Noncommutative Potential Theory 4 Fabio Cipriani Dipartimento di - PowerPoint PPT Presentation

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory Noncommutative Potential Theory 4 Fabio Cipriani Dipartimento di Matematica Politecnico di Milano � � joint works with U. Franz, D. Guido, T. Isola, A. Kula, J.-L. Sauvageot Villa Mondragone Frascati, 15-22 June 2014

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory Themes. Sierpinski Gasket K Harmonic structures and Dirichlet forms on K Dirac operators and Spectral Triples on K Volume functional dimensional spectrum Energy functional dimensional spectrum Dirichlet form as a residue Fredholm modules and pairing with K-theory de Rham cohomology and Hodge Harmonic decomposition on K Potentials of locally exact 1-forms on the projective covering of K

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory References. Kigami, J. Analysis on fractals Cambridge Tracts in Mathematics, 143 (2001) Guido, D.; Isola, T. Dimensions and singular traces for spectral triples, with applications to fractals J. Funct. Anal. 203 (2003) A. Jonsson. A trace theorem for the Dirichlet form on the Sierpinski gasket , Math. Z., 250 (2005), 599–609. Christensen, E.; Ivan, C.; Lapidus, M. L. Dirac operators and spectral triples for some fractal sets built on curves Adv. Math. 217 (2008) Cipriani, F.; Sauvageot, J.-L. Fredholm modules on P.C.F. self-similar fractals and their conformal geometry Comm. Math. Phys. 286 (2009) Christensen, E.; Ivan, C.; Schrohe, E. Spectral triples and the geometry of fractals J. Noncommut. Geom. 6 (2012) Cipriani, F.; Guido, D.; Isola, T.; Sauvageot, J.-L. Integrals and potentials of differential 1-forms on the Sierpinski gasket Adv. Math. 239 (2013) Cipriani, F.; Guido, D.; Isola, T.; Sauvageot, J.-L. Spectral triples for the Sierpinski gasket J. Funct. Anal. 266 (2014)

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple 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 Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory Geometric and analytic features of the Sierpinski gasket K is not a manifold the group of homeomorphisms is finite K is not semi-locally simply connected hence K does not admit a universal cover K-theory group � K 1 ( K ) = Z i ∈ N K-homology group � K 1 ( K ) = Z i ∈ N Volume and Energy are distributed singularly on K existence of localized eigenfunctions

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory Self-similar volume measures and their Hausdorff dimensions The natural 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 d µ = α i ( f ◦ F i ) d µ f ∈ C ( K ) K K i = 1 when α i = 1 3 for all i = 1 , 2 , 3 then µ is the normalized Hausdorff measure on K associated to the restriction of the Euclidean metric: its dimension is d = ln 3 ln 2

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple 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 Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory Dirichlet form 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 operator H µ has discrete ln 9 spectrum with spectral exponent d S = ln 5 / 3 : ♯ { eigenvalue of H µ ≤ λ } ≍ λ d S / 2 λ → + ∞ .

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory Volume and Energy measures Theorem. (Kigami-Lapidus 2001) The self-similar volume measure µ with weights α i = 1 / 3 can be re-constructed as � f d µ = Trace Dix ( M f ◦ H − d S / 2 ) = Res s = d S Trace ( M f ◦ H − s / 2 ) µ µ K Theorem. (M. Hino 2007) The energy measures on K defined by � 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 measures on K .

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory Derivation and Fredholm module on K Theorem. (FC-Sauvageot 2003) There exists a symmetric derivation ( F , ∂, H , J ) , defined on the Dirichlet algebra F , with values in a symmetric C ( K ) -monomodule ( H , J ) such that E [ a ] = � ∂ a � 2 a ∈ F . H In other words, ( F , ∂, H , J ) is a differential square root of H µ : H µ = ∂ ∗ ◦ ∂ . 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 associated phase operator. Then ( F , H ) is a 2-summable (ungraded) Fredholm module over C ( K ) and Trace ( | [ F , a ] | 2 ) ≤ const . E [ a ] a ∈ F .

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory 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 2010) Consider the Dirichlet form on L 2 ( T ) defined on the fractional Sobolev space � � | a ( z ) − a ( w ) | 2 F α := { a ∈ L 2 ( T ) : E α [ a ] < + ∞} . E α [ a ] := dzdw | z − w | 2 α + 1 T T Then H α := L 2 ( T × T ) is a symmetric Hilbert C ( K ) -bimodule w.r.t. actions and involutions given by ( a ξ )( z , w ) := a ( z ) ξ ( z , w ) , ( ξ a )( 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 .

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory Proposition. Spectral Triples on a circle (CGIS 2010) 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 � � ∂ α 0 D α := . ∂ ∗ 0 α � | a ( z ) − a ( w ) | 2 Then A α := { a ∈ C ( T ) : sup z ∈ T | z − w | 2 α + 1 < + ∞} 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 ) .

Overview Sierpinski gasket Differential calculus on Sierpinski gasket Dirac operator and Spectral triple Potential Theory 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 .