Noncommutativity &Geometr y

1. Noncommutativity&Geometry Why we need the non-commutative geometry ? Geometrical approach in physical theories is not unified.

2. Field theory Space – „passive“ scene, on which fields are developing Electromagnetic, weak andstrong interactions General relativity& Field theory OTR • Space-time – dynamical object • Geometry (curvature) – distribution of mass • Gravity

3. Disturbing harmony • Planck lengthL=√(κħ/c3)~ 10-35 m • Quantum gravity ? Renormalisation ! • Unification payed by strong physical hypotheses: string theory (dimension=11), supersymmetry (2 times moreparticles) • Experiment? None!

4. Harmony renewing Unification by geometry ? • Relativity –simple interpretation ofgeometry • Quantum theory – how to introduce geometry ? By algebraic structures!!

5. Geometry & Algebra Geometry - distances || x–y || Algebra - operations αf(x)+βg(x), f(x)g(x) sup |f’(x)|1| f(x) – f(y) |

6. Why is it so – first • Mean value theorem (Lagrange)

7. Distances,surroundings & Operations • Topological structure X … [locally]compactspace • Algebraicstructure ... commutativealgebra [C0(X)]C(X) ... [vanishing]continuous functionsf: XC • Metric structure ||f || = supxX |f(x)|norm  metric  topology • Involution f* = f • Unit [in C0(X)... does not exist], in C(X)...f(x)  1

8. Spaces& Algebras of functions • Space Commutative algebra Compact topological space X  complex commutative algebra of continuous functionson X withunit and norm ||f||=sup {|f(x)||xX} andusual involutionhas the structure of C*- algebra. Anycommutative C*- algebra A  topologicalspace, with the algebra of continuous functions isomorphicwithA (Gelfand – Najmark – Segal construction). • ???????  Non-commutative algebra It will not be possible to construct such X for which the algebra of functionswould represent a given non-commutative algebra, because an algebra of functions is alwayscommutative.

9. Whyjust C*- algebra ? • Basic scheme of aphysical theorygeometricalbackgroundX algebraic calculations, differentiation • Classical theories(locally) compact Hausdorff space X  complex functionscontinuouson X withusul „supreme“ norm andusualinvolutionform acommutative C*- algebra • Quantum theoryHilbert space  bounded linear operators (physical quantities) form non-commuative C*-algebra

10. C*- algebra: list of structures • Algebraic structure • Involution (adjunction) • Topological structure norm  metric  topology • Additionalconditions - for algebraic operations (asociativity, unit) - for involution - for topological structure (continuous operations) • Reprezentation - generalbounded operatorsin Hilbert space

11. C*- algebra: algebraic structure • Operations (structure of an algebra)A addition … vectorspace over C multiplication … (associative, in general non-commutative) ring withunit, 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 algebrasLie algebras: anticommutativity, Jacobi identity

12. 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 twosidedproper*- subagebraB A a.b B , b.a B forarbitrary elements aA,bB ideal cannot containunit • factor algebra A/B

13. C*- algebra: topological structure • Normand metricon algebraA  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 completenesswith respect to norm, every normed algebraA can be completed (complete obal B – Banach algebra - containsA as a densesubalgebra) • Topological structure… induced by norm ||.|| base of topology: U(a,r)={bA | ||b - a|| < r}

14. C*- algebra:continuous involution • Normed*- algebraadditionalcondition ||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 functionscontinuous on acompact Hausdorff space X *… complex conjugation, ||f ||=supxX |f(x)|. locallycompact X … C0(X) … functionsvanishing atinfinity … without unit

15. C*- algebra: examples • Example: operators Non-commtative algebra B(H) of bounded linear operatorsoninfinite-dimensional Hilbert space, * ... adjunction, ||B||=sup{||B|| | H , ||||  1} • Example andcounterexample: matrices Non-commutative algebra of square matrices Mn(C), T* = TT, a) ||T||...square root of maximumeigenvalue of matrixT*T b) ||T|| = sup{Tij} ... condition of C*- algebra is not fulfilled Both norms definethe same topologyon Mn(C).

16. C*- algebra: spectrum • Spectrum of a (a) = {C | (a-I)-1 doesn’t exist} • Rezolvent set of an element, rezolventr(a)={C | (a-I) is invertible}, rezolvent a = (a-I)-1 • Spectral radius(a) = sup{ || |   (a) } • in C*- algebrar(a) is open, (a)is non-empty and closed r(a)    (a-I)-1A analytical function(a)=||a|| … unique norm uniquely givenbythe algebraic structure hermiteanelements … (a) (-||a||, ||a||) pozitiveelements … hermiteanand(a) [0, ), a=b*b

17. C*- algebra: morphisms • * - morpisms linearmappings 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 • * - isomorphismsbijective*- morphisms, -1 authomatically *- morphism isometric *- isomorphisms… the same topologicaland algebraic structure of algebrasAaB

18. C*- algebra: representation • Representation of C*- algebra Apairs (H,) … HHilbert space *- morphism : A  B(H) is*- isomorphism  it is isometry … faithful represent. • Irreducible representationHhas no nontrivial invariant subspacesto action(A ) • Cyclic vector of a representation H… (A )()= { (a)() | a  A } is denseinH example: A= Mn(C), (T)() = T.  , everyvector   0 is cyclic

19. GNS construction – states • State on a C*- algebralinear 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 onA}... convex, tf1+(1-t)f2S(A), 0t1 • Pure andmixed statespure states…extremalpoints of the set S(A) are not of the formtf1+(1- t)f2, 0 < t < 1(e.g. triangleapexes)

20. Why is it so - second • f(I)=1 ? It is so clear! However, how many steps we need for verifying this ? • positive element pA …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

21. Stateson C*-algebra - topology • topologyonS(A)*- weak topology … coarser topology, inwhichlinearfunctionalsâ:S(A)  Care continuous a  â isomorphismA  S(A)  S(A)dual • compactnessS(A)…locallycompact and HausdorffIAS(A)…compact

22. GNS construction–representation • Representations associatedwith statesf  S(A) … (Hf ,f) Nf = {aA| f(a*a) = 0} Nf ... closedleft ideal inA scalarproductinA /Nf … (a+ Nf , b+Nf)  f(a*b)  C independentofthe choice of representatives a, b 0 (a)(b + Nf ) = a.b + Nf … bounded linear operator onA /Nf … independent of the choice of representative b competeness Hf =complete ofA /Nf is Hilbert space f (a) B(Hf) … uniqueextension of operator0 (a) ona bounded linear operator in Hilbert spaceHf

23. GNS construction–representation • GNS – triple … (f , Hf ,f )vector f = I + Nf  Hf is cyclic … the set f (a)(f ) is dense inHf and (f , f (a)(f ))=f(a) for all elements aA||f|| = || f || = 1 • Irreducibilityrepresentation GNS is irreducible  lib.   0 is cyclic  f isa pure state • Equivalenceof representationsGNS triple is given up to an unitary transformationU : Hf Hf, f = Uf ,f(a) = Uf(a)U-1

24. GNS construction–representation • Isometric representation of C*- algebra For every C*- algebra A with unit there exists an isometric representation: A B(H) onsome Hilbert space. H =  Ha … aA runnsthrough nonzero elements ofA (a , Ha , a) … GNS triplecorresponding to functional fa pozitive with theproperty fa (a*a) = ||a||2 (a)() = {a(b)(a ) | 0  a  A } ,  = {a | 0  a  A } Existencetheorem – not practical – spaceHis “too big“, representation is“too reducible“ .

25. Commutativegeometry • Commutative GNS – constructionEvery commutative C*- algebraA withunit is isometrically*- isomorphous toalgebraC (X) of continuous functions on a compact Hausdorffspace X. • CharactersIrreducible representationsof commutative C*- algebraareonedimensional then there existsa linear functionalf : A  C which is a homomorphism, i.e. It holds f(a.b) = f(a) f(b)  f(I)=1

26. 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.

27. Commutativegeometry • Topologyon the set of charactersÂtopology defined bypointwiseconvergence: {f}  f  {f(a)}  f (a) base V={gÂ| g(a1)U1,..., g(ak) Uk, ajA, UjC} space of charactersÂ… locallycompact and Hausdorff,Acontains I Âis compact (a=I, 1C  V=Â) • C*-algebra of functionsonÂâ: Â f  â(f) = f(a)  C,||â|| = sup{|f(a)| | a A}= ||a|| • A  ÂC(Â) A isometric isomorphism

28. Why is it so - fourh • What is the connections between the character spacecompactness with the algebra unit ? • Generally – the character space with the pointwise topologyis 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.

29. Why is it so – fourth • How to separe points • for g, h  Â, 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Â | f(a1)U1 , ... , f(ak)Uk ,f(a)Uh}Wh={fÂ | f(b1)V1 , ... , f(am)Vm,f(a)Uh} gWg,hWh WgWh= Ø

30. Commutativegeometry-alternatively • Equivalent construction … space of maximal ideals of the algebraA kernels of irreducible representations … max. idealsinA f  … A = Ker f  C  Ker f … maximal ideal inA letI … maximal ideal, thennatural representaton ofA inA/I is irreducible onedimensional  A/I  C factor homomorphismA A/Ican be identified with f whereI = Ker f Space of maximal idealswith Jacobson topologyis homeomorphouswith with Gelfand topology.

31. Commutativegeometry-example • ExampleY … (locally) compact Hausdorff space A= C(Y) … C*- algebra of complex functionsonY a : C(Y)  y  a(y)  C norm ||a||=sup{|a(y)| | yY}, involution ... a*(y)= a(y) Â ={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ÂSpace Y andspace of charactersof C*- algebra of its functionsare topologically equivalent.

32. Noncommutative geometry-example • GNS-representationof matrix alg. M2(C) commutativesubalgebra A= {T | diag (, )} charactersg(T)=,f(T)= extension toM2(C) F: M2(C) C, F(I)=1  F11+F22=1, g(T)=a11, f(T)=a22

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

34. Noncommutative geometry-example • Irreducible representationcorresponds topure states f: M2(C)  C, g: M2(C)  C • Ideals for factorisationN1 ={T1M2(C)| a11=a21=0}, N2 ={T2M2(C)| a12=a22=0}

35. Noncommutativegeometry-example • Associated Hilbertspaces 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 Scalarproduct(X , X’) = x1*x1’+ x2*x2’ (Y , Y’) = y1*y1’+ y2*y2’

36. 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

37. Noncommutative geometry-example • Equivalence C*-algebra M2(C)has the unique irreducible representation. This representationistwodimensional. Representatons of C*-algebra M2(C)corresponding to pure states fandgare equivalent.

38. Non-Commutativefinal excuse • To physicists (andsome mathematicians)I knowthat you aredisappointed because you did not see a direct physical application. However, the algebra M2(C)is very close to spinors. • To mathematicians (andsome physicsts)I know that it was too trivial. However, to omit trivial examples is a very bad habit making an understanding difficult.