Noncommutative Geometry of Dirichlet spaces · Intensive Month on Operator Algebra and Harmonic...

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 MatematicaPolitecnico di Milano(

Joint works with J.-L. Sauvageot)

Intensive Month on Operator Algebra and Harmonic Analysis,Madrid, 20 May - 14 June 2013


Noncommutative Geometry underlying Dirichlet forms on singular spaces

Usually one uses tools of NCG to build up hamiltonian on singular spaces

NCG→ Energy functionals

Ground States in QFT (L. Gross)Algebraic QFTQuantum Hall Effect (J. Bellissard)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 structuresunderlying Energy functionals

Energy functionals→ NCG

Compact Quantum Groups (U. Franz, A. Kula)Orbits of Dynamical systems (M. Mauri)Fractals (D. Guido, T. Isola, J.-L. Sauvageot)


Dirichlet forms, Dirichlet spaces

Dirichlet spaces on locally compact Hausdorff spaces were introduced byBeurling-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 oneof potential kernel (Newtonian, Riesz,...) in classical Potential Theory on Rn.

Generalization to C∗-algebras with traces were initiated by

[Gross ’72]: Hypercontractive Markovian semigroups on von Neumann algebraswith finite traces to prove existence/uniqueness of ground states in QFT[Albeverio-Høegh-Krohn ’76]: Dirichlet forms and Markovian semigroups onC∗-algebras with trace[Sauvageot ’88]: dilation of Markovian semigroups on C∗-algebras with trace[Davies-Lindsay ’88]: construction of Dirichlet forms by unbounded derivations onC∗-algebras with trace[Goldstein-Lindsay ’92]: Dirichlet forms on Haagerup’s standard form of vonNeumann algebras[Cipriani ’92,’97]: Dirichlet forms and Markovian semigroups on standard form ofvon Neumann algebras[Cipriani-Sauvageot ’03]: Dirichlet forms and Hilbert bimodule derivations onC∗-algebras with trace[Cipriani ’08]: KMS-symmetric Markovian semigroups on C∗-algebras


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 norm dense in A.

(E ,F) is a complete Dirichlet space if its matrix ampliations

En[(aij)ij] :=∑

ij

E [aij]

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

‖a‖F :=√E [a] + ‖a‖2

L2(A,τ).


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

E [a] = limt→0

1t(a|a− Tta) a ∈ F

or through the self-adjoint generator (L, dom (L))

Tt = e−tL E [a] = ‖√

La‖2L2(A,τ) a ∈ F = dom (

√L) .

These semigroups, can be characterized as those weakly∗−continuous, positivelypreserving, 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 .


Examples: Dirichlet spaces on Riemannian manifold

1. The archetypical Dirichlet form is the Dirichlet integral of a Riemannian manifold

(V, g), A = C0(V), τ(a) =

∫V

a dmg , E [a] =

∫V|∇a|2 dmg ,

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 aRiemannian manifold, the quadratic form of the Bochner Laplacian

EB[a] =

∫V|∇Ba|2 dmg

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 Riemannianmanifold, the quadratic form of the Dirac Laplacian D2 ' ∆HdR

ED[a] =

∫V|Da|2 dmg

is a Dirichlet form if and only if the curvature operator is nonnegative R ≥ 0.Recall Cl0(V, g) ' C0(Λ

∗(V)), D ' d + d∗, D2 ' dd∗ + d∗d ' ∆HdR.


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,+∞),

E [a] =

∫G`(g)|a(g)|2 dg a ∈ L2(G)

is a Dirichlet form,

(Tta)(t) = e−t`(g)a(g) a ∈ L2(G)

is its associated Markovian semigroup and

(La)(g) = `(g)a(g) a ∈ Cc(G)

is the associated generator.


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

(a∗b)(x, g) =

∫G

a(x, h)b(xh, h−1g) dh , a∗(x, g) = a(xg, g−1) a ∈ Cc(V×G) .

The trace, such that L2(A, τ) ' L2(V × G) as Hilbert spaces, is given by

τ(a) =

∫V

a(x, e) mg(dx) a ∈ Cc(V × G) .

Any continuous negative definite function ` : G→ [0,+∞) gives a Dirichlet form by

E [a] =

∫V×G

(|∇a(x, g)|2 + `(g)|a(x, g)|2) mg(dx)dg .

The case where V = S1 × S1 is the 2-torus, the action of G = Z is given by

α((z,w), k) := (z,we2πikθ) (z,w) ∈ V , k ∈ Z ,

for a fixed irrational θ ∈ [0, 1], is the Kronecker foliation with dense leaves .


Differential calculus on Dirichlet spaces

There is a natural differential calculus underlying any Dirichlet form (E ,F): itsexistence 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

‖a‖B = ‖a‖A +√E [a] a ∈ B ,

it is a semisimple Banach algebra.


Theorem. ([C-Sauvageot JFA ’03])

There exists an essentially unique derivation ∂ : B → H, defined on the Dirichletalgebra B, with values in a Hilbertian A-bimoduleH, 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

E [a] = ‖∂a‖2H 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 Hilbert space adjoint map ∂∗ is a called the divergence operator.There exists an antililear symmetry J : H → H with respect to which the derivationis 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.


Example 1.1. When applied to a Riemannian manifold (V, g), the above resultreturns the Sobolev space H1,2(V, g) and the gradient operator∇g from the Dirichletintegral

E [a] =

∫V|∇ga|2 dmg a ∈ H1,2(V, g) .

Example 4.1. In case of Dirichlet forms on group C∗-algebras C∗(G) associated tocontinuous functions ` : G→ [0,+∞) of negative type

E [a] =

∫G`(g)|a(g)|2 dg ,

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

representing `(g) = ‖c(g)‖2K. The Hilbert C∗(G)-bimodule is given by L2(G,KC)

acted on the left by λ⊗ π and on the right id ⊗ ρ. The derivation is given by

∂ : Cc(G)→ L2(G,KC) (∂a)(g) = c(g)a(g) g ∈ G .


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)

There exists a unique derivation ∂X : B[X]→ L2(W ⊗W, τ ⊗ τ) such that

∂XX = 1⊗ 1

∂Xb = 0 b ∈ B .

Under the assumption 1⊗ 1 ∈ dom (∂∗X ) it follows that

(∂X,B[X]) is closable in L2(W∗, τ)

the closure of EX[a] := ‖∂Xa‖2 is a Dirichlet form.


Definition. (Voiculescu ’00)

Under the assumption 1⊗ 1 ∈ dom (∂∗X ) define

J (X : B) := ∂∗X (1⊗ 1) = ∂∗X∂X(X) ∈ L2(W ⊗W, τ ⊗ τ)

nc-Hilbert Transform of X w.r.t. B

Φ(X : B) := ‖J (X : B)‖2 = ‖∂∗X∂X(X)‖2

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 ∂X f coincides with the differencequotient. In case p := dµX

dm ∈ L3(R,m), then J (X : B) is the usual Hilbert transform

Hp(t) := p.v.1π

∫R

p(s)t − s

ds .

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 unitalcovariance and semicircular distribution.


Other applictions

Differential calculus in momentum space of electrons in Quasi-Crystals and inQuantum Hall Effect (by J. Bellissard)

K-theory (by D:V: Voiculescu in Almost normal operators modHilbert-Schmidt and the K-theory of the Banach algebras EΛ(O)arXiv:1112.4930 to appear J. Noncommutative Geometry)


Finite energy states, potentials

Finer properties of the differential calculus underlying a Dirichlet spaces rely onsome 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

L2(A,τ).

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

p ∈ F is called a potential if

(p|a)F ≥ 0 a ∈ F+ := F ∩ L2(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 · ‖a‖F a ∈ F .


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

Let (E ,F) be a Dirichlet form on (A, τ).

Potentials are positive: P ⊂ L2+(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 1. If h ∈ L2+(A, τ) ∩ L1(A, τ) then ωh ∈ A∗+ defined by

ωh(a) := τ(ha) a ∈ A

is a finite energy functional whose potential is given by G(ωh) = (I + L)−1h.

Example 4.2. Let E` be the Dirichlet form on C∗red(Γ), associated to a negativedefinite function ` on a discrete group Γ. Then ω is a finite energy functional iff∑

t∈Γ

|ω(δs)|2

1 + `(s)< +∞

and its potential is given by G(ω)(s) = ω(δs)1+`(s) s ∈ Γ.


Example 1.3. In a d-dimensional Riemannian manifold (V, g), the volume measureµW 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) .

Thenω(b∗b) ≤ ||G(ω)||M ||b||2F b ∈ B .

The embedding F # L1(A, ω) is thus upgraded to an embedding F # L2(A, ω).

Example. Let E` be the Dirichlet form associated to a negative type function ` on adiscrete group Γ. Deny’s theorem applies whenever∑

g1

1+`(g) |ω(δg)|2 < +∞ ω has finite energy∑g

11 + `(g)

ω(g)λ(g) ∈ λ(Γ)′′ ω has bounded potential.

It is not difficult, in concrete examples, to find an ω which is a coefficient on C∗r (G),but not a coefficient of the regular representation (i.e. ω is singular with respect to τ ).


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

ω(b∗

1G(ω)

b)≤ ||b||2F b ∈ F .

In the noncommutative case, since in general the finite energy functional ω is not atrace, the proofs of the above results necessarily require the treatment ofKMS-symmetric Markovian semigroups and Dirichlet forms on Standard Forms ofvon Neumann algebras.


Multipliers of Dirichlet spaces

The following one is the central subject of Potential Theory whose properties will becrucial to investigate geometrical aspects.

Consider the Dirichlet space with its Hilbertian norm ‖a‖F :=√E [a] + ‖a‖2

L2(A,τ).

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

An element of the von Neumann algebra b ∈ L∞(A, τ) is a multiplier of theDirichlet space if

b · F ⊆ F , F · b ⊆ F .Denote the algebra of multipliers byM(F).

By the Closed Graph Theorem, multipliers are bounded operators on the DirichletspaceM(F) ⊂ B(F).

Example. Let F` be the Dirichlet space associated to a negative type function ` on adiscrete group Γ. Then the unitaries δt ∈ λ(Γ)′′ are multipliers and

‖δt‖F` ≤√

2√

1 + `(t) t ∈ Γ.


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

‖(I + L)−1h‖B(F) ≤ 2√

5‖h‖∞ 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 ‖h‖B(F) ≤ 2√

5(1 + λ)‖h‖∞the algebra of finite energy multipliersM(F) ∩ F is a form core

the Dirichlet form is regular on the C∗-algebraM(F) ∩ FM(F) ∩ F is norm dense in the C∗-algebra A provided the semigroup isstrongly continuous on A

Notice that, even if the definition of multiplier of the Dirichlet space F does notinvolve properties of the quadratic form E other that to be closed, proofs of existenceand large supply of multipliers are based on the properties of potentials and finiteenergy states developed in Noncommutative Potential Theory.


Dirac operator

The differential calculus of a Dirichlet space provides a natural definition of a Diracoperator on which the development of Conformal or Metric Geometry can be basedfollowing 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 Hilbertspace L2(A, τ)⊕H

D :=

(0 ∂∗

∂ 0

)dom(D) := F ⊕ F∗ ⊆ L2(A, τ)⊕H

Notice that the square of the Dirac operator is given by

D2 =

(∂∗∂ 0

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. Thisimplies that, far away from zero, the spectrum of the Dirac D operator is discrete too.


Fredholm moduleLet (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 thesymmetry with respect to the graph G(∂∗) of the divergence operator (∂∗,F∗) in theHilbert space HF := L2(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).

Then the commutator [F, a] ∈ B(L2(A, τ)⊕H) is a compact operator for anymultiplier a ∈M(F) of the Dirichlet space (E ,F)

(F,HF) is a Fredholm module on the C∗algebraM(F)

(F,HF) is a Fredholm module on the C∗algebra A provided that the Markovsemigroup is strongly continuous on A


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

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

µn,([F, a]) ≤ 4‖a‖B(F) · (1 + λn(L))−12 n ∈ N . (1)

In particular, if (I + L)−12 is (d,∞)-summable, in the sense that λn(L) ' nd/2, then

TrDix(|[F, a]|d) ≤ 4d+1‖a‖dB(F) · TrDix(I + L)−

d2 < +∞ (2)

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

Definition. Conformal energy (C-Sauvageot ’12)

Assume the Weyl’s asymptotics λn(L) ' nd/2 holds true. The conformal energyfunctional is defined

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

By the previous result, the following bound holds true

Ed[a] ≤ 4d‖a‖dB(F) · TrDix(I + ∆)−

d2 < +∞ . (4)


Example. In case the Dirichlet integral of a Riemannian manifold (V, g)

E [a] =

∫V|∇a|2 dmg a ∈ H1,2(V)

the conformal energy functional reduces with the Sobolev seminorm

Ed[a] =

∫V|∇a|d dmg a ∈ H1,d(V)

which is the fundamental global conformal invariant of (V, g).


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

〈Γ[a], b〉 := (∂a|(∂a)b)H b ∈ B .

Then the space A := {a ∈ B : dΓ[a]dτ ∈ L∞(A, τ)} is an involutive subalgebra of A

and (A,D, L2(A, τ)⊕H) is a Spectral Triple

[D, a] ∈ B(L2(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)⊕Ha∗a ∈ domL∞ (∆) .

Last two conditions do not hold true on self-similar fractal spaces where the energyand volume are distributed singularly.


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Symmetries of Lévy processes, their Markov semigroups and potential theoryon compact quantum groupsarXiv:1210.6768, 50 pages

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