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 3 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 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) ∈ [0, 1] with:

I m(∅) = 0 and m(Γ) = 1.

I Finite additivity.

I Translation invariance: m(gU) = m(U) for all g ∈ 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 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= {e} 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 ∈ B(H) has Hermitian adjoint T : hT ξ, ηi = hξ, T ηi.

I Weak topology : Tn → T weakly iff hTnξ, ηi → hT ξ, ηi, ∀ξ, η ∈ H.

Definition A is a weakly closed unital ∗-subalgebra of B(H).

Examples : • B(H), • L∞(X ) acting as multiplication operators on H = L2(X ).

8/27 Amenable versus non-amenable

1 For group von Neumann algebras L(Γ).

∞ 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)h∈Γ.

I Left translation operators λg ∈ B(H) given by λg δh = δgh.

I Group von Neumann algebra L(Γ) is the von Neumann algebra generated by( λg )g∈Γ.

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 {hgh−1 | h ∈ Γ} 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 ∈ 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

AII1 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(S∞) =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 = ∞.

I Is L(F∞) 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 |Γ1| ≥ 2 and |Γ2| ≥ 3.

15/27 Group measure space construction (Murray - von Neumann)

Data : a countable groupΓ acting on a probability space( X , µ), preserving µ.

∞ Output : a tracial von Neumann algebra M = L (X ) o Γ,

∞ I generated by a copy of A = L (X ) and unitaries( ug )g∈Γ,

I satisfying ug uh = ugh, ∗ I ug F ( · )ug = F (g· ) P  R I τ Fg ug = X Fe dµ.

∞ 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

∞ 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 ∈ X has a trivial stabilizer.

Γ Example : Bernoulli actionΓ y {0, 1} 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 {0, 1} is ergodic. For every free ergodic pmp actionΓ y (X , µ)aII 1 factor ∞ 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), ∞ 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. ∞ ∼ ∞ 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.

∗ ∞ 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

∞ 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) = {u ∈ U(M) | uAu = A} generates M.

We call such A ⊂ M a Cartan subalgebra.

∞ In the classification of L (X ) o Γ, it is crucial to understand the uniqueness of the Cartan subalgebra.

Up to unitary conjugacy : for every u ∈ U(M), we have the “other” Cartan subalgebra uAu∗ ⊂ M.

19/27 Uniqueness of Cartan subalgebras

I Some II1 factors do not have a Cartan subalgebra : L(Fn) (Voiculescu, 1995)

I Some II1 factors have several Cartan subalgebras (Connes-Jones, 1981).

Theorem (Popa-V, 2011-2012)

IfΓ= Fn is the free group, or any hyperbolic group, and ifΓ y (X , µ) is an arbitrary free ergodic pmp action, then L∞(X ) is the unique Cartan ∞ subalgebra of L (X ) o Γ, up to unitary conjugacy.

We say thatΓ is Cartan-rigid.

Theorem (Ioana, 2012)

All free productsΓ=Γ 1 ∗ Γ2 with |Γ1| ≥ 2, |Γ2| ≥ 3, are Cartan-rigid.

20/27 Crossed products with free groups

Corollary (Popa-V, 2011)

If Fn y X and Fm y Y are free ergodic pmp actions and if ∞ ∼ ∞ L (X ) o Fn = L (Y ) o Fm, then n = m.

∞ ∞ I By uniqueness of Cartan, an isomorphism maps L (X ) onto L (Y ).

I It thus induces an orbit equivalence of the actions : isomorphism ∆ : X → Y with ∆(Fn · x) = Fm · ∆(x) for a.e. x ∈ X .

I This implies that n = m by one of Gaboriau’s invariants for countable equivalence relations : cost or the first L2-Betti number.

21/27 Bernoulli actions

Theorem (many hands) Γ Consider the Bernoulli actionΓ y (X , µ) = (X0, µ0) and crossed product ∞ M = L (X ) o Γ.

• ForΓ= Fn, the II1 factors M are exactly classified by n.

• ForΓ= Fn × Fm, the II1 factors M are exactly classified by {n, m} and entropy(µ0).

Γ Noncommutative Bernoulli action : Γ y (Mk (C), Tr( · D)) . The crossed product M need not have a trace.

Theorem (V-Verraedt, 2014)

ForΓ= Fn, the factors M are exactly classified by n and the subgroup of ∗ R+ generated by the ratios between the eigenvalues of D.

22/27 Intermezzo : continuous dimension

I AII1 factor M has a tracial state τ : M → C. −1 I Also Mn(C) has a tracial state τ(A) = n Tr(A). −1 I Note that for a projection p ∈ Mn(C), we have τ(p) = n dim(Im p).

I In a II1 factor, τ(p) can take every value in [0, 1].

View pM as a right M-module.

We declare dimM (pM) = τ(p).

We extend dimM to arbitrary M-modules.

23/27 Jones’ subfactors

Definition

A subfactor is an inclusion N ⊂ M of II1 factors.

Jones index : [M : N] = dimN (M).

Example : forΛ < Γ, we have[ L(Γ), L(Λ)] = [Γ : Λ].

Theorem (Jones, 1982) The index can take exactly the values {4 cos2(π/n) | n ≥ 3} ∪ [4, +∞].

Knots and links (Jones polynomial). Conformal field theory. Low dimensional topology.

24/27 Standard invariant of a subfactor

Let N ⊂ M be a subfactor with[ M : N] < ∞.

Define the M-M-bimodule Xn = M ⊗N M ⊗N · · · ⊗N M.

Principal graph of a subfactor A bipartite graph with

I even vertices : the irreducible N-N-bimodules X appearing in the Xn,

I odd vertices : the irreducible N-M-bimodules Y appearing in the Xn, I a k-fold edge between X and Y if X appears k times in Y .

Temperley-Lieb-Jones subfactors. 2 π  I Principal graph An : • — • — •···• — • with index 4 cos n+1 . I Principal graph A∞ : • — • — • — •··· with any index ≥ 4.

Standard invariant : the entire tensor category, not only the fusion rules.

25/27 Subfactors : amenable versus non-amenable

Definition Let N ⊂ M be a finite index subfactor with principal graph G. Always : kGk2 ≤ [M : N]. Amenable : if equality holds.

Theorem (Popa, 1992) Every amenable standard invariant arises from exactly one subfactor of the hyperfinite II1 factor. So: in the amenable case, the standard invariant is a complete invariant.

I (Bisch-Nicoara-Popa, 2006) infinitely many hyperfinite subfactors with the same non-amenable standard invariant and index 6,

I (Brothier-V, 2013) unclassifiably many hyperfinite subfactors with standard invariant A3 ∗ D4 and index 6.

26/27 Open problems

I Which values of[ M : N] arise for irreducible hyperfinite subfactors 0 (meaning that N ∩ M = C1)?

I Which standard invariants arise from hyperfinite subfactors ?

I At which indices, the A∞ standard invariant arises from a hyperfinite subfactor ?

I Are there infinitely many hyperfinite√ subfactors with standard invariant A3 ∗ A4 and index3+ 5 ?

The standard invariant of a subfactor has a flavor.

The principal graph corresponds to the Cayley graph.

A “ approach” to standard invariants : Popa-V (2015) and Popa-Shlyakhtenko-V (2016).

27/27