Grothendieck's Works On Banach Spaces And Their Surprising .
Grothendieck’s works on Banach spaces andtheir surprising recent repercussions(parts 1 and 2)Gilles PisierIPAM, April 2018Gilles PisierGrothendieck’s works on Banach spaces
Gilles PisierGrothendieck’s works on Banach spaces
Gilles PisierGrothendieck’s works on Banach spaces
PLAN Classical GT Non-commutative and Operator space GT GT and Quantum mechanics : EPR and Bell’s inequality GT in graph theory and computer scienceGilles PisierGrothendieck’s works on Banach spaces
Classical GTIn 1953, Grothendieck published an extraordinary paper entitled“Résumé de la théorie métrique des produits tensorielstopologiques,”now often jokingly referred to as “Grothendieck’s résumé”( !).Just like his thesis, this was devoted to tensor products oftopological vector spaces, but in sharp contrast with the thesisdevoted to the locally convex case, the “Résumé” wasexclusively concerned with Banach spaces (“théorie métrique”).Boll. Soc. Mat. São-Paulo 8 (1953), 1-79.Reprinted in “Resenhas"Gilles PisierGrothendieck’s works on Banach spaces
Initially ignored.But after 1968 : huge impact on the development of “Geometryof Banach spaces"starting withPietsch 1967 and Lindenstrauss-Pełczyński 1968Kwapień 1972Maurey 1974 and so on.Gilles PisierGrothendieck’s works on Banach spaces
The “Résumé" is about the natural -normsGilles PisierGrothendieck’s works on Banach spaces
The central result of this long paper“Théorème fondamental de la théorie métrique des produitstensoriels topologiques”is now calledGrothendieck’s Theorem (or Grothendieck’s inequality)We will refer to it asGTInformally, one could describe GT as a surprising andnon-trivial relation between Hilbert space, or sayL2and the two fundamental Banach spacesL , L1(here L can be replaced by the space C(Ω) of continuousfunctions on a compact set S).Gilles PisierGrothendieck’s works on Banach spaces
Why are L , L1 fundamental ?because they are UNIVERSAL !Any Banach space is isometric to a SUBSPACE of L ( in separable case)Any Banach space is isometric to a QUOTIENT of L1( 1 in separable case)(over suitable measure spaces) Moreover :L is injectiveL1 is projectiveGilles PisierGrothendieck’s works on Banach spaces
L is injectiveXOũ?E#/ L (µ)uExtension Pty : u ũ with kũk kukL1 is projectiveXOQũL1 (µ)u / X /ELifting Pty : u compact ε 0 ũ with kũk (1 ε)kukGilles PisierGrothendieck’s works on Banach spaces
The relationship betweenL1 , L2 , L is expressed by an inequality involving3 fundamental tensor norms :Let X , Y be Banach spaces, let X Y denote their algebraictensor product. Then for anyT Xn1xj yj X YGilles PisierGrothendieck’s works on Banach spaces(1)
T Xn1xj yj(1)(1.“projective norm”)kT k infnXokxj kkyj k(2.“injective norm”)(kT k sup(3.“Hilbert norm”)(kT kH infsupx BX )X x (xj )y (yj ) X 2 x (xj ) 1/2 x BX , y BY supy BY X y (yj ) 2 1/2where again the inf runs over all possible representations (1).Gilles PisierGrothendieck’s works on Banach spaces)
Open Unit Ball of X Y convex hull of rank one tensors x ywith kxk 1 ky k 1 Note the obvious inequalitieskT k kT kH kT k In fact k k (resp. k k ) is the largest (resp. smallest)reasonable -normGilles PisierGrothendieck’s works on Banach spaces
The γ2 -norm eLet T : X Y be the linear mapping associated to T ,Xe (x ) Tx (xj )yje kB(X ,Y ) andThen kT k kTkT kH inf{kT1 kkT2 k}where the infimum runs over all Hilbert spaces H and alle through H :possible factorizations of TT2T1e : X H YTwith T T1 T2 .More generally (with Z in place of X )γ2 (V : Z Y ) inf{kT1 kkT2 k V T1 T2 }ecalled the norm of factorization through Hilbert space of TGilles PisierGrothendieck’s works on Banach spaces(2)
Important observations :k k is injective, meaningX X1 and Y Y1 (isometrically) impliesX Y X1 Y1k k is projective, meaning X1 X and Y1 Y impliesX1 Y1 X Y(where X1 X means metric surjection onto X )but k k is NOT projective and k k is NOT injectiveNote : k kH is injective but not projectiveGilles PisierGrothendieck’s works on Banach spaces
Natural question :Consider T X Y with kT k 1then let us enlarge X X1 and Y Y1 (isometrically)obviously kT kX1 Y1 T kX YQuestion : What is the infimum over all possible enlargementsX1 , Y1kT k6 V\ inf{kT kX1 Y1 }?Answer using X1 Y1 :kT k6 V\ kT k and (First form of GT) :(kT kH ) kT k6 V\ KG kT kH.was probably Grothendieck’s favorite formulationGilles PisierGrothendieck’s works on Banach spaces
One of the great methodological innovations of “the Résumé”was the systematic use of duality of tensor norms : Given anorm α on X Y one defines α on X Y by settingα (T 0 ) sup{ hT , T 0 i T X Y , α(T ) 1}. T 0 X Y In the caseα(T ) kT kH ,Grothendieck studied the dual norm α and used the notationα (T ) kT kH 0 .Gilles PisierGrothendieck’s works on Banach spaces
Gilles PisierGrothendieck’s works on Banach spaces
GT can be stated as follows : there is a constant K such that forany T in L L (or any T in C(Ω) C(Ω)) we haveGT1 :kT k K kT kH(3)Equivalently by duality the theorem says that for any ϕ inL1 L1 we have(GT1 ) :kϕkH 0 K kϕk .The best constant in either (3) or (3)0 is denoted byKG “the Grothendieck constant"(actually KGR and KGC )Exact values still unknownalthough it is known that 1 KGC KGR1.676 KGR 1.782Krivine 1979, Reeds (unpublished) more on this to come.Gilles PisierGrothendieck’s works on Banach spaces(3)0
More “concrete" functional version of GTGT2Let BH {x H kxk 1} n xi , yj BH(i, j 1, · · · , n) φi , ψj L ([0, 1])such that i, jhxi , yj i hφi , ψj iL2sup kφi k sup kψj k KijGilles PisierGrothendieck’s works on Banach spaces
Remark.We may assume w.l.o.g. thatxi yibut nevertheless we cannot (in general) takeφi ψi !!. more on this laterGilles PisierGrothendieck’s works on Banach spaces
GT2 implies GT1 in the form T n n kT k K kT kHT n n is a matrix T [Ti,j ]ThenkT kH 1 iff xi , yj BHTi,j hxi , yj iLetC {[ε0i ε00j ] ε0i 1 ε00j 1]then {T n n kT k 1} convex-hull(C) C But now if kT kH 1 for any b C Z XXX hT , bi Ti,j bi,j hxi , yj ibi,j ϕi ψj bi,j sup kφi k sup kψj k KijConclusion :kT k sup hT , bi Kb C and the top line is proved !Gilles PisierGrothendieck’s works on Banach spaces
But now how do we show :Givenxi , yj BHthere are φi , ψj L ([0, 1])such that i, jhxi , yj i hφi , ψj iL2sup kφi k sup kψj k Kij?Gilles PisierGrothendieck’s works on Banach spaces
Let H 2 . Let {gj j N} be an i.i.d. sequence of standardGaussian randomvariables on (Ω, A, P). PPFor any x xj ej in 2 we denote G(x) xj gj .hG(x), G(y )iL2 (Ω,P) hx, y iH .Assume K R. The following formula is crucial both toGrothendieck’s original proof and to Krivine’s : π hsign(G(x)), sign(G(y ))i .hx, y i sin2 Krivine’s proof of GT with K π(2Log(1 2)) 1Here K π/2a where a 0 is chosen so that sinh(a) 1 i.e. a Log(1 2).Gilles PisierGrothendieck’s works on Banach spaces(4)
Krivine’s proof of GT with K π(2Log(1 2)) 1We view T [Ti,j ]. Assume kT kH 1 i.e.Tij hxi , yj i, xi yj BHWe will prove that kT k K .Since k kH is a Banach algebra norm we havek sin(aT )kH sinh(akT kH ) sinh(a) 1. (here sin(aT ) [sin(aTi,j )]) sin(aTi,j ) hxi0 , yj0 ikxi0 k 1 kyj0 k 1By (4) we have Zπξi ηj dPsin(aTi,j ) sin2where ξi sign(G(xi0 )) and ηj sign(G(yj0 )). We obtainZπξi ηj dPaTi,j 2and hence kaT k π/2, so that we conclude kT k π/2a.Gilles PisierGrothendieck’s works on Banach spaces
Best ConstantsThe constant KG is “the Grothendieck constant.” Grothendieckproved thatπ/2 KGR sinh(π/2)Actually (here g is a standard N(0, 1) Gaussian variable)kgk 21 KGR : kgk1 E g (2/π)1/2C : kgk1 (π/4)1/2and hence KGC 4/π. Note KGC KGR .Krivine (1979) proved that 1.66 KGR π/(2 Log(1 2)) 1.78 . . . and conjectured KGR π/(2 Log(1 2)).C : Haagerup and Davie 1.338 KGC 1.405The best value best of the constant in Corollary 0.4 seems alsounknown in both the real and complex case. Note that in thereal case we have obviously best 2 because the2-dimensional L1 and L are isometric.Gilles PisierGrothendieck’s works on Banach spaces
Disproving Krivine’s 1979 conjectureBraverman, Naor, Makarychev and Makarychev proved in 2011that :The Grothendieck constant is strictly smaller than krivine’sboundi.e.KGR π/(2 Log(1 Gilles Pisier 2))Grothendieck’s works on Banach spaces
Grothendieck’s Questions :The Approximation Property (AP)Def : X has AP if for any Yb X YˇX Yis injectiveAnswering Grothendieck’s main questionENFLO (1972) gave the first example of Banach FAILING APSZANKOWSKI (1980) proved that B(H) fails APalso proved that for any p 6 2 p has a subspace failing AP.Gilles PisierGrothendieck’s works on Banach spaces
NuclearityA Locally convex space X is NUCLEAR if Yb X YˇX YGrothendieck asked whether it suffices to take Y X , i.e.b X XˇX Xbut I gave a counterexample (1981) even among Banach spacesb X Xˇ is onto, this X also fails AP.also X XGilles PisierGrothendieck’s works on Banach spaces
Other questions[2] Solved by Gordon-Lewis Acta Math. 1974. (related to thenotion of Banach lattice and the so-called “local unconditionalstructure")[3] Best constant ? Still open ![5] Solved negatively in 1978 (P. Annales de Fourier) andKisliakov independently : The Quotients L1 /R for R L1reflexive satisfy GT.Gilles PisierGrothendieck’s works on Banach spaces
[4] non-commutative GTIs there a version of the fundamental Th. (GT) for boundedbilinear forms on non-commutative C -algebras ?On this I have a small story to telland a letter from Grothendieck.Gilles PisierGrothendieck’s works on Banach spaces
Gilles PisierGrothendieck’s works on Banach spaces
Gilles PisierGrothendieck’s works on Banach spaces
Gilles PisierGrothendieck’s works on Banach spaces
Dual Form and factorization :Since kϕk n1 n1 kϕk[ n n ] (GT1 ) ϕ n1 n1kϕkH 0 K kϕk is the formulation put forward by Lindenstrauss and Pełczyński(“Grothendieck’s inequality") :TheoremLet [aij ] be an N N scalar matrix (N 1) such thatXaij αi βi sup αi sup βj .i α, β KnjThen for Hilbert space H and any N-tuples (xj ), (yj ) in H wehaveXaij hxi , yj i K sup kxi k sup kyj k.(5)Moreover the best K (valid for all H and all N) is equal to KG .Gilles PisierGrothendieck’s works on Banach spaces
We can replace n n by C(Ω) C(Π) (Ω, Π compact sets)Theorem (Classical GT/inequality)For any ϕ : C(Ω) C(Π) K and for any finite sequences(xj , yj ) in C(Ω) C(Π) we haveXϕ(xj , yj ) K kϕk X xj 2 1/2 X yj 2 1/2 . (6) (We denote kf k sup f (.) for f C(Ω)) Here againΩKbest KG .For later reference observe that here ϕ is a bounded bilinearform on A B with A, B commutative C -algebrasGilles PisierGrothendieck’s works on Banach spaces
By a Hahn–Banach type argument, the preceding theorem isequivalent to the following one :Theorem (Classical GT/factorization)Let Ω, Π be compact sets. (here K R or C) ϕ : C(Ω) C(Π) K bounded bilinear form λ, µprobabilities resp. on Ω and Π, such that (x, y ) C(Ω) C(Π) Z ϕ(x, y ) K kϕk 1/2 1/2 Z2 y dµ x dλ2where constant Kbest KGR or KGCϕeC(Ω) C(Π) x J Jλ y µuL2 (λ) L2 (µ)Gilles PisierGrothendieck’s works on Banach spaces(7)
Note that any L -space is isometric to C(Ω) for some Ω, andany L1 -space embeds isometrically into its bidual, and henceembeds into a space of the form C(Ω) .CorollaryAny bounded linear map v : C(Ω) C(Π) or any boundedlinear map v : L L1 (over arbitrary measure spaces)factors through a Hilbert space. More precisely, we haveγ2 (v ) kv kwhere is a numerical constant with KG .Gilles PisierGrothendieck’s works on Banach spaces
GT and tensor products of C -algebrasNuclearity for C -algebrasAnalogous C -algebra tensor productsA min BandA max BGuichardet, Turumaru 1958, (later on Lance)Def : A C -algebra A is called NUCLEAR (abusively.) if BA min B A max BExample : all commutative C -algebras,K (H) {compact operators on H},C (G) for G amenable discrete ————————————For C -algebras :nuclear ' amenableConnes 1978, Haagerup 1983Gilles PisierGrothendieck’s works on Banach spaces
KIRCHBERG (1993) gave the first example of a C -algebra Asuch thatA min Aop A max AopbutA is NOT nuclearHe then conjectured that this equality holds for the twofundamental examplesA B(H)andA C (F )Gilles PisierGrothendieck’s works on Banach spaces
Why are B(H) and C (F ) fundamental C -algebras ?because they are UNIVERSALAny separable C -algebra EMBEDS in B( 2 )Any separable C -algebra is a QUOTIENT of C (F )Moreover, B(H) is injective (i.e. extension property)and C (F ) has a certain form of lifting propertycalled (by Kirchberg) Local Lifting Property (LLP)Gilles PisierGrothendieck’s works on Banach spaces
With JUNGE (1994) we proved that if A B(H)(well known to be non nuclear, by S. Wassermann 1974)A min Aop 6 A max Aopwhich gave a counterexample to the first Kirchberg conjectureGilles PisierGrothendieck’s works on Banach spaces
The other Kirchberg conjecture has now become the mostimportant OPEN problem on operator algebras :(here F is the free group)If A C (F ),?A min Aop A max Aop ? CONNES embedding problemGilles PisierGrothendieck’s works on Banach spaces
Let (Uj ) be the free unitary generators of C (IF )Ozawa (2013) provedTheoremThe Connes-Kirchberg conjecture is equivalent to n 1 aij CnXaij Ui Uji,j 1 maxnXaij Ui Uji,j 1minGrothendieck’s inequality impliesnXi,j 1 KGCaij Ui UjmaxGilles PisiernXi,j 1aij Ui UjminGrothendieck’s works on Banach spaces
Indeed,nXi,j 1aij Ui Uj sup{ hη,aij ui vj ξi , ξ, η BH }i,j 1max sup{ nXnXaij hui η, vj ξi , ξ, η BH }i,j 1 sup{ nXaij hxi , yj i , xi , yj BH }i,j 1nX KGC sup{ aij hxi , yj i , xi , yj BC }i,j 1 KGCnXi,j 1Gilles Pisieraij Ui UjminGrothendieck’s works on Banach spaces
Theorem (Tsirelson 1980)If aij R for all 1 i, j n. ThenkXi,jaij Ui Uj kmax kXi,jaij Ui Uj kmin kak n1 H 0 n1 .Moreover, these norms are all equal toXaij ui vj ksup kwhere the sup runs over all n 1 and all self-adjoint unitaryn n matrices ui , vj such that ui vj vj ui for all i, j.Gilles PisierGrothendieck’s works on Banach spaces(8)
Non-commutative and Operator space GTTheorem (C -algebra version of GT, P-1978,Haagerup-1985)Let A, B be C -algebras. Then for any bounded bilinear formϕ : A B C there are states f1 , f2 on A, g1 , g2 on B such that (x, y ) A B ϕ(x, y ) kϕk(f1 (x x) f2 (xx ))1/2 (g1 (yy ) g2 (y y ))1/2 .Many applications to amenability, similarity problems,multilinear cohomology of operator algebras (cf. Sinclair-Smithbooks)Gilles PisierGrothendieck’s works on Banach spaces
Operator spacesNon-commutative Banach spaces (sometimes called “quantumBanach spaces".)DefinitionAn operator space E is a closed subspace of a C -algebra, i.e.E A B(H)Any Banach space can appear, butIn category of operator spaces, the morphisms are differentu:E Fkukcb sup k[aij ] [u(aij )]kB(Mn (E) Mn (F ))nB(E, F )is replacedbounded mapsisomorphismsby CB(E, F )are replacedare replaced(Note : kuk kukcb )by completely bounded mapsby complete isomorphismsIf A is commutative : recover usual Banach space theoryGilles PisierGrothendieck’s works on Banach spaces
L is replaced byNon-commutative L : any von Neumann algebraOperator space theory :developed roughly in the 1990’s byEFFROS-RUAN BLECHER-PAULSEN and othersadmits Constructions Parallel to Banach space caseSUBSPACE, QUOTIENT, DUAL, INTERPOLATION, ANALOGUE OF HILBERT SPACE (”OH").Analogues of projective and injective Tensor productsE1 B(H1 )InjectiveE2 B(H2 )E1 min E2 B(H1 2 H2 )Again Non-commutative L and Non-commutative L1are UNIVERSAL objectsGilles PisierGrothendieck’s works on Banach spaces
Theorem (Operator space version of GT)Let A, B be C -algebras. Then for any CB bilinear formϕ : A B C with kϕkcb 1 there are states f1 , f2 on A, g1 , g2on B such that (x, y ) A B hϕ(x, y )i 2 f1 (xx )g1 (y y ))1/2 (f2 (x x)g2 (yy ))1/2 .Conversely if this holds then kϕkcb 4.With some restriction : SHLYAKHTENKO-P (Invent. Math. 2002)Full generality : HAAGERUP-MUSAT (Invent. Math. 2008) and2 is optimal !Also valid for “exact" operator spaces A, B (no Banach spaceanalogue !)Gilles PisierGrothendieck’s works on Banach spaces
GT, Quantum mechanics, EPR and Bell’s inequalityIn 1935, Einstein, Podolsky and Rosen [EPR] published afamous article vigorously criticizing the foundations of quantummechanicsThey pushed forward the alternative idea that there are, inreality, “hidden variables" and that the statistical aspects ofquantum mechanics can be replaced by this concept.In 1964, J.S. BELL observed that the hidden variables theorycould be put to the test. He proposed an inequality (now called“Bell’s inequality") that is a CONSEQUENCE of the hiddenvariables assumption.After Many Experiments initially proposed by Clauser, Holt,Shimony and Holt (CHSH, 1969), the consensus is :The Bell-CHSH inequality is VIOLATED, and in fact themeasures tend to agree with the predictions of QM.Ref : Alain ASPECT, Bell’s theorem : the naive view of anexperimentalist (2002)Gilles PisierGrothendieck’s works on Banach spaces
In 1980 TSIRELSON observed that GT could be interpreted asgiving AN UPPER BOUND for the violation of a (general) Bellinequality,and that the VIOLATION of Bell’s inequailty is related to theassertion thatKG 1!!He also found a variant of the CHSH inequality (now called“Tsirelson’s bound")Gilles PisierGrothendieck’s works on Banach spaces
The experimentGilles PisierGrothendieck’s works on Banach spaces
Gilles PisierGrothendieck’s works on Banach spaces
Outline of Bell’s argument :Hidden Variable Theory :If A has spin detector in position iand B has spin detector in position iCovariance of their observation isZξij Ai (λ)Bj (λ)ρ(λ)dλwhere ρ is a probability density over the “hidden variables"Now if a n1 n1 , viewed as a matrix [aij ], for ANY ρ we have Xaij ξij HV (a)max sup Xaij φi ψj kak φi 1ψj 1Gilles PisierGrothendieck’s works on Banach spaces
But Quantum Mechanics predictsξij tr(ρAi Bj )where Ai , Bj are self-adjoint unitary operators on H(dim(H) ) with spectrum in { 1} such that Ai Bj Bj Ai andρ is a non-commutative probability density,i.e. ρ 0 trace class operator with tr(ρ) 1. This yieldsXX aij ξij QM(a)max sup aij hAi Bj x, xi kakminx BHwith kakmin relative to embedding (here gj free generators) n1 n1 C (Fn ) min C (Fn )ei ej 7 gi gjEasy to show kakmin kakH 0 , so GT implies :kak kakmin KG kak HV (a)max QM(a)max KG HV (a)maxGilles PisierGrothendieck’s works on Banach spaces
But thePcovariance ξij can be physically measured, and hencealso aij ξij for a fixed suitable choice of a, so we obtain anexperimental answerEXP(a)maxand (for well chosen a) it DEVIATES from the HV valueIn fact the experimental data strongly confirms the QMpredictions :HV (a)max EXP(a)max ' QM(a)maxGT then appears as giving a bound for the deviation :HV (a)max QM(a)maxbutGilles PisierQM(a)max KG HV (a)maxGrothendieck’s works on Banach spaces
JUNGE (with Perez-Garcia, Wolf, Palazuelos, Villanueva,Comm.Math.Phys.2008) considered the same problem forthree separated observers A, B, CThe analogous question becomes : Ifa Xaijk ei ej ek n1 n1 n1 C (Fn ) mi
Just like his thesis, this was devoted to tensor products of topological vector spaces, but in sharp contrast with the thesis devoted to the locally convex case, the “Résumé” was exclusively concerned with Banach spaces (“théo
Locally convex quasi *-algebras, in particular Banach quasi *-algebras, . like tensor products (see [5, 36, 37, 41, 43, 52, 53, 59]). In [2] we construct the tensor product of two Banach quasi *-algebras in order to obtain again a Banach quasi *-algebra tensor
2Théorie des topos et cohomologie étale des schémas, Séminaire de géométrie algébrique du Bois-Marie 1963-64, dirigé par M. Artin, A. Grothendieck, J.-L. Verdier, Lec-ture Notes in Math. 269, 270, 305, Springer-Verlag, 1972, 1973. said, “I have a few comments on your text. Could you please come to my place, I will explain them to you.”
[SGA1]A. Grothendieck, Revêtements Étale et Groupe Fondamental, Lecture Notes in Math 224, Springer-Verlag (1971). [SGA4]A. Grothendieck, M. Artin, and J. L. Verdier, Théorie des Topos et Cohomologie Étale des Schémas, I, II, III, Lecture Notes in Math 269, 270, 305, Springer-Verlag (1972- 1973). 1 FINITE MORPHISMS OF SCHEMES (19/10/2016) Definition 1. .
Let X;Y be two locally convex spaces. Grothendieck proved in his famous PhD thesis (Mem. AMS 1953) that there was a minimal tensor product X _ Y and a maximal tensor product X Y Grothendieck observed that there were locally convex spaces X such that for any other such space Y there was only one possi
“Grothendieck’s r esum e”(!). Just like his thesis ([38]), this was devoted to tensor products of topological vector spaces, but in sharp contrast with the thesis devoted to the locally convex case, the “R esum e” was exclusively concern
tensor products, and nuclear spaces. To summarize Grothendieck’s pre-Kansas research in brief, Laurent Schwartz had given him the following subject as a thesis topic: put a good topology on the tensor product of two locally convex spaces. Schwartz was at that time studying D, the sp
or less rich from one epoch to another, which unfurls over the course of generations and centuries, by the . this boy was his child, [25], p. 77, and the name was changed into Alexander Grothendieck. Many interest- . In the Fall of 1933 dramatic things
Signs with blue circles but no red border mostly give positive instruction. One-way traffic (note: compare circular ‘Ahead only’ sign) Ahead only Turn left ahead (right if symbol reversed) Turn left (right if symbol reversed) Keep left (right if symbol reversed) Route to be used by pedal cycles only Segregated pedal cycle and pedestrian route Minimum speed End of minimum speed Mini .
