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Classification of Von Neumann Algebras and Their Quantum Symmetries

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Classification of Von Neumann Algebras and Their Quantum Symmetries Classification of von Neumann algebras and their quantum symmetries Inaugural Lecture Francqui Chair VUB, 22 February 2016 Stefaan Vaes∗ Fondation Francqui-Stichting Fondation d’Utilité Publique Stichting van Openbaar Nut ∗ Supported by ERC Consolidator Grant 614195 1/27 Banach-Tarski paradox Theorem (Banach and Tarski, 1924) It is possible to cut an orange into pieces, move these pieces by translations and rotations, and obtain two oranges with the same radius as the original one. I Obviously false by comparing weights. I But: the partition is into non-measurable sets. I There is no finitely additive, translation and rotation invariant 3 measure on R that gives a finite nonzero weight to the unit ball. 2/27 Let's try it 3/27 Banach-Tarski: amenable versus non-amenable n Group of motions of R : all distance preserving transformations. John von Neumann (1929) Concept of an amenable group. 3 I The group of motions of R is non-amenable. 2 I The group of motions of R is amenable. Consequences. • The unit ball admits a paradoxical decomposition. • The unit disk does not admit a paradoxical decomposition. 4/27 Amenable groups A groupΓ is called amenable if we can assign to every U ⊂ Γ a weight m(U) 2 [0; 1] with: I m(;) = 0 and m(Γ) = 1. I Finite additivity. I Translation invariance: m(gU) = m(U) for all g 2 Γ and U ⊂ Γ. Results. I If an amenable groupΓ acts on X , there is aΓ-invariant mean on X . Application : no paradoxical decomposition for the unit disk. I A non-amenable groupΓ admits a paradoxical decomposition: Γ can be partitioned into finitely many subsets Ai , Bj such that the union of gi Ai equalsΓ, as well as the union of hj Bj . Application : paradoxical decomposition for the unit ball. 5/27 The free groups The free group F2 is defined as \the group generated by a and b subject to no relations". −1 −1 I Elements of F2 are reduced words in the letters a; a ; b; b , like aba−1a−1b, or like bbbbbba−1bbbb. −1 −1 I Reduced means : no aa , no b b, ... in the word, because they \simplify". So bbaa−1a is not reduced. It reduces to bba. I Group operation : concatenation followed by reduction. Similarly, the free group Fn generated by a1;:::; an. Paradoxical decomposition of F2 : write W (a) = words starting with a. Then,Γ= feg t W (a) t W (a−1) t W (b) t W (b−1), but alsoΓ= W (a) t aW (a−1) andΓ= W (b) t bW (b−1). 3 Observe : two generic rotations of R generate a copy of F2. 6/27 Further examples The following groups are amenable. I Finite groups. I Abelian groups. I Stable under subgroups, direct limits and extensions. The following groups are non-amenable. 1 A group of transformations of the interval [0,1]. I The free groups Fn. 1 = e p lo I Groups containing F2. s I Also other examples. 3/8 2 = e p o l s 1/8 1/2 Open problem : = pe slo Is the Thompson group amenable ? 0 1/4 3/8 1 7/27 Von Neumann algebras I B(H) = the space of all bounded operators on a Hilbert space H. ∗ ∗ I Every T 2 B(H) has Hermitian adjoint T : hT ξ; ηi = hξ; T ηi. I Weak topology : Tn ! T weakly iff hTnξ; ηi ! hT ξ; ηi, 8ξ; η 2 H. Definition A von Neumann algebra is a weakly closed unital ∗-subalgebra of B(H). Examples : • B(H), • L1(X ) acting as multiplication operators on H = L2(X ). 8/27 Amenable versus non-amenable 1 For group von Neumann algebras L(Γ). 1 2 For crossed product von Neumann algebras L (X ) o Γ. 3 For subfactors N ⊂ M in the sense of Jones. Also the program of the upcoming lectures. Together with an introduction to the basics of von Neumann algebras. 9/27 Group von Neumann algebras LetΓ be a countable group. 2 I Hilbert space H = ` (Γ) with orthonormal basis( δh)h2Γ. I Left translation operators λg 2 B(H) given by λg δh = δgh. I Group von Neumann algebra L(Γ) is the von Neumann algebra generated by( λg )g2Γ. Observe : M = L(Γ) has a favorite functional τ : M ! C. Given by τ(T ) = hT δe ; δe i. P Satisfying τ( xg λg ) = xe . 10/27 Intermezzo : factors Simplicity assumption : consider von Neumann algebras M that cannot be written as M = M1 ⊕ M2. Equivalently : Z(M) = C1. Definition : a factor is a von Neumann algebra with trivial center. Theorem (Murray - von Neumann, 1943) The group von Neumann algebra L(Γ) is a factor iffΓ has infinite conjugacy classes (icc), meaning that fhgh−1 j h 2 Γg is infinite for every g 6= e. Remark : every von Neumann algebra is a generalized direct sum of factors. 11/27 Intermezzo : II1 factors P Recall : L(Γ) has favorite functional τ( xg λg ) = xe . Note : τ(xy) = τ(yx) for all x; y 2 L(Γ). Tracial state : a functional τ : M ! C that is ∗ I positive : τ(x x) ≥ 0, I tracial : τ(xy) = τ(yx), I normalized : τ(1) = 1. Definition A II1 factor is a factor that admits a tracial state. L(Γ) is a II1 factor for every icc groupΓ. All factors can be built from II1 factors using Tomita-Takesaki theory. 12/27 Amenable versus non-amenable Theorem (Connes, 1976) All L(Γ) withΓ amenable icc are isomorphic ! ∼ Murray-von Neumann : L(S1) =6 L(F2). I A von Neumann algebra M is called hyperfinite if there exists an increasing sequence of finite dimensional ∗-subalgebras Mn ⊂ M such that [Mn is weakly dense in M. A 0 I Example : M2(C) ⊂ M4(C) ⊂ M8(C) ⊂ · · · , where A 7! 0 A . I Murray-von Neumann : there is a unique hyperfinite II1 factor. I Connes : every amenable II1 factor is hyperfinite. I L(Γ) is amenable iffΓ is amenable. 13/27 Open problems The free group factor problem Are the free group factors L(Fn) isomorphic for distinct n ≥ 2 ? I (Voiculescu 1990, Radulescu 1993) They are either all isomorphic, or all non-isomorphic, including n = 1. I Is L(F1) singly generated ? I Is every II1 factor (acting on a separable Hilbert space) singly generated ? Connes' rigidity conjecture ∼ ∼ If L(PSL(n; Z)) = L(Γ) and n ≥ 3, thenΓ = PSL(n; Z). ∼ Known : there are at most countably manyΓ with L(PSL(n; Z)) = L(Γ), ∼ but there are uncountably manyΓ with L(F2) = L(Γ). 14/27 W∗-superrigidity for groups Theorem (Ioana-Popa-V, 2010) There are countable groups G such that L(G) entirely remembers G : ifΛ is an arbitrary countable group with L(G) =∼ L(Λ), then G =∼ Λ. (I ) These groups are of the form G = (Z=2Z) o Γ: Given an actionΓ y I , consider the action ofΓ by automorphisms (I ) of the direct sum( Z=2Z) , and make a semidirect product. Theorem (Berbec-V, 2012) (Γ) The same is true for G = (Z=2Z) o (Γ × Γ), whereΓ × Γ acts onΓ by left and right multiplication, for many groupsΓ, including the free groups and arbitrary free product groupsΓ=Γ 1 ∗ Γ2 with jΓ1j ≥ 2 and jΓ2j ≥ 3. 15/27 Group measure space construction (Murray - von Neumann) Data : a countable groupΓ acting on a probability space( X ; µ), preserving µ. 1 Output : a tracial von Neumann algebra M = L (X ) o Γ, 1 I generated by a copy of A = L (X ) and unitaries( ug )g2Γ, I satisfying ug uh = ugh, ∗ I ug F ( · )ug = F (g· ) P R I τ Fg ug = X Fe dµ. 1 Silly remark : L(Γ) = L ({∗}) o Γ. Γ Example : Bernoulli actionΓ y (X0; µ0) . n n n Example : SL(n; Z) y T = R =Z . 16/27 Free and ergodic actions 1 GivenΓ y (X ; µ), write A = L (X ) and M = A o Γ. Freeness The subalgebra A ⊂ M is maximal abelian, meaning that A0 \ M = A, iff Γ y (X ; µ) is essentially free, meaning that almost every x 2 X has a trivial stabilizer. Γ Example : Bernoulli actionΓ y f0; 1g is essentially free, but not free. Ergodicity Under essential freeness, M is a factor iff Γ Γ y (X ; µ) is ergodic, meaning that A = C1. Γ Example : the Bernoulli actionΓ y f0; 1g is ergodic. For every free ergodic pmp actionΓ y (X ; µ)aII 1 factor 1 L (X ) o Γ. 17/27 Amenable versus non-amenable Theorem (Connes, 1976) For all amenable groupsΓ and all free ergodic pmp actionsΓ y (X ; µ), 1 the crossed products M = L (X ) o Γ are isomorphic. Indeed: M is amenable and thus the unique hyperfinite II1 factor. W∗-superrigidity (Popa 2003-2004, Ioana 2010) LetΓ be a property (T) group, e.g. Γ = SL( n; Z), n ≥ 3, and Γ Γ y (X ; µ) = (X0; µ0) the Bernoulli action. 1 ∼ 1 If L (X ) o Γ = L (Y ) o Λ for any free ergodic pmp actionΛ y (Y ; ν), thenΓ =∼ Λ and the actions are conjugate. ∗ 1 W -superrigidity : the crossed product L (X ) o Γ remembersΓ and the actionΓ y (X ; µ). First such W∗-superrigidity theorems: Popa-V, 2010. 18/27 Cartan subalgebras 1 GivenΓ y (X ; µ), write A = L (X ) and M = A o Γ. The subalgebra A ⊂ M is special. 0 I It is maximal abelian : A \ M = A. I And regular : the group of unitaries ∗ NM (A) = fu 2 U(M) j uAu = Ag generates M.
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