Dixmier traces of nonmeasurable commutators Fun with integrals of nonintegrable functions Heiko Gimperlein 1 (joint with Magnus Goffeng 2 ) 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 ∗ , F 2 = 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 ∗ , F 2 = Id mod compact [ F , π ( a )] compact spectral triple ( A , H , D : D ⊂ H → H ) analogue for unbounded operator D D selfadjoint Fredholm, ( 1 + D 2 ) − 1 compact Lipschitz algebra = { a : [ D , π ( a )] densely defined, bounded } dense in A 1 + D 2 on M = S 1 D Example: D = − i ∂ ϕ , F = √ Fredholm module ( C ( S 1 ) , L 2 ( S 1 ) , F ) spectral triple ( C ( S 1 ) , L 2 ( S 1 ) , D ) Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 2 / 22
For the algebraists D 1 + D 2 on M = S 1 Example: D = − i ∂ ϕ , F = √ Fredholm module ( C ( S 1 ) , L 2 ( S 1 ) , F ) spectral triple ( C ( S 1 ) , L 2 ( S 1 ) , D ) F example of pseudodifferential operator ( Ψ DO) F ∼ 2 Π + − Id , Π + = orthogonal projection onto span { e in ϕ : n ≥ 0 } Study geometry encoded in ( C ( S 1 ) , L 2 ( S 1 ) , F ) and ( C ( S 1 ) , L 2 ( S 1 ) , D ) Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 2 / 22
Integration D Example: D = − i ∂ ϕ , F = 1 + D 2 on M = S 1 √ Fredholm module ( C ( S 1 ) , L 2 ( S 1 ) , F ) spectral triple ( C ( S 1 ) , L 2 ( S 1 ) , D ) C closed, smooth curve parametrisation Z : S 1 → R d � � � S 1 f | Z ′ | d θ ≡ f ds = S 1 f | dZ | C Noncommutative view on differential forms: dZ = [ D , Z ] , [ F , Z ] � � S 1 f | dZ | = Tr ω ( f ( 1 + D 2 ) − 1 2 | [ D , Z ] | ) = Tr ω ( f | [ F , Z ] | ) f ds = C � N 1 Tr ω Dixmier trace, Tr ω A = lim N → ω n = 1 λ n ( A ) for A ≥ 0 log ( N ) 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 � N 1 Tr ω Dixmier trace, Tr ω A = lim N → ω n = 1 λ n ( A ) for A ≥ 0 log ( N ) We need to understand the precise behaviour of λ n ( 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 D 1 = � d j = 1 a j ( x ) ∂ j , a j ∈ L ∞ ( R d ) , f : R d → C Lipschitz = ⇒ [ D 1 , f ] φ = D 1 ( f φ ) − fD 1 φ = � j a j ( ∂ j f ) φ extends to bounded operator on L p order of commutator = (order of D 1 ) − 1 = 0. Calderón (1965) D 1 Ψ DO on R d , order 1 , f : R d → C Lipschitz ⇒ [ D 1 , f ] : L p ( R d ) → L p ( R d ) bounded and � [ D 1 , f ] � L p → L p � � f � Lip = Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 6 / 22
Why commutators? – Harmonic analysis Calderón (1965) D 1 Ψ DO on R d , order 1 , f : R d → C Lipschitz ⇒ [ D 1 , f ] : L p ( R d ) → L p ( R d ) bounded and � [ D 1 , f ] � L p → L p � � f � Lip = Coifman – Meyer (1978) P 0 Ψ DO on R d , order 0 , a : R d → C bounded mean oscillation (or C 0 ) ⇒ [ P 0 , a ] : L p ( R d ) → L p ( R d ) bounded and � [ P 0 , a ] � L p → L p � � a � BMO = Kato – Ponce, Auscher – Taylor, . . . [ P 0 , a ] compact if e.g. a ∈ C 0 � Spectral theory? c . . . related works of Pushnitski, Yafaev, also Frank (2013 –). Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 6 / 22
Dixmier traces of commutators � N 1 Tr ω Dixmier trace, Tr ω A = lim N → ω n = 1 λ n ( A ) for A ≥ 0 log ( N ) We need to understand the precise behaviour of λ n ( 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 ⊂ R 3 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: − ∆ u k = λ k u k in Ω , u | ∂ Ω = 0 . Our central objects of interest: vol (Ω) 2 / 3 k 2 / 3 + o ( k 2 / 3 ) in 3 dimensions c 3 λ k ∼ 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: N Ω ( λ ) = # { k : λ k ≤ λ } ∼ | S d − 1 | | Ω | λ d / 2 − | S d − 2 | | ∂ Ω | 4 ( 2 π ) d − 1 λ ( d − 2 ) / 2 + o ( λ ( d − 2 ) / 2 ) ( 2 π ) d Then he moved on to other areas of mathematics. Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 9 / 22
Dixmier traces of commutators � N 1 Tr ω Dixmier trace, Tr ω A = lim N → ω n = 1 λ n ( A ) for A ≥ 0 log ( N ) We need to understand the precise behaviour of λ n ( 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 � P 0 , D � D 1 A kaleidoscope of commutators – [ P 0 , a ] on S 1 = ∂ D ⊂ C asymptotics of singular values: lim k k 2 α λ k ( A ∗ A ) < ∞ if a ∈ C α , α ∈ ( 0 , 1 ] wide variety of non-convergence, governed by singularities Dixmier traces of products: Tr ω [ P 0 , 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 S 1 , compute Tr ω A A = product of commutators [ P 0 , a ] , a Hölder. � N 1 Tr ω A = lim N → ω n = 1 λ n ( A ) for A ≥ 0 log ( N ) � linear functional Tr ω : L 1 , ∞ → C on weak-Schatten class L 1 , ∞ = { A ∈ K : sup n λ n ( A ∗ A ) 1 / 2 < ∞} n First question: When is A ∈ L 1 , ∞ ? Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 12 / 22
Set-up Aim: On S 1 , compute Tr ω A A = product of commutators [ P 0 , a ] , a Hölder. � N 1 Tr ω A = lim N → ω n = 1 λ n ( A ) for A ≥ 0 log ( N ) � linear functional Tr ω : L 1 , ∞ → C on weak-Schatten class L 1 , ∞ = { A ∈ K : sup n λ n ( A ∗ A ) 1 / 2 < ∞} n First question: When is A ∈ L 1 , ∞ ? Theorem k , a k ] ∈ L 1 , ∞ provided a k ∈ C α k ( S 1 ) , � a) A = Π k n = 1 [ P 0 k α k = 1 . k α k > 1 , then n λ n ( A ∗ A ) 1 / 2 → 0 and hence Tr ω A = 0 . b) If � This is the easy part – an upper bound on λ n ( A ) . Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 12 / 22
Set-up Aim: On S 1 , compute Tr ω A A = product of commutators [ P 0 , a ] , a Hölder. � N 1 Tr ω A = lim N → ω n = 1 λ n ( A ) for A ≥ 0 log ( N ) � linear functional Tr ω : L 1 , ∞ → C on weak-Schatten class L 1 , ∞ = { A ∈ K : sup n λ n ( A ∗ A ) 1 / 2 < ∞} n First question: When is A ∈ L 1 , ∞ ? Theorem k , a k ] ∈ L 1 , ∞ provided a k ∈ C α k ( S 1 ) , � a) A = Π k n = 1 [ P 0 k α k = 1 . k α k > 1 , then n λ n ( A ∗ A ) 1 / 2 → 0 and hence Tr ω A = 0 . b) If � Pathological functionals: C 1 / 2 ( S 1 ) 2 ∋ ( a , b ) → Tr ω [ P 0 , a ][ P 0 , b ] vanishes if a or b ∈ C 1 / 2 + ε ( S 1 ) Heiko Gimperlein (Edinburgh) Dixmier traces of commutators Glasgow 12 / 22
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