UFFE HAAGERUPS FIRST TALK AT MASTERCLASS ON VON NEUMANN ALGEBRAS AND GROUP ACTIONS

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1 UFFE HAAGERUPS FIRST TALK AT MASTERCLASS ON VON NEUMANN ALGEBRAS AND GROUP ACTIONS 1. Von Neumann Algebras Denition 1. Let H be a Hilbert space (usually nite dimensional separable), let B(H) be the *-algebra of bounded operators on H. M B(H) is a von Neumann Algebra i M is a *-subalgebra of B(H) 1 M M is SO-closed Theorem 2 (von Neumann, 1929). M B(H) is a von Neumann algebra i M = M M = M. Denition 3. The center of M is denoted by Z(M) dened as Z(M) = M M. M is a factor i Z(M) = C 1. Type of factor M Model / characterization I n, n N M n (C) I B(H), dim(h) = II 1 dim(m) =, tracial state τ : M C a, b M : τ(ab) = τ(ba) II M = N B(H), dim(h) =, N II 1 factor III Everything else If H is a separable Hilbert space M B(H) a von Neumann algebra, then (1.1) M = Ω M(ω)dω where each M(ω) is a factor. This is the desintegration of M in factors. 2. Projections in a von Neumann algebra Denition 4. If M is a von Neumann algebra, we set (2.1) P (M) = {p M p = p = p 2 } the set of projections in M. Theorem 5. (2.2) M = span(p (M)) Denition 6. For p, q P (M) we say p q i u M : p = u u, q = uu. Denition 7. M is nite i (p P (M) : p 1 p = 1). M is properly innite i p, q P (M) : p q, p q 1 In a von Neumann algebra (but not in a general C*-algebra), this is the same as p, q P (M) : p q, p + q = 1, p q 1. Date : 25/01/

2 2UFFE HAAGERUPS FIRST TALK AT MASTERCLASS ON VON NEUMANN ALGEBRAS AND GROUP ACTIONS Type I n type II 1 factors are nite, type I, type II type III factors are properly innite. 3. Traces A II 1 -factor M has a unique trace state τ: τ is normal τ is faithful ((a 0, τ(a) = 0) a = 0) p, q P (M) : p q τ(p) = τ(q) τ(p (M)) = [0, 1] A I -factor or II factor Mhas a normal faithful seminite trace τ dened on M + (Seminite means that the set {a M τ(a a) < } is SO-dense in M.) τ is unique up to multiplication by λ ]0, [ a M : τ(a a) = τ(aa ) If M is a type II -factor, τ(p (M)) = [0, ] 4. group von Neumann Algebras Denition 8. If G is a discrete group (usually countable).then we dene L(G) as the von Neumann algebra generated by the left regular representation of G. For this dene λ : G B(l 2 (G)) as f l 2 (G) x, y G : (λ(x)f)(y) = f(x 1 y). Set (4.1) L(G) = span{λ(x) x G} SO (4.2) C r (G) = span{λ(x) x G}. Observe C r (G) L(G). L(G) is a nite von Neumann algebra. Denition 9. A group G is an ICC-group i g G, g e : {hgh 1 h H} is innite. Theorem 10. Let G {e} then TFAE G is an ICC-group L(G) is a factor L(G) is a II 1 -factor 5. Injectivity hyperfiniteness of von Neumann algebras Denition 11. M B(H) is injective i p P (B(H)), p = 1 of B(H) onto M. Denition 12. M is hypernite (or AFD) on a separable Hilbert space i M = n=1m n where M 1 M 2... are nite dimensional von Neumann algebras. Theorem 13 (Connes, 1976). If H separable, M B(H). Then M is injective i M is hypernite. Theorem 14 (Murray + von Neumann, 1940). There is up to isomorphism only one hypernite factor of type II 1 on a separable Hilbert space.

3 UFFE HAAGERUPS FIRST TALK AT MASTERCLASS ON VON NEUMANN ALGEBRAS AND GROUP ACTIONS3 6. Amendable groups Denition 15. A discrete group G is amendable if µ : P(G) [0, 1] nite additive measure with µ(g) = 1 such that A G x G : µ(xa) = µ(a). Observation 16. G is amendable i state m : l (G) C such that f l (G) x G : m(xf) = m(f) where x, y G : (xf)(y) = f(x 1 y). Theorem 17 (Sakai, Connes). If G is a discrete countable group TFAE G amendable L(G) is injective L(G) is hypernite where the equivalence of the rst two statements is old, does not require countability. Remark 18. G amendable i C r (G) nuclear. Corollary 19. If G is a amendable, countable ICC-group then L(G) = R the unique hypernite II 1 -factor. Denition 20. A group G is solvable i G 1 G 2 G n = G such that G k+1 /G k is abelian. Denition 21. (6.1) F n = group generated by {g 1,..., g n no relations } (6.2) SL(n, Z) = {g M n (Z) det(g) = 1} { (6.3) P SL(n, Z) = SL(n, Z) SL(n, Z)/{±1} n odd n even (6.4) S n = { permutations of {1,..., n}} (6.5) S = Group nite groups abelian groups solvable groups inductive limits of amendable groups S n=1 F n, 2 n no SL(n, Z), 2 n no P SL(n, Z), 2 n no Remark 22. P SL(n, Z) is an ICC-group. S n Amendable Remark 23. L(F n ) L(S ) = R for 2 n as L(F n ) is not injective, but F n, S are ICC-groups, so L(F n ) is a II 1 -factor not hypernite.

4 4UFFE HAAGERUPS FIRST TALK AT MASTERCLASS ON VON NEUMANN ALGEBRAS AND GROUP ACTIONS 7. The fundamental group of II 1 -factors Recall 24. Let M be a II 1 -factor, then M has unique trace state τ p, q P (M) : p q τ(p) = τ(q) t [0, 1] p t P (M) : t = τ(p t ) Observation 25. p, q P (M) : p q pmp = qmq as von Neumann algebras on ph qh these are II 1 -factors too. Denition 26. If t ]0, 1] put M t = p t Mp t which is welldened up to isomorphism. For general t ]0, [ dene M t = M n (M) t/n where n N, n 2. (Have to check that M t doesn't depend on n up to isomorphism. Proposition 27. (7.1) (M s ) t = M st s, t > 0 (7.2) M n = M n (M) = M M n (C) n N Denition 28. The fundamental group of M is the set (7.3) F(M) = {t ]0, [ M t = M} Remark 29. F(M) is a multiplicative subgroup of (], [, ) (7.4) F(M) [0, 1] = {τ(p) p P (M) : pmp = M} F(M) is the multiplicative group generated by {τ(p) p P (M) : pmp = M} (7.5) F(M) = { τ(p) τ(q) p, q P (M) : pmp = qmq} Observation 30. M n (M) = M i n F(M) ( n N : M n (M) = M) i Q + F(M). Example 31 (Murray + von Neumann 1940). If R is the hypernite II 1 -factor, F(R) =]0, [. Theorem 32 (Connes 1975). If G has Kazhdan's property T (e.g. G = P SL(n, Z), n 3) then F(L(G)) is countable, in particular F(L(G)) ]0, [. Theorem 33 (Radulescu + Dykema 1994). F(L(F )) =]0, [ Problem 34. Compute F(L(F n )) for 2 n <. Using free probability one can prove [Radulescu + Dykema] that L(F n ) = 1 L(F 2 ) tn where t n = n 1. From this follows that F(L(F n )) = F(L(F 2 )) for 2 n <. Theorem 35 (Radulescu + Dykema). Either F(L(F 2 )) = {1} in which case L(F 2 ), L(F 3 ),..., L(F ) are all nonisomorphic II 1 -factors or F(L(F 2 )) =]0, [ in which case L(F 2 ), L(F 3 ),..., L(F ) are all isomorphic II 1 -factors or

5 UFFE HAAGERUPS FIRST TALK AT MASTERCLASS ON VON NEUMANN ALGEBRAS AND GROUP ACTIONS5 8. Interpolated free group factors Denition 36. For t ]1, [ dene L( t ) = L([F 2 ) 1 t 1. Remark 37. L(F 7 6 F(L(F 2 )). = L(P SL(2, Z)) = L(Z 2 Z 3 ) hence F(L(P SL(2, Z)) = Theorem 38 (Sorin + Popa, ). There exists a II 1 -factor M on a separable Hilbert space such that F(M) = {1}. (e.g. M = L(Z 2 SL(2, Z))) Theorem 39 (Popa, Popa + Adrian + Petersen). For all countable subgroups Γ ]0, [ there exists a II 1 -factor M such that {(M) = Γ. Theorem 40 (Popa + Vaes, 2008). There exists a II 1 -factor M such that F(M) ]0, [ but F(M) is not countable. 9. The "Group measure space"construction Denition 41. Let A B(H) be a von Neumann algebra G be a discrete group α : G Aut(A) be an action of G on A (a group homomorphism). Then the crossed product M = A α G acts on l 2 (G, H) M = {π(a) λ(g)} where π : A B(l 2 (G, H)) is given by (9.1) (π(a)ξ)(g) = αg 1 (a)ξ(g) ξ l 2 (G, H), g G where (9.2) l 2 (G, H) = H λ : G B(l 2 (G, H)) is given by g G (9.3) (λ(g)ξ)(h) = ξ(g 1 h) ξ l 2 (G, H), g, h G. (π, λ) is a??? representation of (A, G, α): (9.4) λ(g)π(a)λ(g) 1 = π(α g (a)) a A, g G. A = π(a) A α G A αg is generated by A λ(g) for g G (9.5) λ(g)aλ(g) 1 = α g (a). 10. Special case If H is a seperable Hilbert space A is abelian then A = L (Ω, µ) for some stard Borel space Ω σ-nite measure µ. Note that L (Ω, µ) = L (Ω, ν) when ν [µ] that is when µ ν have the same null sets (ν µ µ ν). Given a group homomorphism σ : G Iso(Ω, [µ]), that is a [µ]-preserving Borel isomorphism of Ω), one can associate an action α : G Aut(L (Ω, µ)) by (10.1) α(f)(ω) = f(σ 1 ω) f L (Ω, µ). Then M = L (Ω) α G is called the von Neumann algebra obtained from the "Group measure space-construction.