Noncommutative Geometry and Potential Theory on the Sierpinski - PowerPoint PPT Presentation

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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, 20-22 September 2012

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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 α i = 1 � � � 3 ( f ◦ F i ) d µ f ∈ C ( K ) f d µ = α i K K i = 1 when α i = 1 3 for all i = 1 , 2 , 3 then µ is the normalized Hausdorff measure associated to the restriction of the Euclidean metric on K Hausdorff dimension d H = ln 3 ln 2 .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory Harmonic structure � := � � � � m := { 1 , 2 , 3 } m , 0 := ∅ , word spaces: m ≥ 0 m length of a word σ ∈ � | σ | := m m : F σ := F i | σ | ◦ . . . F i 1 if σ = ( i 1 , . . . , i | σ | ) iterated contractions: V m := � V ∅ := { p 1 , p 2 , p 3 } , | σ | = m F σ ( V 0 ) vertices sets: 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 � 5 � m � E m [ a ] := E 0 [ 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 | V m = a } a ∈ C ( V m ) .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory Dirichlet form, Laplacian and Spectral Dimension Theorem 1. (Kigami 1986) The quadratic form E : C ( K ) → [ 0 , + ∞ ] defined by m → + ∞ E m [ a | V m ] E [ a ] := lim 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 � 3 E [ a ] = 5 E [ a ◦ F i ] a ∈ C ( K ) . 3 i = 1 It is closed in L 2 ( K , µ ) and the associated self-adjoint (Laplacian) operator ∆ µ has ln 9 discrete spectrum with spectral dimension d S = ln 5 / 3 � = d H : Weyl ′ s asymptotics ♯ { eigenvalues of ´ µ ≤ λ } ≍ λ d S / 2 λ → + ∞ .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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 d µ = Trace Dix ( M f ◦ ∆ − d S / 2 ) = Res s = d S Trace ( M f ◦ ∆ − s / 2 ) µ 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 d Γ( a ) := E ( a | ab ) − 1 2 E ( a 2 | b ) a , b ∈ F K 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.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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 a ∈ F . H In other words, ( F , ∂, H ) is a differential square root of the Laplacian H µ : H µ = ∂ ∗ ◦ ∂ . � K b d Γ[ a ] = ( ∂ a | b ∂ a ) H a ∈ F . Energy measures are represented as 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 ) .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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 E ( a | cdb ∗ ) + E ( abd ∗ | c ) − E ( bd ∗ | a ∗ c ) 2 the derivation ∂ is the quotient of the derivation d The proof uses the following facts the algebra F is dense in C ( K ) 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 , ⇒ E ( a , b ) = 0 . a constant on the support of b

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory A first Fredholm module on K Theorem. (FC-Sauvageot 2009) Let P ∈ Proj ( H ) the projection onto the image Im ∂ of the derivation above P H = 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 .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory 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 ∂ a k ∈ P H = Im ∂ complete orthonormal basis k Since the Green function G ( x , y ) = � ∞ k = 1 λ − 1 k a k ( x ) a k ( y ) is continuous on K ∞ ∞ � � L 2 = 8 � P ⊥ aP � 2 λ − 1 k � P ⊥ aP ( ∂ a k ) � 2 λ − 1 � [ F , a ] � 2 k � ( ∂ a ) a k � 2 L 2 = 8 H ≤ 8 H k = 1 k = 1 � � � ∞ λ − 1 a 2 = 8 k d Γ( a ) = 8 G ( x , x )Γ( a )( dx ) k K K k = 1 a = a ∗ ∈ F . ≤ 8 � G � ∞ E [ a ] Since F is uniformly dense in C ( K ) , we have that [ F , a ] is compact for all a ∈ C ( K ) .