Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop...

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Dualities in Noncommutative Geometry Mohammad Hassanzadeh University of New Brunswick August 2012, IPM, Tehran Mohammad Hassanzadeh (University of New Brunswick ) Dualities in Noncommutative Geometry August 2012, IPM, Tehran 1 / 32

Transcript of Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop...

Page 1: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Dualities in Noncommutative Geometry

Mohammad Hassanzadeh

University of New Brunswick

August 2012, IPM, Tehran

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 1 / 32

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What is Noncommutative Geometry?

In mathematics each object (or subject) can be looked at in twodifferent ways:Geometric or Algebraic

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 2 / 32

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NCG

NCG, created by A. Connes, is the name of a very young and fastdeveloping mathematical theory (1980).

A great idea here is to find algebraic generalizations of most of thestructures currently available in mathematics: measurable,topological, differential, metric, vector bundle, ...

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 3 / 32

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NCG Idea

The fundamental idea, implicitly used in A. Connes’noncommutative geometry is a powerful extension of R. Decartes’ analyticgeometry:

To trade geometrical spaces X of points with their Abelian algebrasof ( complex or real valued) functions f : X −→ C;

Example

Manifold M C∞(M).

Then we find a way to translate the geometrical properties of spacesinto algebraic properties of the associated algebras.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 4 / 32

Page 5: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

NCG Idea

To develop noncommutative geometries, we usually proceed asfollows:

1) First we find a suitable way to codify or translate the geometricproperties of a space X (topology, measure, differential structure,metric, . . . ) in algebraic terms, using a commutative algebra offunctions over X .2) Then we try to see if this codification survives” generalizing to thecase of noncommutative algebras.3) Finally the generalized properties are taken as axioms definingwhat a dual of a noncommutative” (topological, measurable,differential, metric, . . . ) space is, without referring to anyunderlying point space.Of course the process of generalization of the properties from thecommutative to the noncommutative algebra case is highly non trivialand, as a result, several alternative possible axiomatizations arise inthe noncommutative case, corresponding to a unique commutativelimit”.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 5 / 32

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NCG

Original space X ⇐= Algebra

A reverse process: To reconstruct the original geometric space X , atechnique that appeared for the first time in the work of I. Gelfand onAbelian C ∗-algebras. ( 1939).

A similar ideas previously used also in algebraic geometry in P.Cartier, A. Grothendieck’s definition of schemes.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 6 / 32

Page 7: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Heisenberg’s Commutatation Relation- 1925

Why NONCOMMUTATIVE?

Heisenberg’s commutation relation:

classical mechanics quantum mechanics

pq − qp =h

2πi1

commutative algebra functions noncommutative algebra operato

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 7 / 32

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Serre(1955)-Swan(1962) Theorem

R. G. Swan, Transactions of the American Mathematical Society,Vol. 105, No. 2 (Nov., 1962), pp.264-277.

Serre has shown that there is a one-to-one correspondence betweenalgebraic vector bundles over an affine variety and finitely generatedprojective modules over its coordinate ring.

For some time, it has been assumed that a similar correspondenceexists between topological vector bundles over a compact Hausdorff spaceX and finitely generated projective modules over the ring of continuousreal-valued functions on X .

However, no rigorous treatment of the correspondence seems tohave been given. I will give such a treatment here.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 8 / 32

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Serre(1955)-Swan(1962) Theorem

M = compact manifold.

C∞(M) = smooth real-valued functions on M.

V = smooth vector bundle over M.

Γ(V ) = space of smooth sections of V .

Γ is a module over C∞(M)

C∞(M)× Γ −→ Γ, (f ⊲ s)(m) = f (m)s(m)

Swan Γ is finitely generated projective module on C∞(M).

One to one correspondence:{ smooth vector bundles on M}! { F.G.P modules on C∞(M)}

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 9 / 32

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Alain Connes

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 10 / 32

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A Dictionary In NCG

commutative, noncommutativemeasure space, Von Neumann algebralocally compact space, C ∗- algebravector bundle, finitely projective modulevector field,derivationintegral,tracede Rham complex , Hochschild homologyde Rham cohomology, cyclic homolgyclosed de Rham current , cyclic cocyclespin Riemannian manifold, spectral triplegroup, Lie algebra, Hopf algebra, quantum groupsymmetry, action of Hopf algebraprincipal bundle, Hopf Galois extension

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 11 / 32

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References

A. Connes, Noncommutative Geometry, Academic Press (1994).

M. Khalkhali, Basic noncommutative geometry, EuropeanMathematical Society (EMS), (2009)

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 12 / 32

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Gelfand-Naimark Theorem-1943

Theorem

a)

{Locally compact Huausdorf spaces} ∼= {Commutative C ∗ algebras}op

or

{Compact Huausdorf spaces} ∼= {Commutative unital C ∗ algebras}op

b) Every C ∗-algebra is isomorphic to a ∗-subalgebra of the algebra ofbounded operators on a Hilbert space.

C ∼= C ′ ⊆ L(H)

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 13 / 32

Page 14: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

C∗ algebras

Involutive algebra∗ : A −→ A, a 7−→ a∗,

(a + b)∗ = a∗ + b∗, (λa)∗ = λa∗, (ab)∗ = b∗a∗, (a∗)∗ = a.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 14 / 32

Page 15: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

C∗ algebras

Involutive algebra∗ : A −→ A, a 7−→ a∗,

(a + b)∗ = a∗ + b∗, (λa)∗ = λa∗, (ab)∗ = b∗a∗, (a∗)∗ = a.

Normed algebra:Algebra + Normed vector space, ‖ ab ‖≤‖ a ‖‖ b ‖ .

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 14 / 32

Page 16: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

C∗ algebras

Involutive algebra∗ : A −→ A, a 7−→ a∗,

(a + b)∗ = a∗ + b∗, (λa)∗ = λa∗, (ab)∗ = b∗a∗, (a∗)∗ = a.

Normed algebra:Algebra + Normed vector space, ‖ ab ‖≤‖ a ‖‖ b ‖ .

Banach algebra: Complete normed algebra.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 14 / 32

Page 17: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

C∗ algebras

Involutive algebra∗ : A −→ A, a 7−→ a∗,

(a + b)∗ = a∗ + b∗, (λa)∗ = λa∗, (ab)∗ = b∗a∗, (a∗)∗ = a.

Normed algebra:Algebra + Normed vector space, ‖ ab ‖≤‖ a ‖‖ b ‖ .

Banach algebra: Complete normed algebra.

C ∗-algebra : Involutive Banach algebra,

‖ a∗a ‖=‖ a ‖2 .

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 14 / 32

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C∗ algebra

Duality in C ∗ algebra structure! C ∗-norm has algebraic structure

‖ a ‖2= sup{|λ| : a∗a− λ1 is not invertible}

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 15 / 32

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C∗ algebra

Duality in C ∗ algebra structure! C ∗-norm has algebraic structure

‖ a ‖2= sup{|λ| : a∗a− λ1 is not invertible}

Example

C=Complex numbers

(a + bi)∗ = a − bi , ‖ a + bi ‖=√

a2 + b2

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 15 / 32

Page 20: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Example

Example

Mn(C)A∗ = At , ‖ A ‖= sup‖x‖=1 ‖ Ax ‖ .

Note: If A = finite dimensional C ∗-algebra,

A ∼= Mn1 ⊕ · · · ⊕Mnk

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 16 / 32

Page 21: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Example

Example

Mn(C)A∗ = At , ‖ A ‖= sup‖x‖=1 ‖ Ax ‖ .

Note: If A = finite dimensional C ∗-algebra,

A ∼= Mn1 ⊕ · · · ⊕Mnk

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 16 / 32

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Spectrum

Morphism of C ∗ algebras: algebra map f : A −→ B , f (a∗) = f (a)∗

Any C ∗-morphism is continuous =⇒ the norm is unique.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 17 / 32

Page 23: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Spectrum

Morphism of C ∗ algebras: algebra map f : A −→ B , f (a∗) = f (a)∗

Any C ∗-morphism is continuous =⇒ the norm is unique.

Character of algebra A: Nonzero linear algebra map χ : A −→ C.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 17 / 32

Page 24: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Spectrum

Morphism of C ∗ algebras: algebra map f : A −→ B , f (a∗) = f (a)∗

Any C ∗-morphism is continuous =⇒ the norm is unique.

Character of algebra A: Nonzero linear algebra map χ : A −→ C.

A = Banach =⇒ χ continuous and ‖ χ ‖= 1

A = C ∗ − algebra =⇒ χ = C ∗ −morphism

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 17 / 32

Page 25: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Spectrum

Morphism of C ∗ algebras: algebra map f : A −→ B , f (a∗) = f (a)∗

Any C ∗-morphism is continuous =⇒ the norm is unique.

Character of algebra A: Nonzero linear algebra map χ : A −→ C.

A = Banach =⇒ χ continuous and ‖ χ ‖= 1

A = C ∗ − algebra =⇒ χ = C ∗ −morphism

A = Spectrum of Banach algebra A: All characters.

Weak * topology (pointwise convergence Topology) y A

A is locally compact Hausdorff space.

A is compact ⇐⇒ A is unital (Two G-N theorems are equivalent).

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 17 / 32

Page 26: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Duality: Characters and Maximal Ideals

Duality: If A is unital:

A⇆ Maximal ideals of A

χ −→ Kernel(χ)

χ : A→ A/I ∼= C←− I

Gelfand-Mazur:

I = maximal ideal =⇒ A/I ∼= C

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 18 / 32

Page 27: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand-Naimark Theorem

locally compact Hausdorff space ← commutative C ∗ − algebra

A A

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 19 / 32

Page 28: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand- Naimark Theorem

locally compact Hausdorff space commutative C ∗ − algebra

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 20 / 32

Page 29: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand- Naimark Theorem

locally compact Hausdorff space commutative C ∗ − algebra

X = locally compact Hausdorff space.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 20 / 32

Page 30: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand- Naimark Theorem

locally compact Hausdorff space commutative C ∗ − algebra

X = locally compact Hausdorff space.

C0(X ) = algebra of complex-valued continuous functions vanishing atinfinity.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 20 / 32

Page 31: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand- Naimark Theorem

locally compact Hausdorff space commutative C ∗ − algebra

X = locally compact Hausdorff space.

C0(X ) = algebra of complex-valued continuous functions vanishing atinfinity.

C0(X ) is commutative C ∗ algebra.

‖ f ‖∞= sup{|f (x)|; x ∈ X}

f 7−→ f ∗, f ∗(x) = f (x)

C0(X ) is unital ⇐⇒ X is compact.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 20 / 32

Page 32: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand-Naimark Theorem-Gelfand Transform

Characters of C0(X ) ?

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 21 / 32

Page 33: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand-Naimark Theorem-Gelfand Transform

Characters of C0(X ) ?

x ∈ X ,χ = χx : C0(X ) −→ C, χx(f ) = f (x)

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 21 / 32

Page 34: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand-Naimark Theorem-Gelfand Transform

Characters of C0(X ) ?

x ∈ X ,χ = χx : C0(X ) −→ C, χx(f ) = f (x)

All characters are of this form!

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 21 / 32

Page 35: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand-Naimark Theorem-Gelfand Transform

Characters of C0(X ) ?

x ∈ X ,χ = χx : C0(X ) −→ C, χx(f ) = f (x)

All characters are of this form!

Theorem

A = commutative C ∗ algebra with spectrum A,

Γ : A −→ C0(A), a 7−→ a,

where a(χ) = χ(a) is an isomorphism of C ∗ algebras.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 21 / 32

Page 36: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand-Naimark

second part

C ∼= C ′ ⊆ L(H)

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 22 / 32

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State of C ∗-Algebras-Gelfand-Naimark-Segal (GNS)

construction.

State: Linear functional ϕ : A −→ C,

ϕ(aa∗) ≥ 0, ϕ(1) = 1

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 23 / 32

Page 38: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

State of C ∗-Algebras-Gelfand-Naimark-Segal (GNS)

construction.

State: Linear functional ϕ : A −→ C,

ϕ(aa∗) ≥ 0, ϕ(1) = 1

Example

A = Mn(C),

{all states ϕ}! {positive matrices p, Tr(p) = 1}

Hint :ϕ(a) = Tr(ap)

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 23 / 32

Page 39: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

States and Probability Measures

For a locally compact Hausdorff space X there is a 1-1correspondence between states on C0(X ) and Borel probability measureson X .

To a probability measure µ is associated states is defined by

ϕ(f ) =

X

fdµ

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 24 / 32

Page 40: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

A Taste of the proof of Gelfand-Naimark-Segal (GNS)

construction

State Hilbert space

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 25 / 32

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A Taste of the proof of Gelfand-Naimark-Segal (GNS)

construction

State Hilbert space

ϕ : A −→ C state.

〈a, b〉 := ϕ(b∗a) inner product

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 25 / 32

Page 42: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

A Taste of the proof of Gelfand-Naimark-Segal (GNS)

construction

State Hilbert space

ϕ : A −→ C state.

〈a, b〉 := ϕ(b∗a) inner product

N := {a ∈ A;ϕ(a∗a) = 0}.

N closed left ideal of A, A/N.

〈a + N, b + N〉 := 〈a, b〉 inner product

Hϕ := Hilbert space completion of A/N.

GNS RepresentationC ∗-algebra Representation

πϕ : A L(Hϕ)

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 25 / 32

Page 43: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Gelfand-Naimark A Dictionary

Space ⇆ Algebra

compact ⇆ unital

closed subspace ⇆ closed ideal

1-point compactification ⇆ unitization

Borel measure ⇆ positive functional

probability measure ⇆ state

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 26 / 32

Page 44: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Great Identification in NCG

Gelfand-Naimark Thm =⇒

category of C ∗-algebras as category of noncommutative topological spaces

How about category of noncommutative Riemannian manifolds?

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 27 / 32

Page 45: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

What are noncommutative manifolds?(Connes)-1996

The complete answer is NOT known, but (at least in the case ofcompact finite dimensional orientable Riemannian spin manifolds), thenotion of Connes’ spectral triples and Connes(2008, a complete proof)-Rennie-Varilly (2006) reconstruction theorem provide and adequatestarting point, specifying the objects of our noncommutative category.

Motivated by the example of the Atiyah-Singer Dirac operator of acompact spin manifold

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 28 / 32

Page 46: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Connes Spectral Triple

A (compact) spectral triple is given by (A,H,D)

A: is a ∗-algebra.

H: Hilbert space.

D: A self adjoint operator on H.

a representation π : A −→ B(H) of A on H

(D − λ)−1 is a compact operator for every irrational λ.

[D, π(a)] ∈ B(H) for every a ∈ A.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 29 / 32

Page 47: Dualities in Noncommutative Geometrymath.ipm.ac.ir/analysis/Hopf algebras.pdfTo develop noncommutative geometries, we usually proceed as follows: 1) First we find a suitable way to

Example

Example

A = C∞(S1),

H = L2(S1),

D = 1

iddx.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 30 / 32

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Connes Reconstruction Theorem

Theorem

Given a compact oriented spin manifold M

(C∞(M), L2(M,S), ∂)

is a spectral triple.

Theorem

(Conjectured by Connes 1996)Given a spectral triple (A,H,D) where A is a ”commutative” ∗-algebrathen A = C∞(M) where M is a compact oriented spin manifold with aunique Riemannian structure.

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 31 / 32

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END

Thanks

Mohammad Hassanzadeh (University of New Brunswick )Dualities in Noncommutative Geometry August 2012, IPM, Tehran 32 / 32