Dixmier traces of nonmeasurable commutators Fun with integrals of - - PowerPoint PPT Presentation

Dixmier traces of nonmeasurable commutators

Fun with integrals of nonintegrable functions

Heiko Gimperlein1 (joint with Magnus Goffeng2)

1: Heriot–Watt University, Edinburgh 2: Chalmers and University of Gothenburg

Scottish Operator Algebras Research meeting Glasgow, 3 December 2016

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 1 / 22

For the algebraists

Fredholm module (A, H, F)

A (unital) C∗ algebra, π : A → H, F ∈ B(H) F = F∗, F2 = Id mod compact [F, π(a)] compact

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 2 / 22

For the algebraists

Fredholm module (A, H, F)

A (unital) C∗ algebra, π : A → H, F ∈ B(H) F = F∗, F2 = Id mod compact [F, π(a)] compact

spectral triple (A, H, D : D ⊂ H → H)

analogue for unbounded operator D D selfadjoint Fredholm, (1 + D2)−1 compact Lipschitz algebra = {a : [D, π(a)] densely defined, bounded} dense in A

Example: D = −i∂ϕ, F =

D √ 1+D2 on M = S1

Fredholm module (C(S1), L2(S1), F) spectral triple (C(S1), L2(S1), D)

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 2 / 22

For the algebraists

Example: D = −i∂ϕ, F =

D √ 1+D2 on M = S1

Fredholm module (C(S1), L2(S1), F) spectral triple (C(S1), L2(S1), D) F example of pseudodifferential operator (ΨDO) F ∼ 2Π+ − Id, Π+ = orthogonal projection onto span{einϕ : n ≥ 0} Study geometry encoded in (C(S1), L2(S1), F) and (C(S1), L2(S1), D)

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 2 / 22

Integration

Example: D = −i∂ϕ, F =

D √ 1+D2 on M = S1

Fredholm module (C(S1), L2(S1), F) spectral triple (C(S1), L2(S1), D) C closed, smooth curve parametrisation Z : S1 → Rd

• C

f ds =

• S1 f |Z′| dθ ≡
• S1 f |dZ|

Noncommutative view on differential forms: dZ = [D, Z], [F, Z]

• C

f ds =

• S1 f |dZ| = Trω(f (1 + D2)− 1

2 |[D, Z]|) = Trω(f |[F, Z]|)

Trω Dixmier trace, TrωA = limN→ω

1 log(N)

N

n=1 λn(A) for A ≥ 0

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 3 / 22

Integration on quasi-Fuchsian circles

Connes ’94 / Sukochev et al. ’16. Open: Compute normalisation λ

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 4 / 22

Dixmier traces of commutators

Trω Dixmier trace, TrωA = limN→ω

1 log(N)

N

n=1 λn(A) for A ≥ 0

We need to understand the precise behaviour of λn(A) as n → ∞. Related motivation: classical theorems in harmonic analysis: boundedness of commutators spectral theory of Hankel operators: ncg approach minimal regularity assumptions geometry/topology on nonsmooth spaces: integration, cohomology, mapping degrees of non-smooth functions, . . .

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 5 / 22

Why commutators? – Harmonic analysis

D1 = d

j=1 aj(x)∂j, aj ∈ L∞(Rd), f : Rd → C Lipschitz =

⇒ [D1, f]φ = D1(fφ)−fD1φ =

j aj(∂jf)φ extends to bounded operator on Lp

• rder of commutator = (order of D1) −1 = 0.

Calderón (1965)

D1 ΨDO on Rd, order 1, f : Rd → C Lipschitz = ⇒ [D1, f] : Lp(Rd) → Lp(Rd) bounded and [D1, f]Lp→Lp fLip

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 6 / 22

Why commutators? – Harmonic analysis

Calderón (1965)

D1 ΨDO on Rd, order 1, f : Rd → C Lipschitz = ⇒ [D1, f] : Lp(Rd) → Lp(Rd) bounded and [D1, f]Lp→Lp fLip

Coifman – Meyer (1978)

P0 ΨDO on Rd, order 0, a : Rd → C bounded mean oscillation (or C0) = ⇒ [P0, a] : Lp(Rd) → Lp(Rd) bounded and [P0, a]Lp→Lp aBMO Kato – Ponce, Auscher – Taylor, . . . [P0, a] compact if e.g. a ∈ C0

c

Spectral theory? . . . related works of Pushnitski, Yafaev, also Frank (2013 –).

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 6 / 22

Dixmier traces of commutators

Trω Dixmier trace, TrωA = limN→ω

1 log(N)

N

n=1 λn(A) for A ≥ 0

We need to understand the precise behaviour of λn(A) as n → ∞. Related motivation: classical theorems in harmonic analysis: boundedness of commutators spectral theory of Hankel operators: ncg approach minimal regularity assumptions geometry/topology on nonsmooth spaces: integration, cohomology, mapping degrees of non-smooth functions, . . .

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 7 / 22

An urgent challenge challenge from theoretical physics

In 1910, Hendrik Lorentz delivered lectures in Göttingen: “Old and new problems in physics”. An “urgent question” related to the radiation of a black body. Prove that the density of standing electromagnetic waves inside a bounded cavity Ω ⊂ R3 is, at high frequency, independent of the shape of Ω. Arnold Sommerfeld had made a similar conjecture earlier in the same year, for scalar waves: Count the solutions to the Helmholtz equation with Dirichlet boundary conditions: −∆uk = λkuk in Ω, u|∂Ω = 0 . Our central objects of interest: λk ∼

c3 vol(Ω)2/3 k2/3 + o(k2/3) in 3 dimensions

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 8 / 22

A young postdoc

Hermann Weyl was attending Lorentz’s lectures. Within a few months he had proved the two–dimensional scalar

• case. Within a few years, he generalized his results to

three dimensions, elastic and electromagnetic waves. In 1913 Weyl conjectured a 2–term asymptotics in d dimensions: NΩ(λ) = #{k : λk ≤ λ} ∼ |Sd−1| |Ω| (2π)d λd/2 − |Sd−2| |∂Ω| 4(2π)d−1 λ(d−2)/2 + o(λ(d−2)/2) Then he moved on to other areas of mathematics.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 9 / 22

Dixmier traces of commutators

Trω Dixmier trace, TrωA = limN→ω

1 log(N)

N

n=1 λn(A) for A ≥ 0

We need to understand the precise behaviour of λn(A) as n → ∞. Related motivation: classical theorems in harmonic analysis: boundedness of commutators spectral theory of Hankel operators: ncg approach minimal regularity assumptions geometry/topology on nonsmooth spaces: integration, cohomology, mapping degrees of non-smooth functions, . . .

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 10 / 22

Outline of remaining talk

general operators: F P0, D D1 A kaleidoscope of commutators – [P0, a] on S1 = ∂D ⊂ C

asymptotics of singular values: limk k2αλk(A∗A) < ∞ if a ∈ Cα, α ∈ (0, 1] wide variety of non-convergence, governed by singularities Dixmier traces of products: Trω [P0, a]2 = 0 if a ∈ C

1 2 +ε

still: Connes’ theorem and integral formulas

higher dimensional generalisations and applications

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 11 / 22

Set-up

Aim: On S1, compute TrωA A = product of commutators [P0, a], a Hölder. TrωA = limN→ω

1 log(N)

N

n=1 λn(A) for A ≥ 0

linear functional Trω : L1,∞ → C on weak-Schatten class L1,∞ = {A ∈ K : sup

n

nλn(A∗A)1/2 < ∞} First question: When is A ∈ L1,∞?

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 12 / 22

Set-up

Aim: On S1, compute TrωA A = product of commutators [P0, a], a Hölder. TrωA = limN→ω

1 log(N)

N

n=1 λn(A) for A ≥ 0

linear functional Trω : L1,∞ → C on weak-Schatten class L1,∞ = {A ∈ K : sup

n

nλn(A∗A)1/2 < ∞} First question: When is A ∈ L1,∞?

Theorem

a) A = Πk

n=1[P0 k, ak] ∈ L1,∞ provided ak ∈ Cαk(S1), k αk = 1.

b) If

k αk > 1, then nλn(A∗A)1/2 → 0 and hence TrωA = 0.

This is the easy part – an upper bound on λn(A).

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 12 / 22

Set-up

Aim: On S1, compute TrωA A = product of commutators [P0, a], a Hölder. TrωA = limN→ω

1 log(N)

N

n=1 λn(A) for A ≥ 0

linear functional Trω : L1,∞ → C on weak-Schatten class L1,∞ = {A ∈ K : sup

n

nλn(A∗A)1/2 < ∞} First question: When is A ∈ L1,∞?

Theorem

a) A = Πk

n=1[P0 k, ak] ∈ L1,∞ provided ak ∈ Cαk(S1), k αk = 1.

b) If

k αk > 1, then nλn(A∗A)1/2 → 0 and hence TrωA = 0.

Pathological functionals: C1/2(S1)2 ∋ (a, b) → Trω[P0, a][P0, b] vanishes if a or b ∈ C1/2+ε(S1)

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 12 / 22

A Lidskii theorem for Dixmier traces

Aim: On S1, compute TrωA A = product of commutators [P0

k, ak], ak ∈ Cαk(S1), k αk = 1.

TrωA = limN→ω

1 log(N)

N

n=1 λn(A) for A ≥ 0

Theorem

TrωA = limN→ω

1 log(2+N)

N

k=0 λk(A) may be computed by Fourier series:

TrωA = lim

N→ω

1 log(2N + 1)

N

• k=−N

Aeikx, eikx . Ordering of N

k=−N is crucial – divergent series!

Proof combines paradifferential techniques from harmonic analysis with recent advances for Dixmier traces of “modulated operators” in

• perator theory (Kalton, Sukochev, . . . )

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 13 / 22

Examples of nice integral formulas

Proposition

Consider the Szégö projection Π+a =

k≥0 ak eikx. For a, b ∈ C1/2(S1),

Trω Π+[Π+, a][Π+, b] = − lim

N→ω 1 log(N+2) limrր1

• S1×S1 a+(¯

ζ)b−(z)kN(rz, ζ) dζ dz , and Trω [Π+, a][Π+, b] = − lim

N→ω 1 log(N+2) limrր1

• S1×S1
• a+(¯

ζ)b−(z) + b+(¯ ζ)a−(z)

• kN(rz, ζ) dζ dz ,

where kN(z, ζ) = 1 − (zζ)N+1 (1 − zζ)2 . Recall that the right hand sides vanish if a or b ∈ C1/2+ε(S1).

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 14 / 22

Weierstrass functions

1 2 3 4 5 6 7 −1.5 −1 −0.5 0.5 1 1.5 2 2.5

Wα,b,(cn)(z) :=

∞

• n=0

b−αncn(zbn+z−bn) = 2

∞

• n=0

b−αncn cos(bnθ), for z = eiθ. Wα,b,(cn) ∈ Cα(S1) if 0 < α < 1, b ∈ N>1, (cn) ∈ ℓ∞(N).

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 15 / 22

limN→∞ does not exist

Theorem

For b ∈ N>1 and c, d ∈ ℓ∞(N) TrωΠ+[Π+, W1/2,b,(cn)][Π+, W1/2,b,(dn)] = − 1 log(b) lim

N→ω

1 N + 1

N

• k=0

cn dn . In particular, Trω[Π+, W1/2,b,(cn)]2 = − 2 log(b) lim

N→ω

1 N + 1

N

• k=0

c2

n .

Note that e.g. for cn =

• 2 + cos(log(n)) the sequence on the rhs

1 N+1

N

k=0 c2 n ∼ 2 + 1 2 {sin(log(N)) + cos(log(N))} diverges:

Its limit depends on ω. Any behavior in the range of the Cesaro operator

1 N+1

N

k=0 possible.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 16 / 22

Nonmeasurable operators

Hankel measurable operators, hmsk,b := {c ∈ ℓ∞(N) : Trω[P, W1/2k,b,c]2k independent of ω }.

Corollary

hms1,b =

• c = (cn)n∈N ∈ ℓ∞(N) :

lim

N→∞

1 N

N

• n=0

c2

n exists

• .

In particular, the inclusion hms1,b ⊆ ℓ∞(N) is strict and does not depend on b. Therefore we obtain countless simple examples of nonmeasurable

• perators. This makes integral formulas even more surprising.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 17 / 22

Generalisations

higher dimensional Riemannian manifolds non-commutative tori (see also Sukochev et al.) sub-Riemannian manifolds modeled on the Heisenberg group instead of Rn

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 18 / 22

Connes’ residue trace theorem

A classical computation of Dixmier traces again involves pseudodifferential operators, functions Q = q(x, D) of D = −i∂x on a closed manifold M: q ∼ smooth rational function: |∂β

x ∂α ξ q(x, ξ)| ≤ Cα,β(1 + |ξ|)m−|α| ∀α, β

m = order ∈ R, p = symbol Assume q ∼ ∞

j=0 qm−j, qm−j(x, λξ) = λm−jqm−j(x, ξ) ∀λ ≥ 1 ∀|ξ| ≥ 1.

Examples

∆ ΨDO of order m = 2, q(x, ξ) = |ξ|2

g + . . .

any differential operator of order m is a ΨDO of order m, and q(x, ξ) polynomial in ξ ∆−1 ΨDO of order m = −2, q(x, ξ) ∼ |ξ|−2

g

+ . . . ∆α, α ∈ C ΨDO of order m = 2Re α, q(x, ξ) ∼ |ξ|2α

g + . . .

• n S1: Szegö projection Π+einx = einx if n ≥ 0, = 0 else,

is ΨDO of order m = 0, q(x, ξ) ∼ χ[0,∞)(ξ) + . . .

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 19 / 22

Connes’ residue trace theorem

A classical computation of Dixmier traces again involves pseudodifferential operators, functions Q = q(x, D) of D = −i∂x on a closed manifold M: q ∼ smooth rational function: |∂β

x ∂α ξ q(x, ξ)| ≤ Cα,β(1 + |ξ|)m−|α| ∀α, β

m = order ∈ R, p = symbol Assume q ∼ ∞

j=0 qm−j, qm−j(x, λξ) = λm−jqm−j(x, ξ) ∀λ ≥ 1 ∀|ξ| ≥ 1.

Connes’ residue trace theorem for Q of order −dim M

TrωQ = 1 d(2π)d

• M
• |ξ|=1

q−dim M(x, ξ) dσξ dx In particular, independent of ω.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 19 / 22

Connes’ residue trace theorem

Connes’ residue trace theorem for Q of order −dim M

TrωQ = 1 d(2π)d

• M
• |ξ|=1

q−dim M(x, ξ) dσξ dx In particular, independent of ω. Basic principle of quantization: If a ∈ C∞, then [P0, a] order −1, symbol ∼ −i{p0, a} = i ∂p0

∂ξ ∂a ∂x.

Trω([P0, a])dim M = 1 d(2π)d

• M
• |ξ|=1
• i∂p0

∂ξ ∂a ∂x dim M dσξ dx

Theorem

Trω([P0, a])dim M = i d(2π)d

• M
• |ξ|=1

∂p0 ∂ξ ∂a ∂x dσξ dx for a ∈ Lip(M). In particular, independent of ω.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 19 / 22

Connes’ residue trace theorem

Connes’ residue trace theorem for Q of order −dim M

TrωQ = 1 d(2π)d

• M
• |ξ|=1

q−dim M(x, ξ) dσξ dx In particular, independent of ω.

Theorem

Trω([P0, a])dim M = i d(2π)d

• M
• |ξ|=1

∂p0 ∂ξ ∂a ∂x dσξ dx for a ∈ Lip(M). In particular, independent of ω. For a ∈ Cα(M), Trω([P0, a])dim M generally diverges.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 19 / 22

(Implicit) Connes’ theorem in general dimension

Reformulation of Connes’ residue trace theorem, d = dim M

Trω P = lim

N→∞

1 log(2 + N)

N

• k=0

λk(P) = 1 d(2π)d

• M
• |ξ|=1

p−d(x, ξ) dσξ dx

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 20 / 22

(Implicit) Connes’ theorem in general dimension

Reformulation of Connes’ residue trace theorem, d = dim M

Trω P = lim

N→∞

1 log(2 + N)

N

• k=0

λk(P) = 1 d(2π)d

• M
• |ξ|=1

p−d(x, ξ) dσξ dx = lim

N→∞

1 (2π)d log(2 + N)

• M
• |ξ|≤N1/d p−d(x, ξ) dξ dx

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 20 / 22

(Implicit) Connes’ theorem in general dimension

Reformulation of Connes’ residue trace theorem, d = dim M

Trω P = lim

N→∞

1 log(2 + N)

N

• k=0

λk(P) = 1 d(2π)d

• M
• |ξ|=1

p−d(x, ξ) dσξ dx = lim

N→∞

1 (2π)d log(2 + N)

• M
• |ξ|≤N1/d p−d(x, ξ) dξ dx

= lim

N→∞

1 (2π)d log(2 + N)

• M
• |ξ|≤N1/d p(x, ξ) dξ dx

= lim

N→ω

1 (2π)d log(2 + N)

• M
• |ξ|≤N1/d p(x, ξ) dξ dx

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 20 / 22

(Implicit) Connes’ theorem in general dimension

Theorem

Let A be a product of commutators [P0

k, ak], ak ∈ Cαk,

• k αk = d = dim M.

= ⇒ Trω A = lim

N→ω

1 (2π)d log(2 + N)

• M
• |ξ|≤N

pA(x, ξ) dξ dx , where pA is the symbol of the conormal distribution KA(x, y). More useful in terms of the integral kernel kA: Trω A = lim

N→ω

N log(N)

• |x−y|<N−1 kA(x, y) dx dy .

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 20 / 22

Complex analysis: commutators as differential forms

∂Ω ⊂ Cn strictly pseudoconvex, Π+ Szegö ζk,ω : Cn/k

CC (M)⊗2k → C defined by

ζk,ω(a1⊗· · ·⊗a2k) = TrωΠ+[[Π+, a1], [Π+, a2]] · · · [[Π+, a2k−1], [Π+, a2k]] .

Corollaries

a) ζk,ω defines a continuous multilinear functional = 0. b) k > n: If ∃j, β > n

k s.t. aj ∈ Cβ CC(M), then ζk,ω(a1 ⊗ · · · ⊗ a2k) = 0.

c) k = n on Lip(M): Englis, Guo, Zhang (2009, 2010) ζn,ω(a1⊗· · ·⊗a2n) =

1 n!(2π)n

• ∂Ω L(∂ba1, ∂ba2) · · · L(∂ba2n−1, ∂ba2n)η∧(dη)n−1

where ∂b boundary ∂–operator, L dual to Levi form ∂∂̺ of ∂Ω, η contact form.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 21 / 22

Complex analysis: commutators as differential forms

Corollaries

a) ζk,ω defines a continuous multilinear functional = 0. b) k > n: If ∃j, β > n

k s.t. aj ∈ Cβ CC(M), then ζk,ω(a1 ⊗ · · · ⊗ a2k) = 0.

c) k = n on Lip(M): Englis, Guo, Zhang (2009, 2010) ζn,ω(a1⊗· · ·⊗a2n) =

1 n!(2π)n

• ∂Ω L(∂ba1, ∂ba2) · · · L(∂ba2n−1, ∂ba2n)η∧(dη)n−1

where ∂b boundary ∂–operator, L dual to Levi form ∂∂̺ of ∂Ω, η contact form. The map R : Kalg

2 (C1/2(S1)) → C∗,

R(a, b) = exp(TrωΠ+[Π+, a][Π+, b]) , seems to be of interest to topologists and noncommutative geometers.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 21 / 22

References

HG, M. Goffeng, Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds, Forum of Mathematics, Sigma, to appear. HG, M. Goffeng, Commutator estimates on sub-Riemannian manifolds and applications, preprint.

Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 22 / 22