Noncommutativity&Geometry

Why we need the non-commutative geometry ?

Geometrical approach in physical theories is not unified.

General relativity & Field theory

OTR

• Space-time – dynamical object

• Geometry (curvature) – distribution of mass

• Gravity

Field theory

• Space – „passive“ scene, on which fields are developing

• Electromagnetic, weak and strong interactions

Disturbing harmony

• Planck length L=√(κħ/c3) ~ 10-35 m• Quantum gravity ? Renormalisation !• Unification payed by strong physical hypotheses: string

theory (dimension=11), supersymmetry (2 times more particles) • Experiment? None!

Harmony renewing

Unification by geometry ? • Relativity –simple interpretation of geometry • Quantum theory – how to introduce geometry ?

By algebraic structures !!

Geometry & Algebra

Geometry - distances

|| x – y ||

Algebra - operations

αf(x)+βg(x), f(x)g(x)

sup |f’(x)|1 | f(x) – f(y) |

Why is it so – first

• Mean value theorem (Lagrange)

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|||||)()(|sup

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Distances,surroundings & Operations

• Topological structure

X … [locally] compact space

• Algebraic structure ... commutative algebra

[C0(X)] C(X) ... [vanishing] continuous functions f: XC

• Metric structure ||f || = supxX |f(x)| norm metric topology

• Involution f* = f • Unit [in C0(X)... does not exist], in C(X)...f(x) 1

Spaces & Algebras of functions

• Space Commutative algebra Compact topological space X complex commutative algebra

of continuous functions on X with unit and norm ||f||=sup {|f(x)| | xX} and usual involution has the structure of C*- algebra .

Any commutative C*- algebra A topological space, with the algebra of continuous functions isomorphic with A (Gelfand – Najmark – Segal construction).

• ??????? Non-commutative algebra It will not be possible to construct such X for which the algebra

of functions would represent a given non-commutative algebra, because an algebra of functions is always commutative.

Why just C*- algebra ?

• Basic scheme of a physical theory geometrical background X algebraic calculations, differentiation

• Classical theories (locally) compact Hausdorff space X complex functions continuous on X with usul „supreme“ norm and usual involution form a commutative C*- algebra

• Quantum theory Hilbert space bounded linear operators (physical quantities) form non-commuative C*- algebra

C*- algebra: list of structures

• Algebraic structure• Involution (adjunction)• Topological structure

norm metric topology

• Additional conditions - for algebraic operations (asociativity, unit) - for involution - for topological structure (continuous operations)

• Reprezentation - general bounded operators in Hilbert space

C*- algebra: algebraic structure

• Operations (structure of an algebra) A addition … vector space over C

multiplication … (associative, in general non-commutative) ring with unit, the unit can be eventually accomplished: [,a]+[,b] = [+,a+b] [,a].[,b] = [, b+a+a.b], I = [1,0]

example of non-associative, non-commutative algebras Lie algebras: anticommutativity, Jacobi identity

C*- algebra: involution

• Involution (involutive algebra, * - algebra) A a a*A (a+b)*=a*+b*, (a.b)* = b*. a*, (a*)*= a a* ... adjoint to element a, a = a* … hermitean element

• * - ideal twosided proper *- subagebra B A a.b B , b.a B for arbitrary elements aA, bB ideal cannot contain unit

• factor algebra A/B

C*- algebra: topological structure

• Norm and metric on algebra A a ||a||R ||a|| 0, equality a = 0 ||a+b|| ||a||+||b||, ||a|| = || ||a||, ||a.b|| ||a|| ||b||, ||I||=1 when the unit is added ... a unique extension of norm ||[,a]|| = sup{||b+ab|| | ||b||1} … again C*- algebra

• Banach algebra completeness with respect to norm, every normed algebra A can be completed (complete obal B – Banach algebra - contains A as a dense subalgebra)

• Topological structure … induced by norm ||.|| base of topology: U(a,r)={bA | ||b - a|| < r}

C*- algebra:continuous involution

• Normed * - algebra additional condition ||a*|| = ||a|| continuous involution

• C*- algebra Banach *- algebra A with additional condition ||a*a||=||a||2 ||a|| ||a*|| a ||a*|| ||a|| ||a*|| = ||a||

• Example: commutative algebra C (X) of functions continuous on a compact Hausdorff space X * … complex conjugation, ||f ||=supxX |f(x)|. locally compact X … C0(X) … functions vanishing at infinity … without unit

C*- algebra: examples

• Example: operators Non-commtative algebra B(H) of bounded linear operators

on infinite-dimensional Hilbert space, * ... adjunction, ||B||=sup{||B|| | H , |||| 1}

• Example and counterexample: matrices Non-commutative algebra of square matrices Mn(C),

T* = TT, a) ||T||...square root of maximum eigenvalue of matrix T* T b) ||T|| = sup{Tij} ... condition of C*- algebra is not fulfilled Both norms define the same topology on Mn(C).

C*- algebra: spectrum

• Spectrum of a (a) = {C | (a-I)-1 doesn’t exist}

• Rezolvent set of an element, rezolvent r(a)={C | (a-I) is invertible}, rezolvent a = (a-I)-1

• Spectral radius (a) = sup{ || | (a) }

• in C*- algebra r(a) is open, (a) is non-empty and closed r(a) (a-I)-1A analytical function (a)=||a|| … unique norm uniquely given by the algebraic structure hermitean elements … (a) (-||a||, ||a||) pozitive elements … hermitean and (a) [0, ), a=b*b

C*- algebra: morphisms

• * - morpisms linear mappings of algebras : A B , moreover (a.b) = (a) . (b), (a*) = (a)* for arbitrary a,b A

• Continuity and norm morphisms of C*- algebras are authomatically continuous ||(a)|| ||a|| for arbitrary a A , equality … isometry

• * - isomorphisms bijective *- morphisms, -1 authomatically *- morphism isometric *- isomorphisms … the same topological and algebraic structure of algebras A a B

C*- algebra: representation

• Representation of C*- algebra A pairs (H , ) … H Hilbert space *- morphism : A B(H) is *- isomorphism it is isometry … faithful represent.

• Irreducible representation H has no nontrivial invariant subspaces to action (A )

• Cyclic vector of a representation H … (A )() = { (a)() | a A } is dense in H example: A = Mn(C), (T)() = T. , every vector 0 is cyclic

GNS construction – states

• State on a C*- algebra linear functional f : A C, pozitive ... f(a*a) 0 ||f|| = 1 for the norm ||f || = sup{|f(a)| | ||a|| 1} continuous f and the property ||f || = f(I) = 1 S(A) ={f | f state on A}... convex, tf1+(1-t)f2 S(A) , 0t1

• Pure and mixed states pure states…extremal points of the set S(A) are not of the form tf1+(1- t)f2 , 0 < t < 1

(e.g. triangle apexes)

Why is it so - second

• f(I)=1 ? It is so clear! However, how many steps we need for verifying this ?

• positive element pA … p=a*a ... hermitean,(p)[0,) f(p)0 f[r(b*b)I- b*b] 0 r(b*b)f(I) - f(b*b) 0

• [a,b] f(a*b)... positive sesquilinear for |f(a*b)|2 f(a*a) f(b*b) |f(b)|2 f(I)f(b*b) f(I)2 r(b*b) f(I)2 ||b||2 |f(b)| f(I)||b||

• for suprema … ||f || = sup{|f(a)| | ||a|| 1} sup |f(b)| f(I) sup||b|| ||f || f(I), for states 1 f(I)

• on the other hand ||I||=1 f(I) ||f || , i.e. for states 1 f(I)

• indeed ... finally we have f(I) = 1

States on C*-algebra - topology

• topology on S(A) *- weak topology … coarser topology, in which linear functionals â: S(A) C are continuous a â isomorphism A S(A) S(A)dual

• compactness S(A)…locally compact and Hausdorff I A S(A)…compact

GNS construction–representation

• Representations associated with states f S(A) … (Hf , f)

Nf = {aA | f(a*a) = 0} Nf ... closed left ideal in A scalar product in A /Nf … (a+ Nf , b+Nf) f(a*b) C independent of the choice of representatives a, b

0 (a)(b + Nf ) = a.b + Nf … bounded linear operator on A /Nf

… independent of the choice of representative b

competeness Hf =complete of A /Nf is Hilbert space f (a) B(Hf) … unique extension of operator0 (a) on a bounded linear operator in Hilbert space Hf

GNS construction–representation

• GNS – triple … (f , Hf , f ) vector f = I + Nf Hf is cyclic … the set f (a)(f ) is dense in Hf and (f , f (a)(f ))=f(a) for all elements aA ||f || = || f || = 1

• Irreducibility representation GNS is irreducible lib. 0 is cyclic f is a pure state

• Equivalence of representations GNS triple is given up to an unitary transformation U : Hf Hf , f = Uf , f (a) = Uf (a) U-1

GNS construction–representation

• Isometric representation of C*- algebra

For every C*- algebra A with unit there exists an isometric representation : A B(H) on some Hilbert space.

H = Ha … aA runns through nonzero elements of A (a , Ha , a) … GNS triple corresponding to functional fa

pozitive with the property fa (a*a) = ||a||2 (a)() = {a (b)(a ) | 0 a A } , = {a | 0 a A }

Existence theorem – not practical – space H is “too big“, representation is “too reducible“ .

Commutative geometry

• Commutative GNS – construction Every commutative C*- algebra A with unit is isometrically *- isomorphous to algebra C (X) of continuous functions on a compact Hausdorff space X.

• Characters Irreducible representations of commutative C*- algebra are onedimensional then there exists a linear functional f : A C which is a homomorphism, i.e. It holds f(a.b) = f(a) f(b) f(I)=1

Why is it so - third

• Why we can see that irreducible representatons (H, ) are onedimensional ?

• Reducibility of operator (a) ... no nontrivial invariant subspace, i.e. no projector P, for which P(a) = (a)P and (E-P)(a)= (a)(E-P) a PH a (E-P)H are invariant subspaces in H

• Representation is irreducible commutant of the set (A ) contains only multiples of identity operator E.

• For a commutative algebra A - (A ) is a subset of the commutant representation is reducibile with one exception – when it is trivial, i.e. for dim H =1.

Commutative geometry

• Topology on the set of characters  topology defined by pointwise convergence: {f} f {f(a)} f (a)

base V={g | g(a1)U1,..., g(ak) Uk, ajA, UjC}

space of characters  … locally compact and Hausdorff, A contains I  is compact (a=I, 1C V= )

• C*-algebra of functions on  â:  f â(f) = f(a) C, ||â|| = sup{|f(a)| | a A}= ||a||

• A Â C(Â) A isometric isomorphism

Why is it so - fourh

• What is the connections between the character space compactness with the algebra unit ?

• Generally – the character space with the pointwise topology is Hausdorff (various points can be separated by disjunct neighbourhoods) and locally compact (every point has a nighbourhood with compact closure)

• Unit of the algebra compactness: For every open covering of the whole space there exists a finite subcovering.

Why is it so – fourth

• How to separe points • for g, h  , g h , there

exists a A : g(a)h(a), choose open sets in C Ug Uh = Ø, g(a)Ug,,

h(a)Uh

• Wg ={f | f(a1)U1 , ... , f(ak)Uk , f(a)Uh } Wh

={f | f(b1)V1 , ... , f(am)Vm , f(a)Uh } gWg , hWh Wg

Wh = Ø

Commutative geometry-alternatively

• Equivalent construction  … space of maximal ideals of the algebra A kernels of irreducible representations … max. ideals in A

f  … A = Ker f C Ker f … maximal ideal in A let I … maximal ideal, then natural representaton of A in A/I

is irreducible onedimensional A/I C factor homomorphism A A/I can be identified with f where I = Ker f

Space of maximal ideals with Jacobson topology is homeomorphous with  with Gelfand topology.

Commutative geometry-example

• Example Y … (locally) compact Hausdorff space A = C(Y) … C*- algebra of complex functions on Y a : C(Y) y a(y) C norm ||a||=sup{|a(y)| | yY}, involution ... a*(y) = a(y)

 ={ ŷ... homomorphism | ŷ : C(Y) a ŷ (a)=a(y)C} Ker ŷ = {a C(Y) | a(y) = 0} ... maximal ideal in C(Y)

homeomorphism ... : Y y ŷ Â Space Y and space of characters of C*- algebra of its functions are topologically equivalent.

Noncommutative geometry-example

• GNS-representation of matrix alg. M2(C)

commutative subalgebra A = {T | diag (, )} characters g(T)=,f(T)=

extension to M2(C) F: M2(C) C, F(I)=1 F11+F22=1, g(T)=a11, f(T)=a22

Why is it so - fifth

• Why it holds F11+F22 = 1 ?• Linear mapping is given by images of a basis.

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Noncommutative geometry-example

• Irreducible representation corresponds to pure states f: M2(C) C , g: M2(C) C

• Ideals for factorisation N1 ={T1M2(C)| a11=a21=0}, N2 ={T2M2(C)| a12=a22=0}

0

0

0

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22

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a

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Noncommutative geometry-example

• Associated Hilbert spaces H1 =M2(C)/N1 = {T1 | a11= x1 , a21= x2 , a12= a22 =0} C2 H2 =M2(C)/N2 ={T2 | a11= a21= 0 , a12= y1 , a22 = y2} C2

Scalar product (X , X’) = x1

* x1’+ x2

* x2’ (Y , Y’) = y1

* y1’+ y2

* y2’

Yy

y

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Noncommutative geometry-example

• Representation morphisms and cyclic vectors 1: H1 T1 1(T1) H1 , 1 … x1 = 1, x2 = 0 2: H2 T2 2(T2) H2 , 2 … x2 = 0, x2 = 1

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Noncommutative geometry-example

• Equivalence C*-algebra M2(C) has the unique irreducible representation. This representation is twodimensional.

Representatons of C*-algebra M2(C) corresponding to pure states f and g are equivalent.

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Non-Commutative final excuse

• To physicists (and some mathematicians) I know that you are disappointed because you did not see a direct physical application. However, the algebra M2(C) is very close to spinors.

• To mathematicians (and some physicsts) I know that it was too trivial. However, to omit trivial examples is a very bad habit making an understanding difficult.