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) ∈ [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 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= {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 von Neumann algebra 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.