Noncommutative Potential Theory 3 Fabio Cipriani Dipartimento di - PowerPoint PPT Presentation

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Noncommutative Potential Theory 3 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 Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Themes. Noncommutative potential theory: carré du champ, potentials, finite energy states, multipliers Dirac operator, Spectral triple on Lipschiz algebra of Dirichlet spaces Closable derivations on algebras of finite energy multipliers References. Cipriani-Sauvageot Variations in noncommutative potential theory: finite energy states, potentials and multipliers TAMS 2014 V.G. Maz’ya, T.O. Shaposhnikova, Theory of Sobolev multipliers. With applications to differential and integral operators Grundlehren der Mathematischen Wissenschaften 337, Springer Verlag 2009. J. Ferrand-Lelong Invariants conformes globaux sur les varietes Riemannien J. Diff. Geom. (8) 1973.

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms One of the main subject of potential theory of Dirichlet spaces ( E , F ) on C ∗ -algebras with trace ( A , τ ) , is the following class of functionals Definition. (Carré du champ) The carré du champ of a ∈ F is the positive functional Γ[ a ] ∈ A ∗ + Γ[ a ] : A → C � Γ[ a ] , b � := ( ∂ a | ( ∂ a ) b ) H b ∈ A defined using the derivation ( B , ∂, H , J ) representing ( E , F ) . Alternatively, whenever a ∈ B we can set � Γ[ a ] , b � := 1 2 {E ( ab ∗ | a ) + E ( a | ab ) − E ( a ∗ a | b ) } b ∈ B . When E [ a ] represents the energy of a configuration a ∈ F of a system, Γ[ a ] may be interpreted as its energy distribution. Example . In case of the Dirichlet integral on R n , the carré du champ are absolutely continuous with respect to the Lebesgue measure m and reduces to Γ[ a ] = |∇ a | 2 · m a ∈ H 1 ( R n ) . In general the energy distribution Γ[ a ] is not comparable with the volume distribution represented by τ .

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Let ( E , F ) be a Dirichlet form on ( A , τ ) , ( F , ∂, H , J ) its differential square root and ( F ∗ , ∂ ∗ , H , J ) its adjoint. Recall that ( B , ∂, H , J ) is a derivation. Definition. (Dirac operator) The Dirac operator ( D , H D ) of the Dirichlet space is the densely defined, self-adjoint operator acting on H D := L 2 ( A , τ ) ⊕ H as � 0 ∂ ∗ � dom ( D ) := F ⊕ F ∗ ⊆ H D D := ∂ 0 or more explicitly � ∂ ∗ ξ � a � 0 � � a � a � ∂ ∗ � � � ∈ F ⊕ F ∗ . D = = ξ ∂ ξ ∂ a ξ 0 � − I � 0 By definition, the operator is anticommuting with involution γ := : 0 I D γ + γ D = 0 . � ∂ ∗ ∂ � 0 D 2 = Notice that . ∂∂ ∗ 0

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Lipschiz algebra Consider below L 2 ( A , τ ) , H and H D as left A -modules. Lemma. (Bounded commutators) For a ∈ B , the following properties are equivalent [ D , a ] is bounded on H D [ ∂, a ] is bounded from L 2 ( A , τ ) to H Γ[ a ] is absolutely continuous w.r.t. τ with bounded Radon-Nikodym derivative h a ∈ L ∞ ( A , τ ) b ∈ L 1 ( A , τ ) ; � Γ[ a ] , b � = τ ( h a b ) for a ∈ B ∩ dom M ( L ) , these are also equivalent to a ∗ a ∈ dom M ( L ) . Definition. (Lipschiz algebra) The ∗ -subalgebra L ( F ) ⊆ B of elements satisfying the first three properties above, is called the Lipschiz algebra of the Dirichlet space. Example. In case of the Dirichlet integral L ( H 1 ( R n )) coincides with the algebra Lip ( R n ) of Lipschiz functions of the Euclidean metric. Example. In a next lecture, we will see that on p.c.f. fractals, as a rule, the Lipschiz algebra reduces to constants functions.

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Spectral triple, Fredholm module Define the phase F D := D | D | − 1 of the Dirac operator to be zero on ker ( D ) . Theorem. (Spectral triple and Fredholm module of D S ) Assume the spectrum of ( E , F ) on L 2 ( A , τ ) to be discrete. Then ( L ( F ) , D , H D ) is spectral triple in the sense [ D , a ] is bounded on H D for all a ∈ L ( F ) sp ( D ) is discrete away from zero. Moreover, setting F := F D + P ker ( D ) , then ( L ( F ) , F , H D ) is a Fredholm module F 2 = I F = F ∗ , [ F , a ] is compact on H D for all a ∈ L ( F ) .

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Example: Ground State representations of Schrödinger operators H := − ∆ + V be a semibounded Hamiltonian with potential V on L 2 ( R n , m ) assume the spectrum to be discrete sp ( H ) = { E 0 < E 1 < . . . } , ψ 0 ∈ L 2 ( R n , m ) the ground state with lowest eigenvalue E 0 : H ψ 0 = E 0 ψ 0 U : L 2 ( R n , m ) → L 2 ( R n , | ψ 0 | 2 · m ) ground state transformation U ( f ) := ψ − 1 f ∈ L 2 ( R n , m ) 0 f H φ 0 the ground state representation of H : H φ 0 := U ( H − E 0 ) U − 1 e − tH positivity preserving on L 2 ( R n , m ) ⇒ e − tH ψ 0 Markovian on L 2 ( R n , | ψ 0 | 2 · m ) Dirichlet form on L 2 ( R n , | ψ 0 | 2 · m ) � R n |∇ a | 2 · | ψ 0 | 2 · m H ψ 0 a � 2 � E ψ 0 [ a ] = � 2 = a ∈ F ψ 0 derivation ∂ : F ψ 0 → L 2 ( R n , m ) ∂ a = ∇ a Lipschiz algebra L ( F ψ 0 ) = L ( R n ) harmonic oscillator V ( x ) := | x | 2 : spectral dimension of ( C b ( R n ) ∩ Lip ( R n ) , D ψ 0 , L 2 ( R n , | ψ 0 | 2 · m ) ⊕ L 2 ( R n , | ψ 0 | 2 · m )) = 2 n

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Potentials, Finite energy functionals Finer properties of the differential calculus underlying a Dirichlet spaces rely on properties of the basic objects of the Potential Theory of Dirichlet forms. � Consider the Dirichlet space with its Hilbertian norm � a � F := E [ a ] + � a � 2 L 2 ( A ,τ ) . Definition. Potentials, Finite Energy Functionals (CS TAMS 2014) p ∈ F is called a potential if a ∈ F + := F ∩ L 2 ( p | a ) F ≥ 0 + ( A , τ ) Denote by P ⊂ L 2 ( A , τ ) the closed convex cone of potentials. ω ∈ A ∗ + has finite energy if for some c ω ≥ 0 | ω ( a ) | ≤ c ω · � a � F a ∈ F . Example . In a d -dimensional Riemannian manifold ( V , g ) , the volume measure µ W of a ( d − 1 ) -dimensional compact submanifold W ⊂ V has finite energy.

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Theorem. (CS TAMS 2014) Let ( E , F ) be a Dirichlet form on ( A , τ ) . Potentials are positive: P ⊂ L 2 + ( A , τ ) Given a finite energy functional ω ∈ A ∗ + , there exists a unique potential G ( ω ) ∈ P ω ( a ) = ( G ( ω ) | a ) F a ∈ F . Example . If h ∈ L 2 + ( A , τ ) ∩ L 1 ( A , τ ) then ω h ∈ A ∗ + defined by ω h ( a ) := τ ( ha ) a ∈ A is a finite energy functional whose potential is given by G ( ω h ) = ( I + L ) − 1 h . Example . Let E ℓ be the Dirichlet form on A := C ∗ r (Γ) , associated to a negative definite function ℓ on a countable, discrete group Γ . Then ω is a finite energy functional iff | ω ( δ s ) | 2 ω ( δ s ) � 1 + ℓ ( s ) < + ∞ G ( ω )( s ) = s ∈ Γ . with potential 1 + ℓ ( s ) t ∈ Γ

Overview Carré du champ Dirac operator in DS Spectral triple, Fredholm module of D S NC Potential Theory Lipschiz and multipliers seminorms Deformation of group representations √ ℓ ) − 1 is a positive definite, normalized function, there exists a Since ϕ ℓ := ( 1 + state ω ℓ ∈ A ∗ + such that ϕ ℓ ( s ) = ω ℓ ( δ s ) for all s ∈ Γ . Thus ω has finite-energy iff | ω ( δ s ) | 2 | ϕ ℓ ( s ) · ϕ ω ( s ) | 2 < + ∞ . � � √ ℓ ( s )) 2 = ( 1 + s ∈ Γ s ∈ Γ Notice that ϕ ℓ · ϕ ω is a coefficient of a sub-representation of the product π ω ℓ ⊗ π ω of the representations ( π ℓ , H ℓ , ξ ℓ ) and ( π ω , H ω , ξ ω ) associated to ω ℓ and ω . Hence if ω has finite-energy, π ω ℓ ⊗ π ω and λ Γ are not disjoint. Moreover, as ω has finite energy simultaneously with respect to E ℓ and E λ − 2 ℓ for λ > 0, the family of normalized, positive definite functions λ ϕ λ ( s ) = · ϕ ω ( s ) s ∈ Γ , � λ + ℓ ( s ) generates a family of cyclic representations { π λ : λ > 0 } contained in λ Γ , deforming the cyclic representation π ω associated to the finite energy state ω to the left regular representation λ Γ . In fact λ → 0 + ϕ λ = δ e , λ → + ∞ ϕ λ = ϕ ω . lim lim