Some appications of free stochastic calculus to II 1 factors.
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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.
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