Infinite groups, actions on the interval [0,1] and von Neumann algebras Algemeen Wiskundecolloquium Radboud Universiteit Nijmegen, March 2009 Stefaan Vaes 1/21
What are groups ? ◮ Abstract definition. ◮ Groups appear as symmetries of mathematical structures. ◮ Actions of groups on ... Examples (biased towards the topic of this talk) : ◮ Action of Z on the circle, where n ∈ Z acts by rotation of the circle over the angle n α . T n : S 1 → S 1 : T n ( z ) = exp ( in α) z . From now on, we write n · z for the action of n ∈ Z on z ∈ S 1 . ◮ Action of SL ( 2 , Z ) on the torus S 1 × S 1 = R 2 / Z 2 . � � � � � � y a z b a b y . · = y c z d c d z 2/21
A crash course in measure theory What do the examples have in common. ◮ On S 1 we have a notion of length and on S 1 × S 1 a notion of area. Our transformations preserve this length/area. ◮ Both S 1 and S 1 × S 1 are probability spaces : a set with a family of measurable subsets and a measure thereon. New notion : a measurable map is such that the inverse image of a measurable set is again measurable. Never mind, in practice every map that you can write down is measurable. Other notions : measure zero and ‘almost everywhere’. ◮ Both Z ↷ S 1 and SL ( 2 , Z ) ↷ S 1 × S 1 are actions of groups by measurable transformations preserving the probability measure. 3/21
More examples of probability measure preserving group actions 1. Let K be a compact group (e.g. K = U ( n ), O ( n ), S 1 ). ◮ K has a unique K -invariant probability measure, called Haar measure. ◮ Choose Γ ⊂ K countable and dense. ◮ Then, Γ acts on K by translation, preserving the Haar measure. 2. Let Γ be any countable group. ◮ Build the probability space � ( X , µ) = [ 0 , 1 ] Γ = [ 0 , 1 ] with its product measure. Γ ◮ The measure of a ‘rectangle’ is what you expect. ◮ Then, Γ ↷ ( X , µ) by ( g · x ) h = x g − 1 h . This is called the Bernoulli action of Γ . 4/21
The assumption of ergodicity We impose a ‘minimality’ or ‘simplicity’ assumption : we look at Γ ↷ ( X , µ) that cannot be decomposed ‘as the sum of two’, meaning X = X 1 ⊔ X 2 where both X 1 , X 2 are Γ -invariant. However : we do not bother if X 1 or X 2 has measure zero. Formal definition of ergodicity The probability measure preserving action Γ ↷ ( X , µ) is called ergodic, if every globally Γ -invariant measurable subset Y ⊂ X has measure 0 or measure 1. ◮ Equivalent condition : Every Γ -invariant function X → C is constant almost everywhere. ◮ We prove that Z ↷ S 1 : n · z = exp ( in α) z is ergodic if and only if α/ 2 π is irrational. 5/21
Ergodicity of irrational rotation Recall : Z ↷ S 1 : n · z = exp ( in α) z . Proof. Let F : S 1 → C be a bounded measurable Z -invariant function. � � = F ( z ) for all n ∈ Z , z ∈ S 1 . This means that F exp ( in α) z ◮ The Fourier coefficients satisfy ˆ F ( n ) = exp ( in α) ˆ F ( n ) . ◮ For α irrational, we get ˆ F ( n ) = 0 when n ≠ 0. ◮ So, ˆ F ( n ) = ˆ G ( n ) for some constant function G . ◮ Hence, F ( z ) = G ( z ) = constant for almost all z ∈ S 1 . By the way : construct yourself, for α rational, Z -invariant functions. 6/21
Hilbert spaces • Complex vector space H . • Positive-definite scalar product H × H → C : (ξ, η) ֏ � ξ, η � . � • Completeness for the norm � ξ � = � ξ, ξ � . Examples of Hilbert spaces n ∞ � � ◮ C n with � ξ, η � = and ℓ 2 ( N ) with � ξ, η � = ξ k η k ξ k η k . k = 1 k = 0 � 2 π 1 ◮ L 2 ( S 1 ) with � ξ, η � = ξ( exp ( it ))η( exp ( it )) dt . 2 π 0 New notions : Operator = linear map from Hilbert space to Hilbert space. Unitary operator = bijective operator preserving scalar product. 7/21
Unitary representations of groups A unitary representation π of a group Γ on a Hilbert space H is ◮ a map π : Γ → unitary operators on H , ◮ such that π( gh ) = π( g )π( h ) and π( e ) = 1. Again a natural appearance of groups : as symmetries of a Hilbert space. Regular representation of Γ Γ Γ • Hilbert space is ℓ 2 ( Γ ) , with orthonormal basis ( e g ) g ∈ Γ , • Representation g ֏ λ g where λ g e h = e gh . 8/21
Amenability of groups Let π : Γ → unitaries on H be a unitary representation of Γ . A sequence of almost invariant vectors is a sequence ξ n of norm one vectors in H satisfying � π( g )ξ n − ξ n � → 0 for all g ∈ Γ . n � 1 √ Example. The ξ n = e k form a sequence of almost 2 n + 1 k =− n invariant vectors for the regular representation of Z . Definition The group Γ is called amenable if its regular representation admits a sequence of almost invariant vectors. Examples of amenable groups. ◮ Abelian groups, solvable groups. ◮ Closed under extensions, subgroups and direct limits. ◮ Open problem: is the group generated by a , b subject to the relation that ab commutes with aba − 1 and with a 2 ba − 2 , amenable (Thompson’s group F ). 9/21
Amenability of groups Let π : Γ → unitaries on H be a unitary representation of Γ . A sequence of almost invariant vectors is a sequence ξ n of norm one vectors in H satisfying � π( g )ξ n − ξ n � → 0 for all g ∈ Γ . n � 1 Example. The ξ n = √ e k form a sequence of almost 2 n + 1 k =− n invariant vectors for the regular representation of Z . Definition The group Γ is called amenable if its regular representation admits a sequence of almost invariant vectors. Examples of amenable groups. ◮ Abelian groups, solvable groups. ◮ Closed under extensions, subgroups and direct limits. ◮ Open problem: is the group generated by a , b subject to the Theorem (Akhmedov, February 23, 2009) relation that ab commutes with aba − 1 and with a 2 ba − 2 , Thompson’s group F is non-amenable. amenable (Thompson’s group F ). 9/21
Von Neumann algebras The most uninteresting examples : M n ( C ) and L ∞ ( X ) . Notations. ◮ B ( H ) denotes the ∗ -algebra of bounded operators on the If H = C n , then B ( H ) = M n ( C ) . Hilbert space H . ◮ The adjoint ∗ : � ξ, T η � = � T ∗ ξ, η � . ◮ One realizes L ∞ ( X ) ⊂ B ( L 2 ( X )) as multiplication operators. ◮ The maps T ֏ � ξ, T η � induce the weak topology on B ( H ) . Von Neumann algebra : weakly closed ∗ -subalgebra of B ( H ) . Group von Neumann algebra Let Γ be a countable group and g ֏ λ g its regular rep. on ℓ 2 ( Γ ) . ◮ span { λ g | g ∈ Γ } is the group algebra C Γ . ◮ Define L ( Γ ) ⊂ B (ℓ 2 ( Γ )) as the weak closure of C Γ . The L ( Γ ) are our first interesting von Neumann algebras. 10/21
Classification of von Neumann algebras Factor M : von Neumann algebra indecomposable ‘as a sum of two’. Equivalent condition : the center of M is trivial. Replaces the ergodicity assumption. Type classification for a factor M (Murray and von Neumann) ◮ Type I, if M ≅ B ( H ) , ◮ Type II 1 , if M admits a finite trace : τ : M → C , τ( xy ) = τ( yx ) , τ( 1 ) = 1 . replaces ‘preserving a probability measure’. ◮ Type II ∞ , if M admits an infinite trace replaces ‘preserving an infinite measure’. ◮ Type III, if M does not admit a trace. Example. The L ( Γ ) always admit a finite trace and are factorial if and only if Γ has infinite conjugacy classes (ICC) : a lot of II 1 factors. 11/21
Amenability of von Neumann algebras Connes : full classification of amenable von Neumann algebras. Definition A von Neumann algebra M ⊂ B ( H ) is called amenable if ... ◮ Example : L ( Γ ) is amenable iff Γ is amenable. Theorem (Connes, 1975) All amenable II 1 factors are isomorphic. In particular, all L ( Γ ) for Γ amenable and ICC are isomorphic ! There is also uniqueness of amenable factors of type II ∞ and ... type III λ (0 ≤ λ ≤ 1). Finer classification into types III λ is the other seminal work of Connes from the 1970’s. 12/21
Back to group actions We assumed so far : ergodic, prob. measure preserving actions. We add one more condition : freeness. The probability measure preserving action Γ ↷ ( X , µ) is called free, if almost every point x ∈ X has trivial stabilizer. If Γ ↷ ( X , µ) = [ 0 , 1 ] Γ is the Bernoulli action, certain x ∈ X have non-trivial stabilizer, but these x ’s form a set of measure 0. Group measure space construction (Murray, von Neumann, 1943) Let Γ ↷ ( X , µ) be free, ergodic, probability measure preserving. II 1 factor L ∞ ( X ) ⋊ Γ is generated by The ◮ the subalgebra L ∞ ( X ) , ◮ the subalgebra L ( Γ ) ∋ λ g , and, for F ∈ L ∞ ( X ) and g ∈ Γ , λ ∗ g F λ g = F g with F g ( x ) = F ( g · x ) . 13/21
Tons of II 1 factors ... or maybe not L ( Γ ) ICC group Γ II 1 factor L ∞ ( X ) ⋊ Γ Free ergodic p.m.p. Γ ↷ ( X , µ) Note : this was all known to Murray and von Neumann. But : they only knew two non-isomorphic II 1 factors ! � ◮ L ( S ∞ ) where S ∞ = n S n and S n = symmetric group, This II 1 factor has approximately central elements. ◮ L ( F 2 ) where F 2 = free group on 2 generators. This II 1 factor has no approximately central elements. Connes’ theorem : all amenable data lead to the same II 1 factor. Open problem. Is L ( F n ) ≅ L ( F m ) for n ≠ m ? 14/21
Recommend
More recommend
