Noncommutative geometry as a functor

Noncomm utativ e geometry as a functor Igor Nik olaev ∗ Abstract In this note the noncomm utative g eometry is interpreted as a functor, whose range is a fa mily of the operator algebras. Some examples are giv en and a p rogram is sketc hed. Key wor ds and phr ases: functors, op er ator algebr as AMS (MOS) Subj. Class.: 18D; 46L In tro duc tion A p oint ma de in this note is that s ome noncommutativ e spa ces (i.e C ∗ -algebra s, Banach or asso cia tive algebr as) can b e viewed as a genera lized homology in the sense that there exis t functors with the range in the noncommutativ e spaces. The doma in of the functors can be any int er e sting ca tegory , e.g. the Haus- dorff s paces, manifolds, Riemann s urfaces, etc. W e shall give examples of such functors. The ab ove functors ha ve a long histo ry , rather natural and familia r to spe- cialists. A foundationa l example is given b y the Gelfand-Na imark functor, which maps the category of the Hausdorff s pa ces to the category o f commut ative C ∗ - algebras . It was conjectured by No vikov a nd prov ed by Kaspa rov [7] & Mis- chenk o [8], that in man y cases the higher signature s of smo oth n -dimensional manifolds are inv ar iants o f a certain cla ss of the C ∗ -algebra s. The resp ective functor is known a s an assembly ma p. In dy na mics, Cun tz & Krieg er [1] con- structed a functor from the categ ory of top olog ical Ma rko v c hains to a catego ry of the C ∗ -algebra s (Cuntz-Krieger algebra s). There ar e many more examples to add to the list. As long as a functor is constructed, one can calculate the noncommutativ e inv ariants attached to it. On the face of it, the C ∗ -algebra s are a wa y mor e complex than the ab elian gr o ups. Howev er, many imp or tant families of the op erator algebr a s hav e b een lately clas sified in terms of the alg ebraic K -theory [4] and more developmen ts will app ear in the future. The inv ariants of the C ∗ - algebras pro duce new (and old) inv ariants of the o b jects in the initial ca tegory . ∗ Pa rtial ly supported by NSER C. 1 Thu s, the problem o f a n int er pretation of the nonco mm utative inv aria n ts in terms of the initial category a rises. In rela tion to the traditional par ts of noncommutativ e geometry (e.g. the index theory , cyclic coho mology , quantum groups , etc), the functorial approach means a switch from a ‘romantic’ to ‘pra gmatic’ rela tionship, in the sense that the noncommutative spaces b ecome a to olkit in the study of the classica l spaces. The problem has tw o parts: (i) to map a given category into a family of the noncommutativ e spaces and (ii) to prove that the mapping is a functor. Note that (ii) is the hardest par t of the problem. The note is o rganized as follows. In sectio n 1 some examples of functors with the range in a categor y o f the opera tor algebras ar e considered a nd their noncommutativ e inv ariants ar e analyzed. In se c tion 2 draft of a progr am is sketc hed. Ac kno wledgm en ts. I a m grateful to W olfgang Krieger for useful discussions and Ryan M. Rohm for helpful remar ks on the first dr aft of the note. 1 Three examples In this se c tio n, some examples of functors with the range in a family of the noncommutativ e spaces are given. In the tw o of three cases, the functors are non-injective. The list is by no means complete and the reader is enco uraged to add examples of his own. 1.1 Gelfand and Naimark functor A. This is a foundational example. Let X be a lo cally co mpact Hausdorff space. By C ( X ) one understands a co mm utative C ∗ -algebra of all functions f : X → C , which v anis h a t infinit y . The no rm on C ( X ) is the supremum nor m. Recall that every p oint x ∈ X can b e though t of as a linear multiplicativ e functiona l ˆ x : C ( X ) → C . The Gelfand transform F : X → C ( X ) is defined by the form ula x 7→ f , where f ∈ C ( X ) is s uch that ˆ x ( f ) = f ( x ). B. Let h : X → Y b e a cont inuous map b etw een the Hausdor ff spaces X and Y . It c an b e ea sily s hown that the map h ∗ = F − 1 ◦ h ◦ F is a homomorphism from the C ∗ -algebra C ( Y ) to C ( X ). In other w or ds, F is a contra v ariant functor from the lo cally co mpa ct Hausdorff s paces to the commutativ e C ∗ -algebra s: ❄ ❄ ✛ ✲ C ( X ) C ( Y ) X Y F F homomorphism continuous map 2 C. Note that F is an injective functor. The functor F do es not pr o duce new in v ariants of the Hausdorff spa ces, because of the following is omorphism: K alg ( C ( X )) ∼ = K top ( X ), where K alg and K top are the algebra ic and the top o- logical K -theo r y , r esp ectively . 1.2 Anoso v automorphisms of a t wo-dim ensional torus A. Let us consider a non- tr ivial applicatio n of the op erator algebras to a pro blem in top olo gy . Recall that an automor phism φ : T 2 → T 2 of the tw o-dimens io nal torus is called Anosov , if it is g iven by a matr ix A φ =  a 11 a 12 a 21 a 22  ∈ GL (2 , Z ), such that | a 11 + a 22 | > 2. W e w is h to construct a functor (an assembly map) µ : φ 7→ A φ , suc h tha t for every h ∈ Aut ( T 2 ) the following diagr am co mm utes: ❄ ❄ ✲ ✲ A φ ⊗ K µ µ A φ ′ ⊗ K , φ φ ′ = h ◦ φ ◦ h − 1 isomorphism conjugacy where A φ is an AF - algebra and K is the C ∗ -algebra of compact op erator s on a Hilber t space. In other words, if φ, φ ′ are conjugate automorphisms, then the AF -algebr a s A φ , A φ ′ are sta bly isomor phic. B. The map µ : φ 7→ A φ is as follows. F or simplicity , let a 11 + a 22 > 2. Note that without lo ss of gener ality , one can assume that a ij ≥ 0 for a pr op er basis in the ho mology group H 1 ( T 2 ; Z ). Consider an AF -algebr a, A φ , given by the following p erio dic Bra tteli diagram: ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜   ❅ ❅     ❅ ❅ ❅ ❅     ❅ ❅ ❅ ❅     ❅ ❅ ❅ ❅ . . . . . . a 11 a 11 a 11 a 12 a 12 a 12 a 21 a 21 a 21 a 22 a 22 a 22 A φ =  a 11 a 12 a 21 a 22  , Figure 1: The AF -algebr a A φ . where a ij indicate the multiplicit y of the resp ective edges o f the graph. W e encourag e the reader to verify that µ : φ 7→ A φ is a correctly defined function on the set of Anosov automo r phisms given by the hyper b olic matrices with the 3 non-negative entries. Note that µ is not injective, since φ and all its pow er s map to the same AF -alg ebra. C. Let us show that if φ, φ ′ ∈ Aut ( T 2 ) are the conjuga te Anosov auto- morphisms, then A φ , A φ ′ are the stably isomorphic AF -alg ebras. Indeed, let φ ′ = h ◦ φ ◦ h − 1 for a n h ∈ Aut ( X ). Then A φ ′ = T A φ T − 1 for a matrix T ∈ GL (2 , Z ). Note that ( A ′ φ ) n = ( T A φ T − 1 ) n = T A n φ T − 1 , where n ∈ N . W e shall use the following criterio n ([2], Theorem 2.3): the AF -a lgebras A , A ′ are stably isomorphic if and only if their Bratteli diagrams contain a common blo ck of an arbitrar y length. Consider the following sequences of matrices: A φ A φ . . . A φ and T A φ A φ . . . A φ T − 1 , which mimic the B ratteli diagrams o f A φ and A φ ′ . Letting the num b er of blo c ks A φ tend to infinit y , w e conclude that A φ ⊗ K ∼ = A φ ′ ⊗ K . D. The conjuga c y problem for the Ano s ov a utomorphisms can now be recast in terms of the AF -algebr as: find in v ariants of the stable isomorphis m c lasses of the stationary AF - a lgebras. One such inv aria nt is due to Handelman [5]. Consider an eig env a lue pro blem for the hyperb olic matrix A φ ∈ GL (2 , Z ): A φ v A = λ A v A , where λ A > 1 is the Perron-F rob enius eig env a lue and v A = ( v (1) A , v (2) A ) the corres p o nding eigenv ector with the p o sitive entries normalized so that v ( i ) A ∈ K = Q ( λ A ). Denote by m = Z v (1) A + Z v (2) A a Z -mo dule in the num b er field K . Recall that the co efficient ring, Λ, o f mo dule m c o nsists of the elements α ∈ K such that α m ⊆ m . It is known that Λ is an order in K (i.e. a subring of K containing 1 ) and, with no r estriction, o ne can assume that m ⊆ Λ . It fo llows from the definition, that m coincides with an idea l, I , whos e equiv a lence cla ss in Λ we s hall denote by [ I ]. It ha s b een proved by Handelman, that the triple (Λ , [ I ] , K ) is an arithmetic inv ar iant of the stable isomo rphism class of A φ : the A φ , A φ ′ are stably isomor phic AF - algebra s if and only if Λ = Λ ′ , [ I ] = [ I ′ ] and K = K ′ . It is interesting to compar e the op er ator algebr a inv ar ia nt s with those obtained in [9]. E. Let M φ be a ma pping tor us of the Anosov automorphism φ , i.e. a three- dimensional manifold { T 2 × [0 , 1] | ( x, 0) 7→ ( φ ( x ) , 1) ∀ x ∈ T 2 } . The M φ is known as a solvmanifold , since it is the quotien t space of a solv able Lie gro up. It is an easy exercise to show that the homotopy cla sses o f M φ are bijectiv e with the conjugacy classes of φ . Thus, the noncommutativ e inv ariant (Λ , [ I ] , K ) is a homotopy inv aria n t o f M φ . 1.3 Complex tor i and Effros-Shen algebras A. Let us consider an application of the op erator algebra s to a pro blem in conformal geometry . Let τ ∈ H := { z ∈ C | I m ( z ) > 0 } b e a co mplex nu mber. Recall that the quotien t space E τ = C / ( Z + Z τ ) is called a c omplex torus . It is well-known that the complex tori E τ , E τ ′ are iso morphic, whenever τ ′ ≡ τ mod S L (2 , Z ), i.e. τ ′ = a + bτ c + dτ , wher e a, b, c, d ∈ Z a nd ad − bc = 1. B . Let 0 < θ < 1 b e a n irra tional n umber g iven by the r egular co nt inued 4 fraction: θ = a 0 + 1 a 1 + 1 a 2 + . . . By the Effr os-Shen algebr a [3], A θ , one understands an AF -algebr a given by the Bratteli diag ram: ❜ ❜ ❜ ❜ ❜ ❜ ❜  ❅ ❅ ❅  ❅ ❅   . . . . . . a 0 a 1 Figure 2: The E ffros-Shen algebr a A θ . where a i indicate the n umber of edges in the upp er row o f the diag ram. Recall that A θ , A θ ′ are sa id to b e stably isomor phic if A θ ⊗K ∼ = A θ ′ ⊗K . It is known that A θ , A θ ′ are stably isomorphic if θ ′ ≡ θ mod S L (2 , Z ). Comparing the categories of complex tori and Effros-Shen algebras, one cannot fail to observe that for the generic o b jects, the corr e sp o nding morphism ar e isomor phic as groups . Let us show that the obser v a tion ha s a ground – there exists a funct o r , F , which makes the following diagram commute: ❄ ❄ ✲ ✲ A θ F F A θ ′ E τ E τ ′ stably isomorphic isomorphic C. T o construct the map F : E τ 7→ A θ , w e shall use a Hubbard-Masur home- omorphism h : H → Φ T 2 , where Φ T 2 is the space o f measur ed foliations o n the t wo-torus [6]. Each mea sured foliatio n F µ θ ∈ Φ T 2 lo oks like a fa mily of the pa rallel lines of a slope θ endowed with an inv ariant tr ansverse measure µ (Fig.3). If φ is a closed 1 -form on T 2 , then the tra jectories of φ define a mea- sured foliation F µ θ ∈ Φ T 2 and vice versa. It is not hard to see that µ = R γ 1 φ and θ = R γ 2 φ/ R γ 1 φ , wher e { γ 1 , γ 2 } is a ba sis in H 1 ( T 2 ; Z ). Denote by ω N an inv a riant (N´ eron) differential of the complex torus C / ( ω 1 Z + ω 2 Z ). It is well known that ω 1 = R γ 1 ω N and ω 2 = R γ 2 ω N . Let π be a pro jection acting b y the formula ( θ, µ ) 7→ θ . The assembly map F is given by the comp ositio n F = π ◦ h , where h is the Hubbar d-Masur ho meomorphism. In other words, the assembly 5 map E τ 7→ A θ can b e written explicitly as: E τ = E ( R γ 2 ω N ) / ( R γ 1 ω N ) h 7− → F R γ 1 φ ( R γ 2 φ ) / ( R γ 1 φ ) π 7− → A ( R γ 2 φ ) / ( R γ 1 φ ) = A θ . ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ Figure 3: The measured foliation F µ θ on T 2 = R 2 / Z 2 . D. Let us s how that the map F is a cov ariant functor. Indeed, an isomorphis m E τ → E τ ′ is induced by an automo rphism ϕ ∈ Aut ( T 2 ) of the two-torus. Let A ϕ =  a 11 a 12 a 21 a 22  ∈ GL (2; Z ) be a matrix realizing such an automor phis m. F ro m the formulas for F , one gets τ ′ = ( R cγ 1 + dγ 2 ω N ) / ( R aγ 1 + bγ 2 ω N ) = c + dτ a + bτ and θ ′ = ( R cγ 1 + dγ 2 φ ) / ( R aγ 1 + bγ 2 φ ) = c + dθ a + bθ . Thus, F s ends isomorphic complex tor i to the stably isomor phic Effros-Shen algebra s. Mor eov er, the for m ulas imply that F is a cov ar iant functor. Note, that since F c ontains a pro jective map π , F is not an injective functor. E. Finally , le t us consider a noncommut ative in v ariant coming from the functor F . The E C M is said to hav e a c omplex multiplic ation , if the e ndo morphism ring of the lattice Z + Z τ exceeds Z . It is an eas y exercise to show (in vie w of the explicit fromulas for F ) that F ( E C M ) = A θ , wher e θ is a quadratic irrationa lit y . In this case the co ntin ued fraction of θ is even tually p erio dic and we let r be the le ngth of the minimal perio d of θ . Clear ly , the in teger r is a n inv a riant o f the s table isomor phis m class of the AF -alg ebra A θ . T o interpret the noncommutativ e inv a riant r in terms of E C M , reca ll that E C M is iso morphic to a pro jective elliptic curve defined o ver a subfield K = Q ( j ( E C M )) of C , where j ( E C M ) is the j -in v a riant. It is known that the K - r ational po ints of E C M make an ab elian gro up, w ho se infinite part has r ank R ≥ 0. W e conc lude by the following Conjecture 1 F or every el li ptic curve with a c omplex multiplic ation R = r − 1 . 2 Sk etc h of a program One ca n o utline a program by indica ting: (i) an ob ject of s tudy , (ii) a typical problem and (iii) a set of exercis e s. A functoria l noncommutativ e ge o metry (FNCG) ca n b e desc r ib ed as follows. Obje ct of study. The FNCG studies non-trivial functor s fro m a catego r y of the classica l ob jects, G , to a catego ry of the noncommutativ e spaces (ope r ator, 6 Banach or asso cia tive algebras), A . The functor can b e non-injective. The category A is (p ossibly) endow ed with a go o d set of in v ariants. T ypic al pr oblem. The main problem of FNCG is constructio n of new inv ari- ants of the ob jects in G from the k nown noncomm utative in v ar iants o f A . A reconstructio n o f the classical inv aria nt s from the nonco mmutative inv aria nts is regar ded as a par tial solution of the ma in pro blem. Exer cises. Let A b e a catego ry of: (i) the U H F algebr as; (ii) the Cuntz-Krieger algebras O A with det ( A ) = ± 1. Find a catego ry G corres po nding to A and solve the typical proble m. (Hin t: for the C untz-Krieger a lgebras of type (ii), the categor y G co nsists o f the homotopy classes of the torus bundles M A ov er S 1 with the mo no dromy g iven by the matrix A .) References [1] J. Cuntz a nd W. K rieger, A class of C ∗ -algebra s and top ologica l Markov chains, Inven t. Math. 5 6 (1980), 251-26 8. [2] E. G. Effros, Dimensio ns a nd C ∗ -Algebras, in: Co nf. Bo ard of the Ma th. Sciences No.46 , AMS (19 81). [3] E. G. E ffros and C. L. Shen, Approximately finite C ∗ -algebra s and con- tin ued fractions, Indiana Univ. Math. J . 29 (19 80), 191-2 04. [4] G. A. Elliott a nd A. S. T oms , Regularity prop erties in the classification progra m for separable a menable C ∗ -algebra s, B ull. Amer. Ma th. So c. 4 5 (2008), 229 -245 [5] D. Handelman, Positiv e matrices and dimensio n g roups affiliated to C ∗ - algebras and top olog ic al Ma rko v chains, J. Op er ator Theory 6 (1981 ), 55-74 . [6] J. Hubbar d and H. Masur, Q uadratic differentials a nd fo liations, Acta Math. 1 4 2 (1979), 22 1-274 . [7] G. Kaspa r ov, T o po logical in v a riants of elliptic oper ators I: K -homology , Izv. Ak ad. Nauk SSSR, Ser. Math. 39 (1975), 796 -838. [8] A. S. Mischenk o, Infinite-dimensional represe n tatio ns of discrete groups and higher sig natures, Izv. Ak ad. Nauk SSSR, Ser. Math. 38 (197 4), 81-10 6. [9] D. I. W allace, Conjugacy classes of hyperb olic matrices in S L ( n, Z ) and ideal class es in an order , T ra ns. Amer. Math. So c. 283 (198 4 ), 177-18 4. 7 The Fields Institute f or Ma thema tical Sciences, Toronto, ON, Canada, E-mail: igor.v.nik ol aev@gmail.com 8