Some appications of free stochastic calculus to II 1 factors.

Transcription

1 Some appications of free stochastic calculus to II 1 factors. Dima Shlyakhtenko UCLA

2 Free Entropy Dimension (Voiculescu) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

3 Free Entropy Dimension (Voiculescu) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

4 Free Entropy Dimension (Voiculescu) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

5 Free Entropy Dimension (Voiculescu) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

6 Free Entropy Dimension (Voiculescu) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

7 Free Entropy Dimension (Voiculescu) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

8 Denition of δ(g) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

9 Denition of δ(g) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

10 Denition of δ(g) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

11 Denition of δ(g) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k ) : lim k ω k Tr(x k x k) = 0} Γ(G) = τ G -preserving homomorhisms G ω M n n

12 Properties of δ(g) You can think of Γ(G) as an analog of the set A(G) = {homomorphisms G Aut(X, µ)}.

13 Properties of δ(g) You can think of Γ(G) as an analog of the set A(G) = {homomorphisms G Aut(X, µ)}.

14 Properties of δ(g) You can think of Γ(G) as an analog of the set A(G) = {homomorphisms G Aut(X, µ)}.

15 More on δ(g)

16 More on δ(g)

17 More on δ(g)

18 More on δ(g)

19 Vanishing results. There are by now many theorems that imply that under some conditions, δ must be 1.

20 Vanishing results. There are by now many theorems that imply that under some conditions, δ must be 1.

21 Vanishing results. There are by now many theorems that imply that under some conditions, δ must be 1.

22 Vanishing results. There are by now many theorems that imply that under some conditions, δ must be 1.

23 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. To prove the conjecture one needs: G) dim L(G) {space of l 2 (G)-valued cocycles} (i on-cojugate actions in Γ(G), then there are nondim L(G) {space of l 2 (G)-va es, to produce many non-con

24 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. To prove the conjecture one needs: G) dim L(G) {space of l 2 (G)-valued cocycles} (i on-cojugate actions in Γ(G), then there are nondim L(G) {space of l 2 (G)-va es, to produce many non-con

25 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. To prove the conjecture one needs: G) dim L(G) {space of l 2 (G)-valued cocycles} (i on-cojugate actions in Γ(G), then there are nondim L(G) {space of l 2 (G)-va es, to produce many non-con

26 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. To prove the conjecture one needs: G) dim L(G) {space of l 2 (G)-valued cocycles} (i on-cojugate actions in Γ(G), then there are nondim L(G) {space of l 2 (G)-va es, to produce many non-con

27 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. To prove the conjecture one needs: G) dim L(G) {space of l 2 (G)-valued cocycles} (i on-cojugate actions in Γ(G), then there are nondim L(G) {space of l 2 (G)-va es, to produce many non-con

28 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. To prove the conjecture one needs: G) dim L(G) {space of l 2 (G)-valued cocycles} (i on-cojugate actions in Γ(G), then there are nondim L(G) {space of l 2 (G)-va es, to produce many non-con

29 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. On the other hand, the second direction: δ(g) dim L(G) {space of l 2 (G)-valued cocycles} looks constructive/easy. Given c : G l 2 (G) construct elements in Γ(G).

30 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. On the other hand, the second direction: looks constructive/easy. δ(g) dim L(G) {space of l 2 (G)-valued cocycles}

31 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. On the other hand, the second direction: looks constructive/easy. δ(g) dim L(G) {space of l 2 (G)-valued cocycles}

32 δ(g) = β (2) 1 (G) β(2) 0 (G) + 1. On the other hand, the second direction: looks constructive/easy. δ(g) dim L(G) {space of l 2 (G)-valued cocycles}

33 Deformations and estimate on δ. Let S 1,..., S m be a free semicircular family in L(F ).

34 Deformations and estimate on δ. Let S 1,..., S m be a free semicircular family in L(F ).

35 Deformations and estimate on δ. Let S 1,..., S m be a free semicircular family in L(F ).

36 Stochastic Calculus Main input: Brownian motion process: family of random variables B([s, t)), t > s 0 so that B([s, t)) are Gaussian, B([s, t)) + B([t, r)) = B([s, r)) if s < t < r and B([s, t)) is independent from B([s, t )) if [s, t) [s, t ] =. Then one can write B([0, t)) = ˆ t 0 db t. Furthermore, for nice enough probability measures µ, there exists a process X t with the properties that: X t is stationary, i.e., X t has distribution µ for all t X t satises the stochastic dierential equation dx t = ϕ(x t ) db t ζ(x t )dt (Note: db t O(t 1/2 )). Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II 1 factors. 10 / 16

37 dxt = ϕ(xt) dbt ζ(xt)dt Very roughly, this means that In particular, the map X t+ϵ = X t + ϕ(x t )B[t, t + ϵ] ζ(x t )ϵ + O(ϵ 3/2 ) }{{} O(ε 1/2 ) f (X ) f (X t 2) gives rise to an isomoprhism α t : L (R, µ) W (X t ) W (X, B[s, t] : s < t) which satises α t (f (X )) = f (X ) + ϕ(x )f (X )B([0, t 2 )) +O(t 2 ) }{{} O(t) (i.e., it exponentiates the derivation (f ) = ϕf, : polynomials L 2 (R, µ)). Dima Shlyakhtenko (UCLA) Some appications of free stochastic calculus to II 1 factors. 11 / 16

38 Free Stochastic Dierential Equations Main fact: L(F ) is generated by a family of self-adjoint elements S([0, t)) = {Sk ([0, t))} n associated to intervals [0, t) R, t 0. For all t, k=1 S([0, t)) is a free semicircular family and S([0, t)) is free from S([0, t )) S([0, t)) if t > t. S([0, t)) is the free analog of Brownian motion on R n measured at time t.

39 Free Stochastic Dierential Equations Main fact: L(F ) is generated by a family of self-adjoint elements S([0, t)) = {Sk ([0, t))} n associated to intervals [0, t) R, t 0. For all t, k=1 S([0, t)) is a free semicircular family and S([0, t)) is free from S([0, t )) S([0, t)) if t > t.

40 Free Stochastic Dierential Equations Main fact: L(F ) is generated by a family of self-adjoint elements S([0, t)) = {Sk ([0, t))} n associated to intervals [0, t) R, t 0. For all t, k=1 S([0, t)) is a free semicircular family and S([0, t)) is free from S([0, t )) S([0, t)) if t > t.

41 Stationary solutions. If X t is a stationary solution (this means that t τ(f ( X t )) = 0 for all non-commutative polynomials f ), then the map α t 2 : f ( X ) f ( X t 2) extends to an isomorphism from W ( X ) to W (X t ) W ( X, S([s, t)) : s < t). Moreover, α t (X j ) = X j + (X j )#d S t 2 + O(t 2 ). so we can apply the estimate on δ.

42 Existence of stationary solutions.... is a somewhat subtle problem in the non-commutative case.

43 Existence of stationary solutions. at subtle problem in the non-commutative case.

44 Existence of stationary solutions. at subtle problem in the non-commutative case.

45 Existence of stationary solutions. at subtle problem in the non-commutative case.

46 Group case. As a corollary you get: δ(g) = β (2) (G) β(2) (G) provided that H 1 (G; l 2 (G)) can be generated by a cocycle whose values at the generators of G are analytic (i.e., are convergent non-commutative power series rather than arbitrary elements of l 2 ).

47 Group case. As a corollary you get: δ(g) = β (2) (G) β(2) (G) provided that H 1 (G; l 2 (G)) can be generated by a cocycle whose values at the generators of G are analytic (i.e., are convergent non-commutative power series rather than arbitrary elements of l 2 ).

48 Group case. As a corollary you get: δ(g) = β (2) (G) β(2) (G) provided that H 1 (G; l 2 (G)) can be generated by a cocycle whose values at the generators of G are analytic (i.e., are convergent non-commutative power series rather than arbitrary elements of l 2 ).

49 Group case. As a corollary you get: δ(g) = β (2) (G) β(2) (G) provided that H 1 (G; l 2 (G)) can be generated by a cocycle whose values at the generators of G are analytic (i.e., are convergent non-commutative power series rather than arbitrary elements of l 2 ).

50 Deformations The lower estimate we get this way on δ is (with few exceptions) the best known so far.

51 Deformations The lower estimate we get this way on δ is (with few exceptions) the best known so far.

52 Deformations The lower estimate we get this way on δ is (with few exceptions) the best known so far.

53 Deformations The lower estimate we get this way on δ is (with few exceptions) the best known so far.

54 Deformations The lower estimate we get this way on δ is (with few exceptions) the best known so far.