτ ( a ∗ a ) , or shor tly , ϕ ≤ κτ . Recall that the tra ce τ is given b y τ ( a ) = h aδ e , δ e i . There is a Ra don-Nykodim deriv ative for all such ϕ [Araki]. Namely , there exists a n e le men t ρ ϕ ≥ 0 in L Γ such that ϕ ( a ) = τ ( ρ ϕ a ) , with k ρ ϕ k ≤ κ 1 / 2 . Denote by S κ the set S κ = { ϕ ∈ S ( L Γ ) : ϕ ≤ κτ } . First note tha t a state which lies in S κ is necessa rily normal. Indeed, let { p i : i ∈ I } b e an a rbitrary family of pa irwise o rthogona l pr o jections in L Γ . Fix ǫ > 0 and let J ⊂ I b e a finite set such that τ ( P i ∈ I − J p i ) = P i ∈ I − J τ ( p i ) < ǫ/ κ . Then ϕ ( P i ∈ I − J p i ) < ǫ . There fore 0 ≤ ϕ ( P i ∈ I p i ) = P j ∈ J ϕ ( p j ) + ϕ ( P i ∈ I − J p i ) ≤ P j ∈ J ϕ ( p j ) + ǫ . That is, P i ∈ I ϕ ( p i ) = ϕ ( P i ∈ I p i ), and ϕ is no rmal. Also it is apparent that S κ is w ∗ compact and conv ex. Prop ositio n 3.1 The metrics d and d 2 ar e wel l define d on S κ and induc e the w ∗ top olo gy. Pro of. Note that if ϕ ∈ S κ then ϕ ( λ g ) = τ ( ρ ϕ λ g ) = h ρ ϕ δ g , δ e i = ρ ϕ ( g − 1 ) , where ρ ϕ ( g − 1 ) denotes the g − 1 -co ordinate o f ρ ϕ regar ded a s an element of ℓ 2 (Γ). In pa rticular, it follows that the family { ϕ ( λ g ) : g ∈ Γ } is square summa ble . Mor eov er,  X g ∈ Γ | ϕ ( λ g ) | 2  1 / 2 = k ρ ϕ k 2 ≤ k ρ ϕ k ≤ κ 1 / 2 . It follows that if ϕ, ψ ∈ S κ , them d ( ϕ, ψ ) ≤ d 2 ( ϕ, ψ ) =  X e 6 = g ∈ Γ | ϕ ( λ g ) − ψ ( λ g ) | 2 L ( g ) 2  1 / 2 ≤ 2 κ. Suppo se now that ϕ n → ϕ in the w ∗ top ology . Fix ǫ > 0. Then the s e t F = { g ∈ Γ : L ( g ) ≤ 2 κ/ǫ } is finite. Say F = { g 1 , ..., g n } . If g lies outside F one has  X e 6 = g ∈ Γ − F | ϕ ( λ g ) − ψ ( λ g ) | 2 L ( g ) 2  1 / 2 ≤ ǫ 2 κ  X g ∈ Γ − F | ϕ n ( λ g ) − ϕ ( λ g ) | 2  1 / 2 ≤ ǫ. Let C = P n i =1 1 L ( g i ) 2 . Ther e exists n 0 such that if n ≥ n 0 then | ϕ n ( λ g i ) − ϕ ( λ g i ) | < ǫ/ C for i = 1 , ..., n. It follows that d ( ϕ n , ϕ ) ≤ d 2 ( ϕ n , ϕ ) < ǫ if n ≥ n 0 . ✷ 5 Remark 3. 2 1. The first p art of t he pr o of in fact shows t hat if ϕ and ψ ar e normal states of L Γ whose R adon-Nyko dim derivatives with r esp e ct to the tr ac e τ lie in ℓ 2 (Γ) , then d ( ϕ, ψ ) ≤ d 2 ( ϕ, ψ ) < ∞ . 2. It is app ar en t that the metrics d and d 2 ar e also fin ite on the set ∪ κ> 0 S κ References [Araki] Araki, Huzihiro - Some pro pe rties of mo dular conjugation op era tor of V o n Neumann algebras and non-commutativ e Ra don-Nikodym theore m with a chain r ule, Pacific J. Math. 50 No. 2 (1974), 309- 354 [Connes1] Connes, Alain - Compact metric spaces, F redholm mo dules and hyper finiteness, Ergo d. Th. and Dyna mm. Sys (198 9), 9, p 29 7-220 . [Connes2] Connes, Alain AND Lott, John - The metric asp ect of noncomm utative geometr y . New symmetry principle s in Quantum Field Theory , J F rohlich et a l, P lenum Pres s, New Y ork (199 2) p 53-93 [Davidson] Davidson, K enneth R - C ∗ -Algebras by E xample, Fields Inst for Res in Math. Sci, AMS, 19 96. [Gromov] Gromov, M - Groups o f p olynomia l g rowth a nd expanding ma ps, Publ. Math. dl’I.H.E.s. (19 8 0) [GL] Larotonda , Gabriel - Metric a sp ects of noncommutativ e geometry , Preprint, T esis de Licenciatura, Depto. de Matemtica, FCEN, UBA, 1999. [Rieffel] Rieffel, Ma rc A - Metrics on States from action o f Compact Groups, Do cumen ta Mathematica 3 (1998), p 215-229 Instituto de Ciencias Universidad Nacional de Gral. Sarmiento J. M. Gutierrez 1150 (1613) Los Polv orines Argentina e-mail: eandruch@ungs.edu.ar, glaroton@ungs.edu .ar 6