Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Noncommutative Geometry of Dirichlet spaces Fabio Cipriani Dipartimento di Matematica Politecnico di Milano Joint works with J.-L. Sauvageot Intensive Month on Operator Algebra and Harmonic Analysis, Madrid, 20 May - 14 June 2013 Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Noncommutative Geometry underlying Dirichlet forms on singular spaces Usually one uses tools of NCG to build up hamiltonian on singular spaces 8 Ground States in QFT (L. Gross) > > > Algebraic QFT > <> Quantum Hall Effect (J. Bellissard) NCG ! Energy functionals Quasi crystals (J. Bellissard) > Standard Model (A. Connes) > > Action Principle (A. Connes) > : Heat equations on foliations (Sauvageot) reversing the point of view, our goal is to analyze the NCG structures underlying Energy functionals 8 < Compact Quantum Groups (U. Franz, A. Kula) Energy functionals ! NCG Orbits of Dynamical systems (M. Mauri) : Fractals (D. Guido, T. Isola, J.-L. Sauvageot) Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Dirichlet forms, Dirichlet spaces Dirichlet spaces on locally compact Hausdorff spaces were introduced by Beurling-Deny in the fall of ’50 to develop a kernel-free Potential Theory. The idea was to emphasize the rôle of the Energy functional rather than the one n of potential kernel (Newtonian, Riesz,...) in classical Potential Theory on R . Generalization to C∗-algebras with traces were initiated by [Gross ’72]: Hypercontractive Markovian semigroups on von Neumann algebras with finite traces to prove existence/uniqueness of ground states in QFT [Albeverio-Høegh-Krohn ’76]: Dirichlet forms and Markovian semigroups on C∗-algebras with trace [Sauvageot ’88]: dilation of Markovian semigroups on C∗-algebras with trace [Davies-Lindsay ’88]: construction of Dirichlet forms by unbounded derivations on C∗-algebras with trace [Goldstein-Lindsay ’92]: Dirichlet forms on Haagerup’s standard form of von Neumann algebras [Cipriani ’92,’97]: Dirichlet forms and Markovian semigroups on standard form of von Neumann algebras [Cipriani-Sauvageot ’03]: Dirichlet forms and Hilbert bimodule derivations on C∗-algebras with trace [Cipriani ’08]: KMS-symmetric Markovian semigroups on C∗-algebras Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Dirichlet forms, Dirichlet spaces Let (A; τ) be a C∗-algebra endowed with a l.s.c. semifinite, positive trace. Definition. (Dirichlet form, Dirichlet space) A Dirichlet form E : L2(A; τ) ! [0; +1] is a l.s.c., quadratic form such that E[a∗] = E[a] E[a ^ 1] ≤ E[a] the domain F := fa 2 L2(A; τ): E[a] < +1g dense in L2(A; τ) the subspace B := F\ A is norm dense in A. (E; F) is a complete Dirichlet space if its matrix ampliations X En[(aij)ij] := E[aij] ij are Dirichlet forms on A ⊗ Mn(C) for all n ≥ 1 (tacitly assumed since now on) The domain F is called Dirichlet spaces when endowed with the graph norm q 2 kakF := E[a] + kakL2(A,τ) : Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Dynamical semigroups There exists a correspondence among Dirichlet spaces and Markovian semigroups (strongly continuous, symmetric, positively preserving, contractive on L2(A; τ)) Theorem. (Beurling-Deny correspondence) Dirichlet forms (E; F) are in 1:1 correspondence with Markovian semigroups by 1 E[a] = lim (aja − Tta) a 2 F t!0 t or through the self-adjoint generator (L; dom (L)) p p −tL 2 Tt = e E[a] = k LakL2(A,τ) a 2 F = dom ( L) : These semigroups, can be characterized as those weakly∗−continuous, positively preserving, contractive semigroups on the von Neumann algebra L1(A; τ) which are 1 τ − symmetric τ(a(Ttb)) = τ((Tta)b) a; b 2 L (A; τ) ; t > 0 : Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Examples: Dirichlet spaces on Riemannian manifold 1. The archetypical Dirichlet form is the Dirichlet integral of a Riemannian manifold Z Z 2 (V; g); A = C0(V); τ(a) = a dmg ; E[a] = jraj dmg ; V V where the Dirichlet space coincides with the Sobolev space F = H1;2(V) 2. ([Davies-Rothaus JFA ’89] On the Clifford algebra A = Cl0(V; g) of a Riemannian manifold, the quadratic form of the Bochner Laplacian Z 2 EB[a] = jrBaj dmg V is a Dirichlet form (independently upon the curvature of (V; g)). 3. [C-Sauvageot GAFA ’03] On the Clifford algebra A = Cl0(V; g) of a Riemannian 2 manifold, the quadratic form of the Dirac Laplacian D ' ∆HdR Z 2 ED[a] = jDaj dmg V is a Dirichlet form if and only if the curvature operator is nonnegative R ≥ 0. ∗ ∗ 2 ∗ ∗ Recall Cl0(V; g) ' C0(Λ (V)), D ' d + d , D ' dd + d d ' ∆HdR. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Examples: Dirichlet spaces on group C∗-algebras C∗(G) 4. Let G be a locally compact, unimodular group with identity e 2 G, C∗(G) its convolution group C∗-algebra with trace τ(a) = a(e) a 2 Cc(G) and recall that L2(A; τ) ' L2(G). Then, for any continuous, negative definite function ` : G ! [0; +1), Z E[a] = `(g)ja(g)j2 dg a 2 L2(G) G is a Dirichlet form, −t`(g) 2 (Tta)(t) = e a(g) a 2 L (G) is its associated Markovian semigroup and (La)(g) = `(g)a(g) a 2 Cc(G) is the associated generator. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Examples: Dirichlet spaces on C∗-algebras of smooth dynamical systems 5. Let α : V × G ! V be a continuous action of a group of isometries G ⊆ Iso (V; g) ∗ of a Riemannian manifold and A = C0(V) oα G its crossed product C -algebra Z −1 ∗ −1 (a∗b)(x; g) = a(x; h)b(xh; h g) dh ; a (x; g) = a(xg; g ) a 2 Cc(V×G) : G The trace, such that L2(A; τ) ' L2(V × G) as Hilbert spaces, is given by Z τ(a) = a(x; e) mg(dx) a 2 Cc(V × G) : V Any continuous negative definite function ` : G ! [0; +1) gives a Dirichlet form by Z 2 2 E[a] = (jra(x; g)j + `(g)ja(x; g)j ) mg(dx)dg : V×G 1 1 The case where V = S × S is the 2-torus, the action of G = Z is given by 2πikθ α((z; w); k) := (z; we )(z; w) 2 V ; k 2 Z ; for a fixed irrational θ 2 [0; 1], is the Kronecker foliation with dense leaves . Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Differential calculus on Dirichlet spaces There is a natural differential calculus underlying any Dirichlet form (E; F): its existence is suggested by the presence of a natural subalgebra. Theorem. (Lindsay-Davies 92’, C. 06’) The Dirichlet space F and the C∗-algebra A intersect in a form core B := F\ A which is an involutive, dense subalgebra of A, called the Dirichlet algebra of (E; F). When endowed with the norm p kakB = kakA + E[a] a 2 B ; it is a semisimple Banach algebra. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Theorem. ([C-Sauvageot JFA ’03]) There exists an essentially unique derivation @ : B!H, defined on the Dirichlet algebra B, with values in a Hilbertian A-bimodule H, i.e. a linear map satisfying Leibniz rule @(ab) = (@a)b + a(@b) a; b 2 B ; by which the Dirichlet form can be represented by 2 E[a] = k@akH a 2 B : The derivation is closable with respect to the norm topology of A and L2(A; τ). In other words, (B; @; H) is a differential square root of the generator: ∆ = @∗ ◦ @: The Hilbert space adjoint map @∗ is a called the divergence operator. There exists an antililear symmetry J : H!H with respect to which the derivation is symmetric J (a(@b)c) = c∗(@b∗)a∗ a; b; c 2 B : Viceversa, any symmetric derivation closable in L2(A; τ), provide a Dirichlet form. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Example 1.1. When applied to a Riemannian manifold (V; g), the above result 1;2 returns the Sobolev space H (V; g) and the gradient operator rg from the Dirichlet integral Z 2 1;2 E[a] = jrgaj dmg a 2 H (V; g) : V Example 4.1. In case of Dirichlet forms on group C∗-algebras C∗(G) associated to continuous functions ` : G ! [0; +1) of negative type Z E[a] = `(g)ja(g)j2 dg ; G the derivation can be constructed using the orthogonal representation π : G ! B(K) and the 1-cocycle c : G !K c(gh) = c(g) + π(g)c(h) g; h 2 G 2 ∗ 2 representing `(g) = kc(g)kK. The Hilbert C (G)-bimodule is given by L (G; KC) acted on the left by λ ⊗ π and on the right id ⊗ ρ. The derivation is given by 2 @ : Cc(G) ! L (G; KC)(@a)(g) = c(g)a(g) g 2 G : Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References Application: Noncommutative Hilbert’s transform in Free Probability 6.