AMS Memphis Oct 18, 2015 1 NONCOMMUTATIVE Sponsoring GEOMETRY of FRACTALS Jean BELLISSARD Georgia Institute of Technology, Atlanta School of Mathematics & School of Physics e-mail: jeanbel@math.gatech.edu
AMS Memphis Oct 18, 2015 2 Collaborations J. Pearson, (Gatech, Atlanta, GA) J. Savinien, (U. Metz, Metz, France) A. Julien, (U. Victoria, Victoria, BC) I. Palmer, (NSA, Washington DC) R. Parada, (Gatech, Atlanta, GA)
AMS Memphis Oct 18, 2015 3 Main References A. Beurling, J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A., 45, (1959), 208-215. M. Fukushima, Dirichlet forms and Markov processes, North-Holland Math. Lib., 23., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. A. Connes, Noncommutative Geometry, Academic Press, 1994. M. A. Rieffel, Metrics on state space, Doc. Math., 4, 1999, 559-600. G. Michon, Les Cantors réguliers, C. R. Acad. Sci. Paris Sér. I Math., (19), 300, (1985) 673-675. J. Pearson, J. Bellissard, Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets, J. Noncommutative Geometry, 3, (2009), 447-480. A. Julien, J. Savinien, Transverse Laplacians for substitution tilings, Comm. Math. Phys., 301, (2011), 285-318. I. Palmer, Noncommutative Geometry and Compact Metric Spaces, PhD Thesis, Georgia Institute of Technology, May 2010
AMS Memphis Oct 18, 2015 4 Content 1. Spectral Triple 2. Compact Metric Spaces 3. The Pearson Laplacian
AMS Memphis Oct 18, 2015 5 I - Spectral Triples
AMS Memphis Oct 18, 2015 6 Spectral Triples A spectral triple is a family (H, A, D), such that H is a Hilbert space D is a self-adjoint operator on H with compact resolvent. A is a unital C -algebra with a faithful representation π into H There is a core D in the domain of D, and a dense -subalgebra A 0 A such that if a A 0 then π(a)d D and [D, π(a) < }. (H, A, D) is called even if there is G B(H) such that G = G = G 1 [G, π( f )] = 0 for f A GD = DG
AMS Memphis Oct 18, 2015 7 Example Let T = R/Z be the torus, A = C(T), H = L 2 (T), where A acts by pointwise multiplication and D = ıd/dx Then, if x y = inf l Z x y + l, d f [D, f ] = ess-sup dx = sup x y f (x) f (y) x y = f Lip and x y = sup { f (x) f (y) ; f Lip 1 }
AMS Memphis Oct 18, 2015 8 Example Let M be a spin c Riemannian manifold, A = C(M), H the space of L 2 -sections of the spin bundle and D the corresponding Dirac operator, where A acts by pointwise multiplication. Theorem (Connes) The family X M = (A, H, D) above is a spectral triple. The geodesic distance between x, y M can be recovered through d(x, y) = sup{ f (x) f (y) ; f A, [D, f ] 1} Actually [D, f ] = f L = f CLip and C 1 (X) = Lip(M). Hence the algebra A encodes the space, the Dirac operator D encodes the metric. H is needed to define D.
AMS Memphis Oct 18, 2015 9 Spectral Metric Spaces Definition A spectral metric space is a spectral triple (H, A, D) such that (i) the commutant A = {a A ; [D, π(a)] = 0} is trivial, A = C1 (ii) the Lipshitz ball B Lip = {a A ; [D, π(a)] 1} is precompact in A/A Theorem [Rieffel 99] A spectral triple (H, A, D) is a spectral metric space if and only if the Connes metric, defined on the state space of A by d C (ω, ω ) = sup{ ω(a) ω (a) ; [D, π(a)] 1} is well defined and equivalent to the weak -topology
AMS Memphis Oct 18, 2015 10 ζ-function and Spectral Dimension Definition A spectral metric space (H, A, D) is called summable is there is p > 0 such that Tr ( D p ) <. Then, the ζ-function is defined as ( ) 1 ζ(s) = Tr D s The spectral dimension is s D = inf { s > 0 ; Tr ( ) 1 D s < } Then ζ is holomorphic in Re(s) > s D Remark For a Riemannian manifolds s D = dim(m)
AMS Memphis Oct 18, 2015 11 Connes trace & Volume Form The spectral metric space is spectrally regular if the following limit is unique ( ) 1 1 ω D (a) = lim s sd ζ(s) Tr D s π(a) a A Then ω D is a trace called the Connes trace. Remark (i) By compactness, limit states always exist, but the limit may not be unique. (ii) Even if unique this state might be trivial. (iii) In the example of compact Riemannian manifold the Connes state exists and defines the volume form.
AMS Memphis Oct 18, 2015 12 Hilbert Space If the Connes trace is well defined, it induces a GNS-representation as follows The Hilbert space L 2 (A, ω D ) is defined from A through the inner product a b = ω D (a b) The algebra A acts by left multiplication.
AMS Memphis Oct 18, 2015 13 Laplacian? How could one define a Laplacian? The following might be a way. If the quadratic form Q(a, b) = lim s sd 1 ζ(s) Tr ( ) 1 D s [D, π(a)] [D, π(b)] extends to L 2 (A, ω D ) as a closable quadratic form, then, it defines a positive operator which generates a Markov semi-group and is a candidate for being the analog of the Laplace-Beltrami operator.
AMS Memphis Oct 18, 2015 14 Laplacian? The example of the Sierpinsky gasket shows that it should be rather something like Q(a, b) = lim s sd 1 ζ(s) Tr ( ) 1 D s [Dα, π(a)] [D α, π(b)] for a suitable value of the exponent α. Problem: Can one define geometrically the exponent α for a compact metric space?
AMS Memphis Oct 18, 2015 15 II - Compact Metric Spaces I. Palmer, Noncommutative Geometry of compact metric spaces, PhD Thesis, May 3rd, 2010.
AMS Memphis Oct 18, 2015 16 Open Covers Let (X, d) be a compact metric space with an infinite number of points. Let A = C(X). An open cover U is a family of open sets of X with union equal to X. Then diamu = sup{diam(u) ; U U}. All open covers used here will be at most countable A resolving sequence is a family (U n ) n N such that lim n diam(u n) = 0 A resolving sequence is strict if all U n s are finite and if diam(u n ) < inf{diam(u) ; U U n 1 } n
AMS Memphis Oct 18, 2015 17 Choice Functions Given a resolving sequence ξ = (U n ) n N a choice function is a map τ : U(ξ) = n U n X X such that τ(u) = (x U, y U ) U U there is C > 0 such that diam(u) d(x U, y U ) diam(u) 1 + C diam(u) U U(ξ) The set of such choice functions is denoted by Υ(ξ).
AMS Memphis Oct 18, 2015 18 A Family of Spectral Triples Given a resolving sequence ξ, let H ξ = l 2 (U(ξ)) C 2 For τ a choice let D ξ,τ be the Dirac operator defined by D ξ,τ ψ (U) = 1 d(x U, y U ) [ 0 1 1 0 ] ψ(u) ψ H For f C(X) let π ξ,τ be the representation of A = C(X) given by π ξ,τ ( f )ψ (U) = [ f (xu ) 0 0 f (y U ) ] ψ(u) ψ H
AMS Memphis Oct 18, 2015 19 Regularity Theorem Each T ξ,τ = (H ξ, A, D ξ,τ, π ξ,τ ) defines a spectral metric space such that A 0 = Lip(X, d) is the space of Lipshitz continuous functions on X. Such a triple is even when endowed with the grading operator [ ] 1 0 Gψ(U) = ψ(u) ψ H 0 1 In addition, the family {T ξ,τ ; τ Υ(ξ)} is regular in that d(x, y) = sup{ f (x f (y) ; sup [D ξ,τ, π ξ,τ ( f )] 1} τ Υ(ξ)
AMS Memphis Oct 18, 2015 20 Summability Theorem There is a resolving sequence leading to a family T ξ,τ of summable spectral metric spaces if and only if the Hausdorff dimension of X is finite. If so, the spectral dimension s D satisfies s D dim H (X). If dim H (X) < there is a resolving sequence leading to a family T ξ,τ of summable spectral triples with spectral dimension s D = dim H (X).
AMS Memphis Oct 18, 2015 21 Hausdorff Measure Theorem There exist a resolving sequence leading to a family T ξ,τ of spectrally regular spectral metric spaces if and only if the Hausdorff measure of X is positive and finite. In such a case the Connes state coincides with the normalized Hausdorff measure on X. Then the Connes state is given by the following limit independently of the choice τ X f (x)hs D (dx) H s D(X) = lim s sd 1 ζ ξ,τ (s) Tr ( ) 1 D ξ,τ s π ξ,τ( f ) f C(X)
AMS Memphis Oct 18, 2015 22 III - The Pearson Laplacian
AMS Memphis Oct 18, 2015 23 Directional Derivative, Tangent Space If τ(u) = (x U, y U ) then [D ξ,τ, π ξ,τ ( f )] ψ (U) = f (x U) f (x U ) d(x U, y U ) [ 0 1 1 0 ] ψ(u) The commutator with the Dirac operator is a coarse grained version of a directional derivative. In particular τ(u) can be interpreted as a coarse grained version of a unit tangent vector at U. the set Υ of all possible choices, can be seen as the set of sections of the tangent sphere bundle. [D ξ,τ, π ξ,τ ( f )] could be written as τ f.
AMS Memphis Oct 18, 2015 24 Choice Averaging The choice space Υ is given by U Υ(U) where Υ(U) is a compact subset of U U. Let ν U be the probability measure on Υ(U) induced by the Hausdorff measure µ H µ H on U U. This leads to the probability ν = U ν U Hence ν U can be interpreted as the average over the tangent unit sphere at U.
AMS Memphis Oct 18, 2015 25 The Pearson Quadratic Form The Pearson quadratic form is defined by (if f, g C(X)) Q s ( f, g) = Υ dν(τ) Tr ( ) 1 D ξ,τ s[d ξ,τ, π ξ,τ ( f )] [D ξ,τ, π ξ,τ (g)] Theorem: If (X, d) is an ultrametric Cantor set, having a positive finite Hausdorff measure µ H, for each s R, the quadratic forms Q s is densely defined, closable in L 2 (X, µ H ) and is a Dirichlet form. The corresponding positive operator s has pure point spectrum. It is bounded if and only if s > dim H (X) + 2 and has compact resolvent otherwise. The eigenspaces are common to all s s and can be explicitly computed.
AMS Memphis Oct 18, 2015 26 Jump Process s generates a Markov semigroup, thus a stochastic process (X t ) t 0 where the X t s takes on values in X. It can be shown to be a jump process. In addition for most examples computed so far, like the triadic Cantor set (Pearson, JB, 08) of for the tiling space of a substitution tiling (Julien, Savinien 11), the diffusion corresponding to s = dim H (X) + 2 satisfies E ( d(x t+δ, X t ) 2) δ 0 const. δ ln(1/δ)
AMS Memphis Oct 18, 2015 27 Thanks for Listening!