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Free Probability, Free Entropy and Applications to von Neumann Algebras 789Type III: Every (non zero) projection is infinite.Examples of von Neumann algebras: 1) Let A = CG for some discrete groupG, % be l 2 (G) and r be the vector state given by the vector that takes value 1 atg and 0 elsewhere. Then r is a trace, l 2 (G) = L 2 (A,T) and the weak (or strong)operator closure of A is called the group von Neumann algebra, denoted by CG-We have that CQ is a factor if and only if each conjugacy class of G other than theidentity is infinite (i.e.a). For example, the free group F n (n > 2, on n generators)is such an i.c.c. group.2) Suppose (ii,p) is a measure space with a a-finite measure p, G is a groupand a is a measurability preserving action of G on Q. Formally, we have an algebraL°°(Q,p)G (= Â) similar to the group algebra definition: (ip(x)g)(ip(x)fi) =(fi(x)ip(g^1 (xj)gh, for >p,ip £ L°°(Q,p) and g,h £ G. Assume that G acts freely(i.e., for any g in G with j / e , the set {x £ 0 : g(x) = x} has measure zero).Define an action of A on the Hilbert space ©„ G G L 2 (Q,p)g by left multiplication( which is induced by the multiplication in A), where L 2 (Q,p)g is an isomorphiccopy of L 2 (Q, p). Then the von Neumann algebra generated by A is called the crossproduct von Neumann algebra, denoted by L°°(Q,p) x Q G. This cross product isa factor if and only if a is ergodic. If a is ergodic and not a measure preservingaction, then L°° (ii, p) x a G is a factor of type III. Type II factors are obtained frommeasure preserving actions (with Q a non atomic measure space) and the finitenessof p gives rise to type Hi factors.The above 1) and 2) are the two basic constructions of von Neumann algebras(given by Alurray and von Neumann [MN]). A. Connes [C] shows that thereare finite factors that cannot be constructed by 1). It was a longstanding openproblem whether every (finite) von Neumann algebra can be obtained by using theconstruction in 2). Using free probability theory, especially the notion of free entropy,Voiculescu [V2] gives a negative answer to this question. We shall discusssome details later in the talk.In recent years, the focus of studies of von Neumann algebras is centered onfactors of type Hi • Many of the unsolved problems in operator algebras are alsoreduced to this class. We end this section with two of the (still) open problems fromthe list of 20 questions asked by Kadison in 1967.1. The weak-operator closure of the left regular representation of the free (nonabelian)group on two or more generators is a factor of type Hi • Are these factorsisomorphic for different numbers of generators?2. Is each factor generated by two self-adjoint operators? —each von Neumannalgebra? —the factor arising from the free group on three generators? —is eachvon Neumann algebra finitely generated?Three of those 20 problems were answered in the last ten years by using freeprobability and free entropy. We explain some of the theory involved in the followingtwo sections.3. Free independence
790 Liming GèSuppose (A,T) is a C*-probability space. We assume that r is a trace. Afamily A,,, i £ I, of unital * subalgebras of A are called free with respect to r ifT(AIA 2 • • • A n ) = 0 whenever Aj £ A h , ti fi 1 • • • fi 1 t n (*i and t 3 may be the same)and T(AJ) = 0 for 1 < j < n and every n in N. A family of subsets (or elements)of A are said to be free if the unital * subalgebras they generate are free.Note that freeness is a highly noneommutative notion, the non-commutativity(or algebraic freeness) of free random variables is encoded in the definition. Recallsome basic concepts in free probability theory.Semicircular elements: The Gaussian laws in classical theory is replaced bythe semicircular laws. The semicircular law centered at a and of radius r is thedistribution p a>r : C[x] —¥ C such thatoi-a+rßa,r(v>(xj) = —2 /
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Beijing 2002August 20-28Proceedings
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ContentsSection 1. LogicE. Bouscare
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R. Pandharipande: Three Questions i
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Section 1. LogicE. Bouscaren: Group
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4 E. Bouscaren2. Different notions
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6 E. BouscarenWe now consider an al
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8 E. Bouscarendimension would need
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10 E. Bouscaren1. Either G is not o
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12 E. Bouscaren[7] E. Bouscaren & F
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14 J. Denef F. Loeserthis theory to
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16 J. Denef F. Loeserwill require m
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18 J. Denef F. Loeserseries P P (T)
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20 J. Denef F. Loeserfor all m> n.
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22 J. Denef F. Loeseri > 1. One ver
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ICM 2002 • Vol. II • 25-33Autom
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Automorphism Groups of Saturated St
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ICM 2002 • Vol. II • 37-45Repre
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3. The Blanchfield pairingRepresent
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Representations of Braid Groups 41n
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Representations of Braid Groups 43A
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Representations of Braid Groups 45T
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48 A.Bondal D.Orlovis believed to b
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50 A.Bondal D.OrlovDefinition 4 [BK
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52 A.Bondal D.OrlovLet Y be a smoot
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54 A.Bondal D.OrlovIn particular, w
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56 A.Bondal D.Orlov[BVdB] Bondal A.
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58 M. Levine(PB) Let £ be a rank r
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60 M. Levine2. Let 1Z d%m (X) be th
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62 M. LevineNow, if P = P(ci,. •
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64 M. LevineProof. For CH*, this us
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66 M. Levineis an isomorphism. Sinc
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68 Cheryl E. Praegeranalysis. Analo
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70 Cheryl E. Praegers-arc-transitiv
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72 Cheryl E. Praegercase a good kno
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74 Cheryl E. Praeger5. Simple group
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76 Cheryl E. Praeger[15] C. H. Li,
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78 Markus Rost1. Norm varietiesAll
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80 Markus RostWe apply the degree f
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82 Markus Rost4. Hubert's 90 for sy
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84 Markus RostFor n = 2 one can tak
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ICM 2002 • Vol. II • 87^92Dioph
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Diophantine Geometry over Groups an
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Diophantine Geometry over Groups an
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ICM 2002 • Vol. II • 93^103None
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Noncommutative Projective Geometry
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106 Dimitri Tamarkinalgebra and def
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108 Dimitri TamarkinThe product on
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110 Dimitri TamarkinProof. Let F =
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112 Dimitri Tamarkin2.5.9.Similarly
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114 Dimitri TamarkinComputeJ2,jl(a
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116 Dimitri Tamarkin3.2.1.To descri
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ICM 2002 • Vol. II • 119-128Con
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Converse Theorems, Functoriality, a
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ICM 2002 • Vol. II • 129^138Con
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Constructing and Counting Number Fi
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ICM 2002 • Vol. II • 139-148Ana
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Analyse p-adique et Représentation
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ICM 2002 • Vol. II • 149-162Equ
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Equiv. Bloch-Kato Conjecture and No
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[17;[is;[19;[20;[21[22[23;[24;[25;[
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ICM 2002 • Vol. II • 163-171Tam
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Tamagawa Number Conjecture for zeta
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174 S. S. Kudlaseries with represen
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t.176 S. S. Kudlais the exponential
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178 S. S. Kudlawhere Z = J2ì n ìP
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180 S. S. Kudla(ii) T £ Sym 2 (Z)>
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182 S. S. KudlaTheorem 6. ([13], [9
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ICM 2002 • Vol. II • 185-195Ell
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Elliptic Curves and Class Field The
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198 E. UllmoSoit X une variété al
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200 E. UllmoThéorème 2.4 Soit X u
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202 E. UllmoQuestion 4.1 Soit x n =
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204 E. UllmoNotons que cet énoncé
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206 E. Ullmo[17] H. Oh. Uniform Poi
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208 T. D. WooleyWaring's problem as
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210 T. D. Wooleycircumstances one h
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212 T. D. Wooleywith 1 < z,w < F, M
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214 T. D. Wooleyand also byl f \K(a
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216 T. D. Wooleytions involving nor
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Section 4. Differential GeometryB.
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222 B. Andrewspotentially applicabl
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224 B. Andrewssurface can look metr
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226 B. Andrewsboundary of the set {
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228 B. Andrewseach t > 0, to either
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230 B. AndrewsUSSR Izv. 20 (1983).[
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232 Robert Bartnikprovides an impor
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234 Robert BartnikTheorem 2 If (M,
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236 Robert BartnikThe horizon condi
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238 Robert BartnikTheorem 8 Suppose
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240 Robert BartnikPhysics, LNP 212.
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242 P. Biran2. Various intersection
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244 P. BiranTheorem E'. Let (M,oj)
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246 P. Biransymplectic packings in
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248 P. Birangroup of Hamiltonian di
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250 P. Birannumber of steps it take
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252 P. Biranbe used to obtain many
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254 P. Biran[6] P. Biran, A stabili
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ICM 2002 • Vol. II • 257-271Bla
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Black Holes and the Penrose Inequal
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[23;[24;[25;[26;[27[28;[29'[30^Blac
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274 Xiuxiong ChenFor simplicity, in
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276 Xiuxiong ChenTheorem 0.1. [14]
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278 Xiuxiong ChenQuestion 0.6. (Don
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280 Xiuxiong Chencurvature in S 2 c
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282 Xiuxiong Chen[12] X. X. Chen an
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284 Weiyue DingSehrödinger flows a
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286 Weiyue DingIn order to prove Th
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288 Weiyue DingFor the first term i
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290 Weiyue Dingwhich satisfies the
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ICM 2002 • Vol. II • 293^302Dif
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Differential Geometry via Harmonic
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then eitherorDifferential Geometry
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ICM 2002 • Vol. II • 303-313Ind
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Index Iteration Theory for Symplect
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316 Anton PetruninTheorem A. Given
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318 Anton PetruninRiemannian manifo
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320 Anton Petruninspace. In the cas
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ICM 2002 • Vol. II • 323-338Col
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Collapsed Riemannian Manifolds with
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ICM 2002 • Vol. II • 339-349Com
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Complex Hyperbolic Triangle Groups
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352 Paul Seideldeformation spaces.
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354 Paul Seidelforgetting all the p
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356 Paul SeidelTheorem 3. Si,...,S
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358 Paul Seidelwhich must be of ord
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360 Paul Seidel[10] E. Getzler, Bat
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362 Weiping Zhang(iii) The heat ker
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364 Weiping Zhang3. The index theor
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366 Weiping Zhangwhere ch(g) is the
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- T p ( P d M , > 0 , 9 P d M , > 0
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Section 5. TopologyMladen Bestvina:
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374 Mladen Bestvinaof Out(F n ) coc
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376 Mladen Bestvina3. Culler-Vogtma
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378 Mladen BestvinaDefinition 8. A
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380 Mladen Bestvinaand then gluing
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382 Mladen BestvinaAnal. 10 (2000),
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384 Mladen Bestvina63 John R. Stall
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386 Yu. V. Chekanovplu I,Figure 1:
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388 Yu. V. Chekanovwithout loss of
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390 Yu. V. ChekanovFigure 2: Lagran
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392 Yu. V. ChekanovFigure 5: Local
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394 Yu. V. Chekanovderived from the
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396 M. Furatadimensional vector spa
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398 M. FurataHowever in the Seiberg
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400 M. Furata6. Seiberg-Witten-Floe
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402 M. Furata[14] K. Fukaya & K. On
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ICM 2002 • Vol. II • 405-114Gé
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Géométrie de Contact 407- la sph
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Géométrie de Contact 4093) chaque
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412 E. Giroux- en tout point p où
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414 E. Giroux[EU] Y. ELIASHBERG, Cl
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416 L. Hesselholt1. Algebraic üC-t
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418 L. Hesselholt(i) a pro-log diff
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420 L. HesselholtThe canonical map
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422 L. Hesselholttogether with a mu
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424 L. Hesselholtwhere \'- GK —^
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ICM 2002 • Vol. II • 427-436Sym
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Symplectic Sums and Gromov-Witten I
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ICM 2002 • Vol. II • 437-146Kno
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Knots, von Neumann Signatures, and
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ICM 2002 • Vol. II • 447-156Str
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Strings and the Stable Cohomology o
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ICM 2002 • Vol. II • 457-168Non
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Non-zero Degree Maps between 3-Mani
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Section 6. Algebraic and Complex Ge
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472 Hélène Esnaultto ^klX)/k carr
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474 Hélène Esnaultf*c 2 (E, V) is
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476 Hélène EsnaultTheorem 3.1 ([5
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478 Hélène Esnaultno longer the c
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480 Hélène EsnaultQuestion 5.4. W
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ICM 2002 • Vol. II • 483-494Hil
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Hilbert Schemes of Points on Surfac
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and use this to define operatorsHil
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ICM 2002 • Vol. II • 495-502Vec
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Vector Bundles on a K3 Surface 497s
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Vector Bundles on a K3 Surface 499A
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Vector Bundles on a K3 Surface 501a
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ICM 2002 • Vol. II • 503-512Thr
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Three Questions in Gromov-Witten Th
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ICM 2002 • Vol. II • 513^524Upd
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Update on 3-folds 515in the hyperbo
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Update on 3-folds 517In many contex
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Update on 3-folds 5193.3. Explicit
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Update on 3-folds 521The modem view
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Update on 3-folds 523is entertainin
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ICM 2002 • Vol. II • 525-532Sur
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Sur les Algèbres Vertex Attachées
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ICM 2002 • Vol. II • 533-541Top
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Topology of Singular Algebraic Vari
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ICM 2002 • Vol. II • 545-554Har
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Harmonie Analysis on Real Reductive
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[BS3][BS4][Be]Harmonie Analysis on
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ICM 2002 • Vol. II • 555-570On
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On the Dynamical Yang-Baxter Equati
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ICM 2002 • Vol. II • 571-582Geo
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Geometrie Langlands Correspondence
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ICM 2002 • Vol. II • 583-597On
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On the Local Langlands Corresponden
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600 A. KlyachkoAmong commonly known
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602 A. KlyachkoBy Theorem 2.3 this
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604 A. Klyachkois stable iff for ev
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606 A. KlyachkoUnitary spectral pro
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608 A. Klyachkospectra. By Metha-Se
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610 A. KlyachkoAgain our experience
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612 A. Klyachko[9] G. Faltings, Mum
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ICAl 2002 • Vol. II • 615-627Br
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Branching Problems of Unitary Repre
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630 Vikram Bhagvandas AlehtaDefinit
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632 Vikram Bhagvandas Alehtais in t
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634 Vikram Bhagvandas AlehtaTheorem
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ICAl 2002 • Vol. II • 637^642Cl
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Clifford Algebras and the Duflo Iso
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Clifford Algebras and the Duflo Iso
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ICAl 2002 • Vol. II • 643^654Re
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Representations of Yangians 6451.2.
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Representations of Yangians 6478 9
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Representations of Yangians 6492. T
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Representations of Yangians 651irre
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Representations of Yangians 6532.4.
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ICAl 2002 • Vol. II • 655-666Au
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Automorphic L-functions and Functor
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ICAl 2002 • Vol. II • 667^677Mo
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Modular Representations 669(l.b.2)
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Modular Representations 671e) Any p
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Modular Representations 673conjugac
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Modular Representations 675A genera
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Modular Representations 677^-repres
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ICAl 2002 • Vol. II • 681-690Va
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Value Distribution and Potential Th
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ICAl 2002 • Vol. II • 691-700Th
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The Branch Set of a Quasiregular Al
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ICAl 2002 • Vol. II • 701-709Ha
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Harmonie Aleasure and "Locally Flat
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712 N. Lerner1. From Hans Lewy to N
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714 N. Lernergood cases in (1.5) (w
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716 N. Lerner2. Counting the loss o
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718 N. LernerSolvability with loss
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720 N. Lerner6] L.Hörmander, On th
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722 C. ThieleA basic point of singu
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724 C. ThieleFigure 1: "Cone"< en I
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726 C. Thieleis negative. If all of
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728 C. Thielealso in these degenera
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730 C. ThieleThus he needed good co
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732 C. Thiele[6] Gilbert J., Nahmod
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I p : — \ J ( Z i , . . . , Z m )
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Page 710 and 711: 738 S. Zelditchfor U C
Page 712 and 713: 740 S. Zelditch[Pi,Pj] = 0 and whos
Page 714 and 715: 742 S. Zelditch[11] V. F. Lazutkin,
Page 716 and 717: 744 Xiangyu Zhou+00 at the boundary
Page 718 and 719: 746 Xiangyu Zhouconnected component
Page 720 and 721: 748 Xiangyu ZhouWightman [32], Jost
Page 722 and 723: 750 Xiangyu ZhouBrief proof is as f
Page 724 and 725: 752 Xiangyu Zhou[11] Al. Jarnicki,
Page 726 and 727: Section 9. Operator Algebras andFun
Page 728 and 729: 758 Semyon Alesker0.1.1 Definition,
Page 730 and 731: 760 Semyon Alesker2. Translation in
Page 732 and 733: 762 Semyon Alesker2.3. Valuations i
Page 734 and 735: 764 Semyon AleskerReferences[1] Ale
Page 736 and 737: 766 P. Bianewith classical cumulant
Page 738 and 739: 768 P. BianeObserve thatr(ai ... a
Page 740 and 741: 770 P. Biane4. Noncrossing cumulant
Page 742 and 743: 772 P. Bianethis gives the order of
Page 744 and 745: 774 P. Biane[20] R. Speicher, Combi
Page 746 and 747: 776 D. Bischsubfactors has develope
Page 748 and 749: 778 D. Bisch(for all k > 0), where
Page 750 and 751: 780 D. BischObserve that by constru
Page 752 and 753: 782 D. BischIt should be evident th
Page 754 and 755: 784 D. Bisch[4] D. Bisch, Bimodules
Page 756 and 757: ICAl 2002 • Vol. II • 787^794Fr
Page 760 and 761: Free Probability, Free Entropy and
Page 762 and 763: Free Probability, Free Entropy and
Page 764 and 765: ICAl 2002 • Vol. II • 795^812Ba
Page 766 and 767: Banach KK-theory and the Baum-Conne
Page 782 and 783: ICAl 2002 • Vol. II • 813-822On
Page 784 and 785: On Some Inequalities for Gaussian A
Page 792: Author IndexAlesker, Semyon 757Andr
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