Noncommutative Geometry and Potential Theory on the Sierpinski Gasket

Transcription

1 Noncommutative Geometry and Potential Theory on the Sierpinski Gasket Fabio Cipriani Dipartimento di Matematica Politecnico di Milano ( Joint works with D. Guido, T. Isola, J.-L. Sauvageot ) Neapolitan workshop on Noncommutative Geometry, Napoli, September 2012

2 Noncommutative Geometry underlying Dirichlet forms on singular spaces Usually one uses tools of NCG to build up hamiltonian on singular spaces Ground States; Algebraic QFT; Quantum Hall Effect; NCG Energy functionals Quasi crystals; Standard Model; Action Principle; Heat equations on foliations; reversing the point of view, our goal is to analyze the NCG structures underlying Energy functionals Compact Quantum Groups (with Franz, Kula); Energy functionals NCG Orbits of Dynamical systems (with Mauri); Fractals (with Guido, Isola, Sauvageot). Today we concentrate on a fractal set: Sierpinski gasket K where the C -algebra of observables is commutative = C(K) Energy functionals = Dirichlet forms: a generalized Dirichlet integral on K

3 Sierpinski gasket K C: self-similar compact set vertices of an equilateral triangle {p 1, p 2, p 3} contractions F i : C C F i(z) := (z + p i)/2 K C is uniquely determined by K = F 1(K) F 2(K) F 3(K) as the fixed point of a contraction of the Hausdorff distance on compact subsets of C: Duomo di Amalfi: Chiostro, sec. XIII

4 Topological, geometric and analytic features of the Sierpinski gasket K is not a manifold K does not admit a universal cover the Hausdorff dimension is non integer 1 < d H < 2 Volume and Energy are distributed singularly on K Volume Zeta functions exhibits complex dimensions We will construct families of Spectral Triples on K reproducing Hausdorff volume measure µ H Hausdorff dimension d H Euclidean geodesic metric as Connes distance Energy functional Energy dimension Volume dimension

5 Self-similar volume measures and their Hausdorff dimensions The natural volume measures on K are the self-similar ones for some fixed (α 1, α 2, α 3) (0, 1) 3 such that 3 i=1 αi = 1 K f dµ = 3 α i (f F i) dµ f C(K) i=1 K when α i = 1 for all i = 1, 2, 3 then µ is the normalized Hausdorff measure 3 associated to the restriction of the Euclidean metric on K Hausdorff dimension d H = ln 3 ln 2.

6 Harmonic structure word spaces: length of a word σ m : 0 :=, m := {1, 2, 3}m, σ := m := m 0 m iterated contractions: F σ := F i σ... F i1 if σ = (i 1,..., i σ ) vertices sets: V := {p 1, p 2, p 3}, V m := σ =m Fσ(V0) consider the quadratic form E 0 : C(V 0) [0, + ) of the Laplacian on V 0 E 0[a] := a(p 1) a(p 2) 2 + a(p 2) a(p 3) 2 + a(p 3) a(p 1) 2 Theorem. (Kigami 1986) The sequence of quadratic forms on C(V m) defined by E m [a] := ( 5 ) me0[a F σ] a C(V m) 3 σ =m is an harmonic structure in the sense that E m [a] = min{e m+1 [b] : b Vm = a} a C(V m).

7 Dirichlet form, Laplacian and Spectral Dimension Theorem 1. (Kigami 1986) The quadratic form E : C(K) [0, + ] defined by E[a] := lim m + E m [a Vm ] a C(K) is a Dirichlet form, i.e. a l.s.c. quadratic form such that E[a 1] E[a] a C(K), which is self-similar in the sense that E[a] = E[a F i] a C(K). i=1 It is closed in L 2 (K, µ) and the associated self-adjoint (Laplacian) operator µ has discrete spectrum with spectral dimension d S = ln 9 ln 5/3 dh: Weyl s asymptotics {eigenvalues of µ λ} λ d S/2 λ +.

8 Volume and Energy measures Volume can be reconstructed as residue of Zeta functionals of the Laplacian Theorem. (Kigami-Lapidus 2001) The self-similar volume measure µ with weights α i = 1/3 can be reconstructed as f dµ = Trace Dix(M f d S/2 µ ) = Res s=ds Trace(M f s/2 µ ) K Volume and Energy are distributed singularly on K Theorem. (Kusuoka 1989, Ben Bassat-Strichartz-Teplyaev 1999) Energy measures on K defined by (Le Jan 1985) b dγ(a) := E(a ab) 1 2 E(a2 b) K a, b F are singular with respect to all the self-similar volume measures on K. Energy/Volume singularity has a simple algebraic interpretation any sub-algebra contained in the domain of the Laplacian is trivial.

9 Differential calculus on K K is not a manifold so that a differential calculus has to be introduced in a unconventional way, more specifically there exists a natural differential calculus on K underlying the Dirichlet form (E, F). Theorem. (FC-Sauvageot 2003) There exists a derivation (F,, H), defined on the Dirichlet algebra F, with values in a Hilbertian C(K)-module H such that E[a] = a 2 H a F. In other words, (F,, H) is a differential square root of the Laplacian H µ: Energy measures are represented as H µ =. K b dγ[a] = ( a b a)h a F. Energy/Volume singularity has another algebraic interpretation: cyclic representations associated to vectors a H are not weakly contained in the representation L 2 (K, µ).

10 Construction of module and derivation Consider the universal derivation d : F Ω 1 (F) over the algebra F the tangent C(K)-module H is the quotient/completion of the F-bimodule Ω 1 (F) under the inner product (a b c d) = 1 2 the derivation is the quotient of the derivation d The proof uses the following facts the algebra F is dense in C(K) [ ] E(a cdb ) + E(abd c) E(bd a c) positive linear maps on the commutative C -algebra C(K) are automatically completely positive efforts are required to prove that the quotient/completion of F is not only an F-bimodule but actually a C(K)-module coincidence of the left and right actions of C(K) on H is a consequence of the strong locality of the Dirichlet form a, b F, a constant on the support of b E(a, b) = 0.

11 A first Fredholm module on K Theorem. (FC-Sauvageot 2009) Let P Proj(H) the projection onto the image Im of the derivation above PH = Im and let F := P P the symmetry with respect to Im. Then (F, H) is an (ungraded) Fredholm module over C(K) (F, H) is 2-summable and densely defined on F Trace( [F, a] 2 ) const. E[a] a F.

12 Proof By the Leibnitz rule for P ap( b) = P ( (ab) ( a)b) = P (( a)b) and P ap( b) ( a)b a, b F 0 < λ 1 λ 2 <... eigenvalues a 1, a 2, L 2 (K, µ) normalized eigenfunctions ξ k := λ 1/2 k a k PH = Im complete orthonormal basis Since the Green function G(x, y) = [F, a] 2 L 2 = 8 P ap 2 L 2 = 8 = 8 k=1 k=1 k=1 λ 1 k λ 1 k a 2 k dγ(a) = 8 K 8 G E[a] a = a F. a k(x)a k(y) is continuous on K λ 1 k P ap( a k) 2 H 8 K G(x, x)γ(a)(dx) k=1 λ 1 k ( a)a k 2 H Since F is uniformly dense in C(K), we have that [F, a] is compact for all a C(K).

13 Multipliers of the Dirichlet space To construct finer Fredholm modules over K, more sensible to the volume measure, we will use the following functions which lies at the core of the Potential Theory of the Dirichlet space (E, F). Definition. (Mokobodzki 2005) A multiplier a C(K) is a continuous function such that b F ab F. It is somewhat non trivial to exhibit non trivial multipliers: however, a tour de force in Potential Theory involving properties of finite energy measures and potentials of the Dirichlet space, allows to prove Theorem. (FC-Sauvageot arxiv: v1) The algebra M(F) of multipliers of the Dirichlet space F is dense in C(K); (I + µ) 1 b C(K) is a multiplier for all a L (K, µ); eigenfunctions are multipliers.

14 A finer Fredholm module on K Theorem. (FC-Sauvageot arxiv: v1) Let F be the symmetry with respect to the graph G( ) L 2 (K, µ) H of the divergence operator. Then (F, L 2 (K, µ) H) is a Fredholm module over C(K); it is (d S, )-summable and densely defined on the multiplier algebra M(F) Trace Dix( [F, a] d S ) 4 ds a d S M(F) TraceDix(I + µ) d S/2 a M(F) ; the energy functional E d S [a] := Trace Dix( [F, a] d S ) a M(F) is a self-similar conformal invariant in the sense that E d S [a] = 2 E d S [a F i] a M(F). i=0

15 Dirac operator associated to the Dirichlet form Definition. (FC-Sauvageot arxiv: v1) The Dirac operator associated to the Dirichlet form (E, F) is defined as ( ) 0 D = 0 acting on the C(K)-module L 2 (K, µ H) H. Even if D does not gives rise to a Spectral Triple because [D, a] bounded a = constant D helps to construct the previous Fredholm module because spectrum D = spectrum µ, away from zero [F, a] factorizes through D for all multipliers a M(F).

16 Quasi-circles We will need to consider on the 1-torus T = {z C : z = 1} structures of quasi-circle associated to the following Dirichlet forms and their associated Spectral Triples for any α (0, 1). Lemma. Fractional Dirichlet forms on a circle (CGIS arxiv: Feb 2012) Consider the Dirichlet form on L 2 (T) defined on the fractional Sobolev space E α[a] := T T a(z) a(w) 2 z w 2α+1 dzdw F α := {a L 2 (T) : E α[a] < + }. Then H α := L 2 (T T) is a symmetric Hilbert C(K)-bimodule w.r.t. actions and involutions given by (aξ)(z, w) := a(z)ξ(z, w), (ξa)(z, w) := ξ(z, w)a(w), (J ξ)(z, w) := ξ(w, z). The derivation α : F α H α associated to E α is given by α(a)(z, w) := a(z) a(w) z w α+1/2.

17 Proposition. Spectral Triples on a circle (CGIS arxiv: Feb 2012) Consider on the Hilbert space K α := L 2 (T T) L 2 (T), the left C(T)-module structure resulting from the sum of those of L 2 (T T) and L 2 (T) and the operator ( ) 0 α D α := α. 0 a(z) a(w) 2 z w 2α+1 Then A α := {a C(T) : sup z T < + } is a uniformly dense T subalgebra of C(T) and (A α, D α, K α) is a densely defined Spectral Triple on C(T).

18 Dirac operators on K. Identifying isometrically the main lacuna l of the gasket with the circle T, consider the Dirac operator (C(K), D, K ) where K := L 2 (l l ) L 2 (l ) D := D α the action of C(K) is given by restriction π (a)b := a l. Fix c > 1 and for σ consider the Dirac operators (C(K), π σ, D σ, K σ) where K σ := K D σ := c σ D α the action of C(K) is given by contraction/restriction π σ(a)b := (a F σ) l b. Finally, consider the Dirac operator (C(K), π, D, K) where K := σ K σ π := σ π σ D := σ D σ Notice that dim Ker D = + and that D 1 will be defined to be zero on Ker D.

19 Volume functionals and their Spectral dimensions Theorem. (CGIS arxiv: Feb 2012) The zeta function Z D of the Dirac operator (C(K), D, K), i.e. the meromorphic extension of the function C s Trace( D s ) is given by Z D(s) = 4 z(αs) 1 3c s where z denotes the Riemann zeta function. The dimensional spectrum is given by S dim = { 1 α } { ln 3 ( 1 + 2πi ) ln c ln 3 k : k Z} and the abscissa of convergence is d D = max(α 1, ln 3 ). When 1 < c < ln c 3α there is a simple pole in d D = ln 3 and the residue of the meromorphic extension of ln c C s Trace(f D s ) gives the Hausdorff measure of dimension d = ln 3 ln 2 Trace Dix(f D s ) = Res s=dd Trace(f D s ) = 4d z(d) f dµ. ln 3 (2π) d Notice the complex dimensions and the independence of the residue Hausdorff measure upon c > 1. K

20 Spectral Triples and Connes metrics on the Sierpinski gasket Spectral Triples on the Sierpinski Gasket constructed by [Guido-Isola 2005], [Christensen-Ivan-Schrohe arxiv: v2] were able to reproduce the volume measure. The following one gives back volume measure, distance and energy. Theorem. (CGIS arxiv: Feb 2012) (C(K), D, K) is a Spectral Triple for any 1 < c 2. In particular we have the commutator estimate for Lipschiz functions with respect to the Euclidean metric [D, a] (1/2)(1 α) (1 α) 1/2 sup σ ( c 2 )σ a Lip(lσ) a Lip(K). For c = 2 the Connes distance is bi-lipschitz w.r.t. the geodesic distance on K induced by the Euclidean metric (1 α) 1/2 2 (1 α) d geo(x, y) d D(x, y) (1 + α) 1/2 2 (3/2) 3 α d geo(x, y).

21 Energy functionals and their Spectral dimensions By the Spectral Triple it is possible to recover, in addition to dimension, volume measure and metric, also the energy form of K Theorem. (CGIS arxiv: Feb 2012) Consider the Spectral Triple (C(K), D, K) for α α 0 := ln 5 ( 1 ln 3 1) 0, 87 ln 4 2 ln 2 and assume a F. Then the abscissa of convergence of the energy functional is δ D := max(α 1, 2 C s Trace( [D, a] 2 D s ) ln 5/3 ln c ). If δ D > α 1 then s = δ D is a simple pole and the residue is proportional to the Dirichlet form Res s=δd Trace( [D, a] 2 D s ) = const. E[a] a F ; if δ D = α 1 then s = δ D is a pole of order 2 but its residue of order 2 is still proportional to the Dirichlet form. energy dimension and volume dimension differ: d D δ D.

22 Pairing with K-Theory Trying to construct a Fredholm module from the Dirac operator one may consider F := D D 1 to be the phase of the Dirac operator. ( I 0 ε 0 := σε σ 0 where ε σ 0 := 0 I ( 0 ε 1 := σε σ 1 where ε σ ivσ 1 := ivσ 0 where V σ : L 2 (l ) L 2 (l l ) is the partial isometry between Im ( σ σ) L 2 (l ) and Im ( σ σ) L 2 (l l ). ) on K σ = L 2 (l l ) L 2 (l ) ) on K σ = L 2 (l l ) L 2 (l ) However (C(K), F, K, ε 0, ε 1) is not a 1-graded Fredholm module because [F, π(a)] K(K) for all a C(K) F = F, ε 0F + Fε 0 = 0, but dim Ker F = + and (F 2 I) / K(K) ε 0 is unitary but ε 1 is a partial isometry only ε 0ε 1 + ε 1ε 0 = 0, ε 2 0 = I but ε 2 1 I [ε 0, π(a)] = 0 but [ε 1, π(a)] / K(K) in general

23 Theorem. (CGIS arxiv: Feb 2012) Consider the operator F 1 := iε 1F and the projection P := F 1+F Then [F 1, π(a)] is compact for all a C(K) PaP is a Fredholm operator for all invertible u C(K) a nontrivial homomorphism on K 1 (K) is determined by u Index PuP the nontrivial element of K 1(K) = σ Z determined by F 1 is (1, 1,... ): Index Pu σp = +1 σ where u σ is the unitary associated to the lacuna l σ.

24 Proofs are based on the harmonic structure defining the Dirichlet form E a result A. Jonsson: A trace theorem for the Dirichlet form on the Sierpinski gasket, Math. Z. 250 (2005), no. 3, by which the restriction to a lacuna l σ of a finite energy function a F belongs to the fractional Sobolev space F α, if α < α 0. These properties allow to develop integration of 1-forms ω H in the tangent module associated to (E, F) along paths γ K characterize exact and locally exact 1-forms in terms of their periods around lacunas define a de Rham cohomology of 1-forms and prove a separating duality with the Cech homology group. represent the integrals of 1-forms around cycles in K by potentials, i.e. affine functions on the abelian universal projective covering L of K

25 Elementary 1-forms and elementary paths Let (E, F) be the standard Dirichlet form on K and (F,, H) the associated derivation with values in the tangent module H (whose elements are understood as square integrable 1-forms). exact 1-forms Ω 1 e(k) := Im = { a H : a F} elementary 1-forms Ω 1 (K) := { n i=1 ai bi H : ai, bi F} locally exact 1-forms Ω 1 loc(k) are those ω Ω 1 (K) which admit a primitive U A F on a suitable neighborhood A K of a any point of K ω = U A on A K. n-exact 1-forms are those ω Ω 1 (K) which are exact on any cell C σ with σ =n an elementary path γ K is a path which is a finite union of edges of K. If it is contained in a cell C σ K we say that γ has depth σ N. Lemma. (CGIS 2010) A 1-form is locally exact if and only if it is n-exact form some n N.

26 Integration of elementary 1-forms along elementary paths Theorem. (CGIS 2010) The integral of an elementary 1-form ω = n i=1 ai bi Ω1 (K) along an elementary path γ K is well defined as γ ω := n a idb i where the aidbi is understood as a Lebesgue-Stieltjes integral. γ Proof is a based on an embedding F H α (γ) of the Dirichlet space into a fractional Sobolev space with α > 1/2. Potentials i=1 A continuous function U C(A) defined on subset A K is a local potential on A of a 1-form ω Ω 1 (K) if for all elementary path γ A ω = U(γ(1)) U(γ(0)). γ γ

27 Proposition. (CGIS 2010) Local potentials of a 1-form on A K have finite energy on A the class of potentials U C(K) of an exact 1-form ω Ω 1 e(k) coincides with the class of its primitives U F on K ω = U(γ(1)) U(γ(0)) ω = U. γ

28 A system of locally exact 1-forms associated with lacunas Cells and lacunas The lacuna l K (depth 0) is defined as the boundary of the triangle K \ (F 1(K) F 2(K) F 3(K)) the lacunas l σ (depth n) are defined as its successive contraction: l σ := F σ(l ) Theorem. (CGIS 2010) For any σ there exists only one ( σ + 1)-exact 1-form dz σ Ω 1 loc(k) having minimal norm ω H among those with unit period on the lacuna l σ l σ ω = 1 ω σ Lin {dz τ : τ σ } such that for all elementary paths γ K ω σ = winding number of γ around l σ. γ

29 Theorem. (a la de Rham) (CGIS 2010) Any elementary 1-form ω Ω 1 (K) can be uniquely decomposed as ω = U + σ k σdz σ the convergence taking place in a topology where integrals are continuous the coefficients k σ only depend upon the periods l σ ω around lacunas the form is locally exact ω Ω 1 loc(k) iff the k σ s are eventually zero the form is exact ω Ω 1 e(k) iff all the periods, hence all k σ s, are zero Cells and lacunas There exist elementary forms which are not closed: f 0 f 1.

30 Theorem. (a la Hodge) (CGIS 2010) The forms {dz σ : σ } H are pairwise orthogonal Any 1-form ω H can be orthogonally decomposed as ω = U + σ k σdz σ the forms {dz σ.σ } are co-closed (dz σ) = 0 so that ω = U = U

31 Theorem. (a de Rham cohomology Theorem) (CGIS 2010) The pairing γ, ω = ω between elementary paths γ K and locally exact forms γ ω Ω 1 loc(k) gives rise to a nondegenerate pairing between the Cech homology group H 1(K, R) and the cohomology group H 1 dr(k) := Ω1 loc(k) Ω 1 e(k) in which the classes of the co-closed forms {dz σ : σ } H is a system of generators parameterized by lacunas.

32 Coverings Let T C the closed triangle envelope of the gasket K and T n := σ =n Fσ(T). The embedding i n : K T n give rise to regular coverings ( K n, K) whose group of deck transformations can identified with π 1(T n). Its abelianization Γ n can be identified with the homology group H 1(T n). Defining L n := K n/[γ n, Γ n] one has that ( L n, K) form coverings whose group of deck transformations can be identified with Γ n. Theorem. () (CGIS 2010) The family {( L n, K) : n 0} is projective and the group Γ of deck transformations of its limit L = lim L n can be identified with the Cech homology group of the gasket H 1(K) Γ := lim Γ n. The covering L has the unique lifting property.

33 Potentials of 1-forms Definition. Affine functions A continuous function f C( L) is said to be affine if f (gx) = f (x) + φ(g) x L, g H 1(K) for some continuous group homomorphisms φ : H 1(K) C. Denote by A( L, Γ) the space of affine functions. Theorem. (Potentials of locally exact forms) (CGIS 2010) Any locally exact form ω Ω 1 loc(k) admits a potential U A( L, Γ) in the sense that ω = U( γ(1)) U( γ(0)). γ An affine function U A( L, Γ) is a potential of a locally exact form iff it has finite energy ( 5 ) n E Γ[U] := lim U(ẽ n +) U(ẽ ) 2. 3 e E n

34 References A. Jonsson. A trace theorem for the Dirichlet form on the Sierpinski gasket, Math. Z., 250 (2005),