Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative 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  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   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, +∞] is a l.s.c., quadratic form such that E[a∗] = E[a] E[a ∧ 1] ≤ E[a] the domain F := {a ∈ L2(A, τ): E[a] < +∞} dense in L2(A, τ) the subspace B := F ∩ A is 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 (a|a − Tta) a ∈ F t→0 t or through the self-adjoint generator (L, dom (L)) √ √ −tL 2 Tt = e E[a] = k LakL2(A,τ) a ∈ F = dom ( L) . These semigroups, can be characterized as those weakly∗−continuous, positively preserving, contractive semigroups on the von Neumann algebra L∞(A, τ) which are

∞ τ − symmetric τ(a(Ttb)) = τ((Tta)b) a, b ∈ 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 1. The archetypical Dirichlet form is the of a Riemannian manifold Z Z 2 (V, g), A = C0(V), τ(a) = a dmg , E[a] = |∇a| 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] = |∇Ba| 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] = |Da| 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 ∈ G, C∗(G) its convolution group C∗-algebra with trace

τ(a) = a(e) a ∈ Cc(G)

and recall that L2(A, τ) ' L2(G). Then, for any continuous, negative definite function ` : G → [0, +∞), Z E[a] = `(g)|a(g)|2 dg a ∈ L2(G) G is a Dirichlet form,

−t`(g) 2 (Tta)(t) = e a(g) a ∈ L (G)

is its associated Markovian semigroup and

(La)(g) = `(g)a(g) a ∈ 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 ∈ 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 ∈ Cc(V × G) . V Any continuous negative definite function ` : G → [0, +∞) gives a Dirichlet form by Z 2 2 E[a] = (|∇a(x, g)| + `(g)|a(x, g)| ) 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) ∈ V , k ∈ Z , for a fixed irrational θ ∈ [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 ∈ B ,

it is a semisimple . 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 ∈ B ,

by which the Dirichlet form can be represented by

2 E[a] = k∂akH a ∈ 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 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 ∈ 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 operator ∇g from the Dirichlet integral Z 2 1,2 E[a] = |∇ga| dmg a ∈ H (V, g) . V Example 4.1. In case of Dirichlet forms on group C∗-algebras C∗(G) associated to continuous functions ` : G → [0, +∞) of negative type Z E[a] = `(g)|a(g)|2 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 ∈ 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 ∈ 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. Let (M, τ) be a nc-probability space and consider 1 ∈ B ⊂ M a ∗-subalgebra X ∈ M a nc-random variable, algebraically free with respect to B B[X] ⊂ M ∗-subalgebra generated by X and B (regarded as nc-polynomials in the variable X with coefficients in B W ⊂ M the von Neumann subalgebra generated by B[X].

Theorem. (Voiculescu ’00) 2 There exists a unique derivation ∂X : B[X] → L (W ⊗ W, τ ⊗ τ) such that

∂XX = 1 ⊗ 1

∂Xb = 0 b ∈ B . ∗ Under the assumption 1 ⊗ 1 ∈ dom (∂X ) it follows that 2 ∗ (∂X, B[X]) is closable in L (W , τ) 2 the closure of EX[a] := k∂Xak is a Dirichlet form. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Definition. (Voiculescu ’00) ∗ Under the assumption 1 ⊗ 1 ∈ dom (∂X ) define ∗ ∗ 2 J (X : B) := ∂X (1 ⊗ 1) = ∂X ∂X(X) ∈ L (W ⊗ W, τ ⊗ τ) nc-Hilbert Transform of X w.r.t. B 2 ∗ 2 Φ(X : B) := kJ (X : B)k = k∂X ∂X(X)k relative free information of X w.r.t. B.

∞ ∞ In case M = L (R, m), B = C, X ∈ M has distribution µX one has W = L (R, µX), C[X] is the algebra of polynomials on R and ∂Xf coincides with the difference dµX 3 quotient. In case p := dm ∈ L (R, m), then J (X : B) is the usual Hilbert transform 1 Z p(s) Hp(t) := p.v. ds . π t − s R

Theorem. (Biane ’03) ∗ Under the assumption 1 ⊗ 1 ∈ dom (∂X ) we have

the Dirichlet form EX is the Hessian of the Free Entropy

EX satisfies a Poincaré inequality (spectral gap) iff X is centered, it has unital covariance and semicircular distribution. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Other applictions

Differential calculus in momentum space of electrons in Quasi-Crystals and in Quantum Hall Effect (by J. Bellissard) K-theory (by D:V: Voiculescu in Almost normal operators mod Hilbert-Schmidt and the K-theory of the Banach algebras EΛ(O) arXiv:1112.4930 to appear J. Noncommutative Geometry) Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Finite energy states, potentials

Finer properties of the differential calculus underlying a Dirichlet spaces rely on some properties of the basic objects of the Potential Theory of Dirichlet forms.

q 2 Consider the Dirichlet space with its Hilbertian norm kakF := E[a] + kakL2(A,τ).

Definition. (C-Sauvageot ’12 arXiv:1207.3524) p ∈ F is called a potential if

2 (p|a)F ≥ 0 a ∈ F+ := F ∩ L (A, τ)

Denote by P ⊂ L2(A, τ) the closed convex cone of potentials. ∗ ω ∈ A+ has finite energy if for some c ≥ 0

|ω(a)| ≤ cω · kakF a ∈ F . Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Theorem. (C-Sauvageot ’12 arXiv:1207.3524) Let (E, F) be a Dirichlet form on (A, τ). 2 Potentials are positive: P ⊂ L+(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 .

2 1 ∗ Example 1. If h ∈ L+(A, τ) ∩ L (A, τ) then ωh ∈ A+ defined by

ωh(a) := τ(ha) a ∈ A

−1 is a finite energy functional whose potential is given by G(ωh) = (I + L) h. ∗ Example 4.2. Let E` be the Dirichlet form on Cred(Γ), associated to a negative definite function ` on a discrete group Γ. Then ω is a finite energy functional iff

2 X |ω(δs)| < +∞ 1 + `(s) t∈Γ

ω(δs) and its potential is given by G(ω)(s) = 1+`(s) s ∈ Γ. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Example 1.3. In a d-dimensional Riemannian manifold (V, g), the volume measure µW of a (d − 1)-dimensional compact submanifold W ⊂ V has finite energy.

Theorem. Deny’s embedding (C-Sauvageot ’12 arXiv:1207.3524) ∗ Let ω ∈ A+ be a finite energy functional with bounded potential

G(ω) ∈ P ∩ L∞(A, τ) .

Then ∗ 2 ω(b b) ≤ ||G(ω)||M ||b||F b ∈ B . 1 2 The embedding F # L (A, ω) is thus upgraded to an embedding F # L (A, ω).

Example. Let E` be the Dirichlet form associated to a negative type function ` on a discrete group Γ. Deny’s theorem applies whenever P 1 2 g 1+`(g) |ω(δg)| < +∞ ω has finite energy

X 1 00 ω(g)λ(g) ∈ λ(Γ) ω has bounded potential. 1 + `(g) g ∗ It is not difficult, in concrete examples, to find an ω which is a coefficient on Cr (G), but not a coefficient of the regular representation (i.e. ω is singular with respect to τ). Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Theorem. Deny’s inequality (C-Sauvageot ’12 arXiv:1207.3524) ∗ Let ω ∈ A+ be a finite energy functional with potential G(ω) ∈ P. Then the following inequality holds true 1 ωb∗ b ≤ ||b||2 b ∈ F . G(ω) F

In the noncommutative case, since in general the finite energy functional ω is not a trace, the proofs of the above results necessarily require the treatment of KMS-symmetric Markovian semigroups and Dirichlet forms on Standard Forms of von Neumann algebras. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Multipliers of Dirichlet spaces The following one is the central subject of Potential Theory whose properties will be crucial to investigate geometrical aspects.

q 2 Consider the Dirichlet space with its Hilbertian norm kakF := E[a] + kakL2(A,τ).

Definition. (C-Sauvageot ’12 arXiv:1207.3524) An element of the von Neumann algebra b ∈ L∞(A, τ) is a multiplier of the Dirichlet space if b · F ⊆ F , F· b ⊆ F . Denote the algebra of multipliers by M(F). By the Closed Graph Theorem, multipliers are bounded operators on the Dirichlet space M(F) ⊂ B(F).

Example. Let F` be the Dirichlet space associated to a negative type function ` on a 00 discrete group Γ. Then the unitaries δt ∈ λ(Γ) are multipliers and √ p kδtkF` ≤ 2 1 + `(t) t ∈ Γ. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Theorem. Existence and abundance of multipliers (C-Sauvageot ’12 arXiv:1207.3524)

Let I(A, τ) ⊂ L∞(A, τ) be the norm closure of the ideal L1(A, τ) ∩ L∞(A, τ). Then (I + L)−1h is a multiplier for any h ∈ I(A, τ) √ −1 k(I + L) hkB(F) ≤ 2 5khk∞ h ∈ I(A, τ) .

bounded eigenvectors h ∈ Lp(A, τ) ∩ L∞(A, τ) of the generator on Lp(A, τ)

Lh = λh √ are multipliers and khkB(F) ≤ 2 5(1 + λ)khk∞ the algebra of finite energy multipliers M(F) ∩ F is a form core the Dirichlet form is regular on the C∗-algebra M(F) ∩ F M(F) ∩ F is norm dense in the C∗-algebra A provided the semigroup is strongly continuous on A

Notice that, even if the definition of multiplier of the Dirichlet space F does not involve properties of the quadratic form E other that to be closed, proofs of existence and large supply of multipliers are based on the properties of potentials and finite energy states developed in Noncommutative Potential Theory. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Dirac operator The differential calculus of a Dirichlet space provides a natural definition of a Dirac operator on which the development of Conformal or Metric Geometry can be based following the Spectral approach introduced by A. Connes.

Definition. Dirac operator (C-Sauvageot ’12 arXiv:1207.3524) Let (E, F) be a Dirichlet form on (A, τ) and consider its derivation ∂ : F → H. The Dirac operator is defined as the densely defined, closed operator on the Hilbert space L2(A, τ) ⊕ H

 0 ∂∗  D := dom(D) := F ⊕ F ∗ ⊆ L2(A, τ) ⊕ H ∂ 0

Notice that the square of the Dirac operator is given by

 ∂∗∂ 0  D2 = , 0 ∂∂∗

where L = ∂∗∂ is the self-adjoint generator L2(A, τ) whose quadratic form is (E, F). Since now on, assume that the spectrum of (E, F) on L2(A, τ) is discrete. This implies that, far away from zero, the spectrum of the Dirac D operator is discrete too. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Fredholm module Let (E, F) be a Dirichlet form on (A, τ) and consider its derivation ∂ : F → H.

Definition. The symmetry of a Dirichlet space (C-Sauvageot ’12 arXiv:1207.3524)

The symmetry (F, HF) of a Dirichlet space (E, F) on (A, τ) is defined as the symmetry with respect to the graph G(∂∗) of the divergence operator (∂∗, F ∗) in the 2 Hilbert space HF := L (A, τ) ⊕ H. In other words

F := P − P⊥

∗ where P ∈ B(HF) is the projection onto G(∂ ).

Theorem. Commutator compactness (C-Sauvageot ’12) Suppose that the Dirichlet form (E, F) has discrete spectrum on L2(A, τ). 2 Then the commutator [F, a] ∈ B(L (A, τ) ⊕ H) is a compact operator for any multiplier a ∈ M(F) of the Dirichlet space (E, F) ∗ (F, HF) is a Fredholm module on the C algebra M(F) ∗ (F, HF) is a Fredholm module on the C algebra A provided that the Markov semigroup is strongly continuous on A Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Theorem. Summability (C-Sauvageot ’12 arXiv:1207.3524)

The singular values µn([F, a]) of the commutator [F, a] with any multiplier, are bounded above in terms of the eigenvalues λn(L) of the generator L as follows

1 − 2 µn,([F, a]) ≤ 4kakB(F) · (1 + λn(L)) n ∈ N . (1)

− 1 d/2 In particular, if (I + L) 2 is (d, ∞)-summable, in the sense that λn(L) ' n , then

d d+1 d − d (|[F, a]| ) ≤ kak · (I + L) 2 < +∞ TrDix 4 B(F) TrDix (2)

where TrDix denotes the Dixmier trace associated to any ultrafilter over the integers.

Definition. Conformal energy (C-Sauvageot ’12)

d/2 Assume the Weyl’s asymptotics λn(L) ' n holds true. The conformal energy functional is defined

d d d E : M(E, F) → [0, +∞) E [a] := TrDix(|[F, a]| ) . (3)

By the previous result, the following bound holds true

d d d − d E [a] ≤ kak · (I + ∆) 2 < +∞ . 4 B(F) TrDix (4) Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Example. In case the Dirichlet integral of a Riemannian manifold (V, g) Z 2 1,2 E[a] = |∇a| dmg a ∈ H (V) V the conformal energy functional reduces with the Sobolev Z d d 1,d E [a] = |∇a| dmg a ∈ H (V) V which is the fundamental global conformal invariant of (V, g). Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

Spectral Triple

Theorem. Spectral Triple of a Dirichlet space (C-Sauvageot ’12 arXiv:1207.3524) ∗ Define the carré du champ Γ[a] ∈ A+ as follows

hΓ[a], bi := (∂a|(∂a)b)H b ∈ B .

dΓ[a] ∞ Then the space A := {a ∈ B : dτ ∈ L (A, τ)} is an involutive subalgebra of A and (A, D, L2(A, τ) ⊕ H) is a Spectral Triple

2 [D, a] ∈ B(L (A, τ) ⊕ H)

The above condition on the carré du champ has an algebraic interpretation.

Theorem. (C-Sauvageot ’12 arXiv:1207.3524)

For a ∈ B ∩ domL∞ (∆) the following conditions are equivalent the commutator [D, a] is a bounded operator on L2(A, τ) ⊕ H ∗ a a ∈ domL∞ (∆) .

Last two conditions do not hold true on self-similar fractal spaces where the energy and volume are distributed singularly. Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

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F. Cipriani - J.L. Sauvageot Derivations as square roots of Dirichlet forms J. Funct. Anal. 201 (2003), no. 1, 78–120. Noncommutative potential theory and the sign of the curvature operator in Riemannian geometry Geom. Funct. Anal. 13 (2003), no. 3, 521–545. Strong solutions to the Dirichlet problem for differential forms: a quantum dynamical semigroup approach Contemp. Math, A.M.S. 335 (2003), 109-117. Fredholm modules on p.c.f. self-similar fractals and their conformal geometry Comm. Math. Phys. 286, 2009, 541-558 Variations in noncommutative potential theory: finite energy states, potentials and multipliers arXiv:1207.3524, 29 pages Overview Dirichlet spaces Differential calculus in Dirichlet spaces Noncommutative Potential Theory Dirac operator, Fredholm modules, Spectral Triple References

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