Noncommutative Geometry

Nigel HigsonPenn State University

Noncommutative Geometry

Alain Connes

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What is Noncommutative Geometry?

� Geometric spaces approached through theiralgebras of functions.� The spaces are often very singular (defined byequivalence relations, or even groupoids).� The function algebras are typically noncommutative.� The algebras/spaces are analyzed using Hilbertspace tools.� In particular, spectral properties of algebras,viewed as algebras of operators on Hilbertspace, are crucial. One might call the subjectspectral geometry.

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What are its Origins?

Werner Heisenberg

What Heisenberg understood . . . is that[the] Ritz-Rydberg combination principle actuallydictates an algebraic formula for the product ofany two observable physical quantities . . .

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Heisenberg wrote down the formula for theproduct of two observables;��� �������� ���� � ���������������� ��and he noticed of course that this algebra isno longer commutative,� � �� ��� �. . . The right way to think about this newphenomenon is to think in terms of a newkind of space in which the coordinatesdo not commute. The starting point ofnoncommutative geometry is to take this newnotion of space seriously.

Alain ConnesNoncommutative geometry, Year 2000

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Commentary of Riemann

. . . it seems that the empirical notions onwhich the metric determinations of spaceare based . . . lose their validity in theinfinitely small; it is therefore quite definitelyconceivable that the metric relations of spacein the infinitely small do not conform to thehypotheses of geometry; and in fact one oughtto assume this as soon as it permits a simplerway of explaining phenomena.

Bernhard RiemannOn the Hypotheses which lie at the Foundations of Geometry

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What Has Noncommutative GeometryAccomplished?

� Manifold topology (progress on the Novikovconjecture, Gromov-Lawson conjecture, etc).� Harmonic analysis, especially of discrete groups.� Models in physics (notably of the quantum Halleffect).� Foliation theory and Atiyah-Singer index theory,on singular spaces, or parametrized by singularspaces.� In addition, NCG may offer the prospect forprogress in fundamental physics, arithmetic, . . .

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Spectral Theory and Hilbert Space

David Hilbert in 1900

In the winter of 1900-1901 the Swedishmathematician Holmgren reported in Hilbert’sseminar on Fredholm’s first publications onintegral equations, and it seems that Hilbertcaught fire at once . . .

Hermann WeylDavid Hilbert and his mathematical work

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Helmholtz Equation

Hilbert saw two things: (1) after havingconstructed Green’s function � for a givenregion and for the potential equation !#" � $. . . , the equation! % & '(% � $for the oscillating membrane changes into ahomogeneous integral equation

% �*)+� & ' � �*)-,/.0� % ��.0�213. � $with the symmetric � , � ��.-,4)+� � � �*)-,/.+�

. . . ;(2) the problem of ascertaining the “eigenvalues” ' and “eigen functions” % �*)+�

of thisintegral equation is the analogue for integralsof the transformation of a quadratic form of 5variables onto principal axes.

Hermann WeylDavid Hilbert and his mathematical work

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Problem of H.A. Lorentz

. . . there is a mathematical problem whichwill perhaps arouse the interest of mathematic-ians . . . In an enclosure with a perfectlyreflecting surface there can form standingelectromagnetic waves analogous to tones ofan organ pipe . . . there arises the mathematicalproblem to prove that the number of sufficientlyhigh overtones which lie between 6 and 6 7 1 6is independent of the shape of the enclosureand is simply proportional to its area.

H.A. LorentzWolfskehl Lecture, 1910

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Reformulation

!#"98 � ':8+";8" 8=<?>A@ � $B � ' �C� # eigenvalues of ! less than or equal to ' .

limDFEHG B � ' �' � Area� �

constant�

This is equivalent to the asymptotic relation

lim8 EHG ' 85 � constantArea

� � �10

The idea was one of many, as they probablycome to every young person preoccupiedwith science but while others soon burstlike soap bubbles, this one soon led, as ashort inspection showed, to the goal. I wasmyself rather taken aback by it as I had notbelieved myself capable of anything like it.Added to that was the fact that the result,although conjectured by physicists some timeago, appeared to most mathematicians assomething whose proof was still far in thefuture.

Hermann WeylGibbs Lecture, 1948

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Compact OperatorsDefinition. A bounded linear operator I J�K L K ona Hilbert space is compact if it maps the closed unitball of Hilbert space to a (pre)compact set.

Example. If I is a norm-limit of finite-rank operatorsthen I is compact.

Elementary calculus M the maximum value of thefunction N �POQ�R� S I OTSVU on the closed unit ball of K isan eigenvalue for IXWYI .

Theorem (Hilbert et al). If I JZK L K is acompact and selfadjoint operator then there is anorthonormal basis for K comprised of eigenvectorsfor I . Thus

I [ \\ '^] ' U ':_. . .

`` �Theorem (Rellich Lemma). !ba ] is a compactoperator.

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Spectral Theory for the Laplacian

c d ! � e W eThe Laplace operator.

Theorem. There is an orthonormal basis for f U � c �consisting of functions gF8 for which

!hgV8 � ':8igV8in the distributional sense. The eigenvalues 'j8 arepositive and converge to infinity.

Spec� ! �

Remark. In fact one can show that gF8 k l G � c �.

This follows from elliptic regularity.

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Singular Values

Definition. The singular values m ] � I �n, m U � I �n,o�p�q�of

a bounded operator I are the scalars

mi8 � I �r�inf

dim��sr�ut 8 a ] supvnw s S I OTSSYOTS �

Observe that m^] � I ��x m U � I �yx �q�z�and thatI is compact { lim8 EHG m 8 � I �r� $2�

Now let I be compact, self-adjoint, and positive(meaning |�I O},�O=~ x $

). List the eigenvalues ' 8 � I �in decreasing order, and with multiplicity.

Theorem. If I is compact, self-adjoint, and positivethen m�8 � I �r� ':8 � I �

.

Proof. I � \\ ' ] ' U ' _. . .

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Trace Class OperatorsLemma.mi8 � I ] 7 I U ��� mi8 � I ] � 7 mi8 � I U ��� m U 8 � I ] 7 I U �V�mi8 ��� I �n, mi8 � I �Q�y� S��TS mi8 � I �o�Definition. The trace ideal in � � K �

is� ] � K �C� � I < mi8 � I �Q� � �=�Definition. If I k � ] � K �

then

Tr� I �r� G�Pt ] | O � , I O � ~��

The sum is over an orthonormal basis. Note: if� O ] ,��q�z��,*Oj��� is an orthonormal set then�8 t ] < | O 8 , I O 8 ~ < �

�8 t ] m 8 � I �V�

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As with the usual trace,� k � � K �F, I k � ] � K � M Tr��� I ���

Tr� I �Q�V�

Example. If � is smooth onc � c

and if

�9g ���}�C� � � ����,��H� g �P����1���,then � is a trace-class operator, and

Tr� � �y� � � ���y,��}��1��}�

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Dixmier Trace

Definition.� ] � G � K �C� I ���� sup 8 5 �im 8 � I �r� � �

Observe that� ] � K ��� � ] � G � K ��� � � K �

.

Definition. If I k � ] � G � K �is positive and LIM � is a

Banach limit then define

Tr � � I ���LIM � �

log� B ��  8�¡ � mi8 � I �

�LIM � �

log� B �   8:¡ � ' 8 � I �V�

Theorem (Dixmier). If LIM � has the property

LIM � �Y¢ ] ,£¢ U ,�¢ _ ,��p�q�z�C� LIM � �/¢ ] ,�¢ ] ,�¢ U ,�¢ U ,£�z�q�p�then Tr � � I ] 7 I U �y� Tr � � I ] � 7 Tr � � I U � .

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Integration

Now back to Weyl’sTheorem . . .':8 � ! � [ ¤ � vol

� c � 5 ¥ ¦ c dWeyl’s Theorem shows that

Tr � � ! a ¦¥ �C� ¤ � vol� c �o�

More generally, given g3J c L § we get

Tr � � g���! a ¦¥ �C� ¤ � � g 1 vol�

This suggests that ! a ¦¥ is some sort of ‘volumeelement’ for the manifold

c. . .

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Spectral Triples

Definition. A spectral triple is a triple��� , K ,V¨ �

consisting of a separable Hilbert space K , analgebra

�of bounded operators on K , and a

(typically unbounded) selfadjoint operator¨

on K ,for which:� the operator

�Y© 7 ¨ U � a ] is compact, and� if ª k �then the commutator « ¨ , ªi¬ � ¨ ª & ª ¨

extends to a bounded operator on K .

According to Connes, spectral triples constitute anextension of the notion of Riemannian geometricspace which is broadly applicable to problems infundamental physics, number theory, . . . .

Remark. If�

has no unit, replace�A© 7 ¨ U � a ] withª �A© 7 ¨ U � a ] in the above.

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The Standard Example

Basic ideas:� Regard¨

as a ‘square root’ of ! .� Think of « ¨ , ªi¬ as a gradient of ª k �.

The simplest case is

� � l G­ ��® �F, K � f U ��® �F, ¨ � �¯ & �11i� �

The theory of Dirac-type operators in geometryprovides further ‘commutative’ examples in thecontext of Riemannian manifolds.¨ � 1 7 1 W ¨ U � e W e on forms¨ �

Dirac Operator¨ U � e W e on spinors

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The Operator F

Write ! � ¨ Uand

¨ � ° ! ±¥ ,so that ° � ° W and

° U [ ©4�In the simplest case this is the Hilbert transform:° g ���}�C� �²r³ ´ g �P���� & � 1��H�The operator

°is important!

Roughly speaking the distinction between¨

and! ±¥ corresponds to the distinction between densitiesand differential forms (on manifolds).

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Groupoids and Quotients

Let µ be a smooth etale groupoid:

µsourcerange

Obj� µ �

(source and range are local diffeomorphisms).

Let� � l G­ � µ �

and define

g ]i¶ g U �¸·h�C� ¹ ± ¹ ¥ t ¹ g ]�¸· ] � g U �¸· U �V�

We obtain the convolution algebra of µ .

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Examples

Heisenberg example, º » ¼ ½¿¾ÁÀÃÂ

Kronecker foliation, º » ÄoÅHÆÈÇ�É�ÅCÇ » Ê ¥ÌËÎÍÐÏ Ç^ÅZÑ23

Infinitesimals and Differentials

Definition. An infinitesimal of order � is a compactoperator I for which m�8 � I ��� Ò � 5 a � .Definition. For ª k �

define1 ª � « °Ó, ªi¬ .

Example. In the basic case,

1 g �operator with integral kernel

g ���}� & g �P���� & � ,

give or take a factor of ²r³ .If we grade differentials ª=ÔÕ« °Ó, ª ] ¬ �q�p� « °Ó, ªiÖ�¬ accordingto degree and use the graded commutator then1 U�× Ø $=,as in de Rham theory.

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Connes’ dictionary

Geometric space Spectral triple�Ì� , K ,V¨ �

Complex variable Operator in�

Real variable Selfadjoint operator in�

Infinitesimal Compact operator

Differential1 ª Commutator « °Ó, ªi¬

Integral Ù ª Dixmier trace Tr � � ª �... ...

... ...

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Zeta Functions

Theorem. If) Ú dU then ! a:Û is

a trace-class operator. ÜÞÝProof. Follows from Rellich Lemma.

Theorem (Minakshisundaram and Pleijel). Thezeta function ß �È)à�C�

Tr� ! a(ᥠ�

is meromorphic on § with only simple poles.

Actual PolesResidues Vanish

§

Singularities of âz¾äãå for a closed surface.

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Weyl’s Theorem

Actual PolesResidues Vanish

§

Abelian-Tauberian Theorem. Let I be a positive,invertible operator and assume that I a:Û is trace-class for all

) Ú � . Then

limDFEHG B æ � ' �' � l { limÛ�ç ] �*) & � � Tr� I a:Û � � l �

See Hardy, Divergent Series. The theorem says

limDFEHG �' D/è ¡ D � � l { limÛ�ç ] �È) & � � 8 ' a:Û8 � l �27

The proof of meromorphicity uses pseudodifferentialoperators� [ Ié! a áPê è¥ , I differential of order 5 �Lemma (Guillemin). Suppose that for every holo-morphic family

� Û there are pseudodifferential ë ��ì�] ,í ���Û and î Û�a ] such that�Y1 7 )+�ï� Û � «ðë ��ì�] , í ���Û ¬27 î Û�a ] �Then Tr

��� Û � is meromorphic, with simple poles.

Proof. If Re�È)à�Tñ $

then

Tr��� Û �C� �1 7 ) ò Tr

�   «ðë ��ì�] , í ���Û ¬ � 7 Tr� î Û�a ] ��ó� �1 7 ) Tr

� î Û�a ] �Hence Tr

��� Û �y� �/1 7 )à� a ] Tr� î Û�a ] � .

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The poles of Tr��� Û � are located at & 1ô, � & 1ô,/õ & 1ô,��q�z�

.

§

Domain of Tr��� Û �

Domain of Tr� î Û�a ] �

& 1 � & 1

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Lemma. If¨

is (pseudo)differential of order 5 thend�t ] « ¨ ,Î� ¬ öö � � 5 ¨ & î ,where î has order 5 & � , and hence

�/1 7 5 �ï¨ � d�t ]÷ ¨ ,ï� öö � Èø & d�t ]

÷ � ¨ , öö � �ø 7 î �Proof. This follows from the Heisenberg relations÷ öö � ,ï� � ø � ù ¸� ©F�

As a result, Weyl’s Theorem follows from Guillemin’sLemma.

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Cyclic Cocycles

Let��� , K ,4¨ �

be a spectral triple.

Proposition. The formula

N � ª Ô , ª ] ,��q�p��, ª 8 �C� Tr�/ú ª Ô « °Ó, ª ] ¬û�z�q�Ó« °Ó, ª 8 ¬ �

defines a multlinear functional on�

with thefollowing properties:� N � ª Ô , ª ] ,£�z�q�Õ, ª 8 �C� � & � � 8 N � ª ] ,��q�z��, ª 8 , ª Ô �� ü N � ªjÔ ,£�z�q�Õ, ª 80ý ] �C� $

, whereü N � ª Ô ,��q�p��, ª 80ý ] �C� N � ª Ô ª ] ,��q�p��, ª 80ý ] �& N � ª Ô , ª ] ª U ,��q�z��, ª 80ý ] �7 �p�q�7 � & � � 8+ý ] N � ª 80ý ] ª Ô ,£�p�q�þ, ª 8 �V�

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Cyclic Cohomology

Lemma. Let N be a cyclic 5 -linear functional. Then� ü N is a cyclic� 5 7 � � -linear functional, and� ü U N � $

.

Definition. Let�

be an algebra. The 5 th cycliccohomology group of

�is

K l 8 �� �y� cyclic 5 -cocycles

modulo cyclic coboundaries.

The cocycle N is a sort of ‘fundamental class’ for thespectral triple

��� , K ,V¨ �. It reflects information from

index theory:

Index� ÿT°zÿ#�T� � & � � è ¥ N �äÿ�,�ÿH,��p�p�Õ,Èÿ#�o�

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Continuation of the Dictionary

... ...

Differential1 ª Commutator « °Ó, ªi¬

de Rham theory Cyclic cohomology

... ...

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Local Index Formula

Theorem (Connes and Moscovici). Let��� , K ,V¨ �

be a spectral triple with simple dimension spectrum.The local formula

� Ö � ª Ô ,��q�z��, ª Ö �C� �� Ô ¤ Ö �� Res Û t Ô Tr ò ú ª Ô « ¨ , ª ] ¬ �ð ± � �q�p�à« ¨ , ª Ö ¬ �ð ���� ! a � ¥ a�� � a:Û ó ,where

¤ Ö �� � � & � � � �� ]� �q�p� � Ö � � � < � < 7 Ö U �� � ] 7 � ��� � U 7 õ � �q�p� � � Ö 7 ÿX�

defines a cocycle which is cohomologous to thefundamental cocycle N .

Notation. I �ð �� � ��¨ U ,�� ¨ U ,��q�p� « ¨ U , I ¬ �q�p�� �� .

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Comments

� The formula is a starting point for Atiyah-Singerindex theory in noncommutative geometry.� In the classical case (Riemmanian manifolds) theresidues are computable from the coefficients of¨

(Seeley, Wodzicki, et al).� Small (smoothing operator) changes in¨

leavethe index formula invariant.� In the noncommutative world, local meansconcentrated at

�in momentum space (c.f.

Fourier theory).

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